IMO2006-A1 2006 · Algebra · IMO/代数
A sequence of real numbers a 0 , a 1 , a 2 , … \htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{0}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{a}_{\htmlData{tutor-start=10,tutor-end=11}{1}}\htmlData{tutor-start=12,tutor-end=13}{,} \htmlData{tutor-start=14,tutor-end=15}{a}_{\htmlData{tutor-start=17,tutor-end=18}{2}}\htmlData{tutor-start=19,tutor-end=20}{,} \htmlData{tutor-start=21,tutor-end=27}{\ldots} a 0 , a 1 , a 2 , … a i + 1 = ⌊ a i ⌋ ⋅ ⟨ a i ⟩ for i ≥ 0 ; \htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{i}\htmlData{tutor-start=4,tutor-end=5}{+}\htmlData{tutor-start=5,tutor-end=6}{1}}\htmlData{tutor-start=7,tutor-end=8}{=}\left\lfloor \htmlData{tutor-start=21,tutor-end=22}{a}_{\htmlData{tutor-start=24,tutor-end=25}{i}}\right\rfloor \htmlData{tutor-start=40,tutor-end=45}{\cdot}\left\langle \htmlData{tutor-start=58,tutor-end=59}{a}_{\htmlData{tutor-start=61,tutor-end=62}{i}}\right\rangle \quad \text { \htmlData{tutor-start=91,tutor-end=92}{f}\htmlData{tutor-start=92,tutor-end=93}{o}\htmlData{tutor-start=93,tutor-end=94}{r} } \quad \htmlData{tutor-start=103,tutor-end=104}{i} \htmlData{tutor-start=105,tutor-end=110}{\geq }\htmlData{tutor-start=110,tutor-end=111}{0} \text {\htmlData{tutor-start=119,tutor-end=120}{;} } a i + 1 = ⌊ a i ⌋ ⋅ ⟨ a i ⟩ f o r i ≥ 0 ; a 0 \htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{0}} a 0 ⌊ a i ⌋ \left\lfloor \htmlData{tutor-start=13,tutor-end=14}{a}_{\htmlData{tutor-start=16,tutor-end=17}{i}}\right\rfloor ⌊ a i ⌋ a i \htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{i}} a i ⟨ a i ⟩ = a i − ⌊ a i ⌋ \left\langle \htmlData{tutor-start=13,tutor-end=14}{a}_{\htmlData{tutor-start=16,tutor-end=17}{i}}\right\rangle\htmlData{tutor-start=31,tutor-end=32}{=}\htmlData{tutor-start=32,tutor-end=33}{a}_{\htmlData{tutor-start=35,tutor-end=36}{i}}\htmlData{tutor-start=37,tutor-end=38}{-}\left\lfloor \htmlData{tutor-start=51,tutor-end=52}{a}_{\htmlData{tutor-start=54,tutor-end=55}{i}}\right\rfloor ⟨ a i ⟩ = a i − ⌊ a i ⌋ a i = a i + 2 \htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{i}}\htmlData{tutor-start=5,tutor-end=6}{=}\htmlData{tutor-start=6,tutor-end=7}{a}_{\htmlData{tutor-start=9,tutor-end=10}{i}\htmlData{tutor-start=10,tutor-end=11}{+}\htmlData{tutor-start=11,tutor-end=12}{2}} a i = a i + 2 i \htmlData{tutor-start=0,tutor-end=1}{i} i 题解状态： 标准答案与规范题解待补充
题目标签：2006 IMO Shortlist A1
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2006-A2 2006 · Algebra · IMO/代数
The sequence of real numbers a 0 , a 1 , a 2 , … \htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{0}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{a}_{\htmlData{tutor-start=10,tutor-end=11}{1}}\htmlData{tutor-start=12,tutor-end=13}{,} \htmlData{tutor-start=14,tutor-end=15}{a}_{\htmlData{tutor-start=17,tutor-end=18}{2}}\htmlData{tutor-start=19,tutor-end=20}{,} \htmlData{tutor-start=21,tutor-end=27}{\ldots} a 0 , a 1 , a 2 , … a 0 = − 1 , ∑ k = 0 n a n − k k + 1 = 0 for n ≥ 1 \htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{0}}\htmlData{tutor-start=5,tutor-end=6}{=}\htmlData{tutor-start=6,tutor-end=7}{-}\htmlData{tutor-start=7,tutor-end=8}{1}\htmlData{tutor-start=8,tutor-end=9}{,} \quad \sum_{\htmlData{tutor-start=22,tutor-end=23}{k}\htmlData{tutor-start=23,tutor-end=24}{=}\htmlData{tutor-start=24,tutor-end=25}{0}}^{\htmlData{tutor-start=28,tutor-end=29}{n}} \frac{\htmlData{tutor-start=37,tutor-end=38}{a}_{\htmlData{tutor-start=40,tutor-end=41}{n}\htmlData{tutor-start=41,tutor-end=42}{-}\htmlData{tutor-start=42,tutor-end=43}{k}}}{\htmlData{tutor-start=46,tutor-end=47}{k}\htmlData{tutor-start=47,tutor-end=48}{+}\htmlData{tutor-start=48,tutor-end=49}{1}}\htmlData{tutor-start=50,tutor-end=51}{=}\htmlData{tutor-start=51,tutor-end=52}{0} \quad \text { \htmlData{tutor-start=67,tutor-end=68}{f}\htmlData{tutor-start=68,tutor-end=69}{o}\htmlData{tutor-start=69,tutor-end=70}{r} } \quad \htmlData{tutor-start=79,tutor-end=80}{n} \htmlData{tutor-start=81,tutor-end=86}{\geq }\htmlData{tutor-start=86,tutor-end=87}{1} a 0 = − 1 , ∑ k = 0 n k + 1 a n − k = 0 f o r n ≥ 1 a n > 0 \htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{n}}\htmlData{tutor-start=5,tutor-end=6}{>}\htmlData{tutor-start=6,tutor-end=7}{0} a n > 0 n ≥ 1 \htmlData{tutor-start=0,tutor-end=1}{n} \htmlData{tutor-start=2,tutor-end=7}{\geq }\htmlData{tutor-start=7,tutor-end=8}{1} n ≥ 1 题解状态： 标准答案与规范题解待补充
题目标签：2006 IMO Shortlist A2
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2006-A3 2006 · Algebra · IMO/代数
The sequence c 0 , c 1 , … , c n , … \htmlData{tutor-start=0,tutor-end=1}{c}_{\htmlData{tutor-start=3,tutor-end=4}{0}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{c}_{\htmlData{tutor-start=10,tutor-end=11}{1}}\htmlData{tutor-start=12,tutor-end=13}{,} \htmlData{tutor-start=14,tutor-end=20}{\ldots}\htmlData{tutor-start=20,tutor-end=21}{,} \htmlData{tutor-start=22,tutor-end=23}{c}_{\htmlData{tutor-start=25,tutor-end=26}{n}}\htmlData{tutor-start=27,tutor-end=28}{,} \htmlData{tutor-start=29,tutor-end=35}{\ldots} c 0 , c 1 , … , c n , … c 0 = 1 , c 1 = 0 \htmlData{tutor-start=0,tutor-end=1}{c}_{\htmlData{tutor-start=3,tutor-end=4}{0}}\htmlData{tutor-start=5,tutor-end=6}{=}\htmlData{tutor-start=6,tutor-end=7}{1}\htmlData{tutor-start=7,tutor-end=8}{,} \htmlData{tutor-start=9,tutor-end=10}{c}_{\htmlData{tutor-start=12,tutor-end=13}{1}}\htmlData{tutor-start=14,tutor-end=15}{=}\htmlData{tutor-start=15,tutor-end=16}{0} c 0 = 1 , c 1 = 0 c n + 2 = c n + 1 + c n \htmlData{tutor-start=0,tutor-end=1}{c}_{\htmlData{tutor-start=3,tutor-end=4}{n}\htmlData{tutor-start=4,tutor-end=5}{+}\htmlData{tutor-start=5,tutor-end=6}{2}}\htmlData{tutor-start=7,tutor-end=8}{=}\htmlData{tutor-start=8,tutor-end=9}{c}_{\htmlData{tutor-start=11,tutor-end=12}{n}\htmlData{tutor-start=12,tutor-end=13}{+}\htmlData{tutor-start=13,tutor-end=14}{1}}\htmlData{tutor-start=15,tutor-end=16}{+}\htmlData{tutor-start=16,tutor-end=17}{c}_{\htmlData{tutor-start=19,tutor-end=20}{n}} c n + 2 = c n + 1 + c n n ≥ 0 \htmlData{tutor-start=0,tutor-end=1}{n} \htmlData{tutor-start=2,tutor-end=7}{\geq }\htmlData{tutor-start=7,tutor-end=8}{0} n ≥ 0 S \htmlData{tutor-start=0,tutor-end=1}{S} S ( x , y ) \htmlData{tutor-start=0,tutor-end=1}{(}\htmlData{tutor-start=1,tutor-end=2}{x}\htmlData{tutor-start=2,tutor-end=3}{,} \htmlData{tutor-start=4,tutor-end=5}{y}\htmlData{tutor-start=5,tutor-end=6}{)} ( x , y ) J \htmlData{tutor-start=0,tutor-end=1}{J} J x = ∑ j ∈ J c j , y = ∑ j ∈ J c j − 1 \htmlData{tutor-start=0,tutor-end=1}{x}\htmlData{tutor-start=1,tutor-end=2}{=}\sum_{\htmlData{tutor-start=8,tutor-end=9}{j} \htmlData{tutor-start=10,tutor-end=14}{\in }\htmlData{tutor-start=14,tutor-end=15}{J}} \htmlData{tutor-start=17,tutor-end=18}{c}_{\htmlData{tutor-start=20,tutor-end=21}{j}}\htmlData{tutor-start=22,tutor-end=23}{,} \htmlData{tutor-start=24,tutor-end=25}{y}\htmlData{tutor-start=25,tutor-end=26}{=}\sum_{\htmlData{tutor-start=32,tutor-end=33}{j} \htmlData{tutor-start=34,tutor-end=38}{\in }\htmlData{tutor-start=38,tutor-end=39}{J}} \htmlData{tutor-start=41,tutor-end=42}{c}_{\htmlData{tutor-start=44,tutor-end=45}{j}\htmlData{tutor-start=45,tutor-end=46}{-}\htmlData{tutor-start=46,tutor-end=47}{1}} x = ∑ j ∈ J c j , y = ∑ j ∈ J c j − 1 α , β \htmlData{tutor-start=0,tutor-end=6}{\alpha}\htmlData{tutor-start=6,tutor-end=7}{,} \htmlData{tutor-start=8,tutor-end=13}{\beta} α , β m , M \htmlData{tutor-start=0,tutor-end=1}{m}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{M} m , M ( x , y ) \htmlData{tutor-start=0,tutor-end=1}{(}\htmlData{tutor-start=1,tutor-end=2}{x}\htmlData{tutor-start=2,tutor-end=3}{,} \htmlData{tutor-start=4,tutor-end=5}{y}\htmlData{tutor-start=5,tutor-end=6}{)} ( x , y ) m < α x + β y < M \htmlData{tutor-start=0,tutor-end=1}{m}\htmlData{tutor-start=1,tutor-end=2}{<}\htmlData{tutor-start=2,tutor-end=9}{\alpha }\htmlData{tutor-start=9,tutor-end=10}{x}\htmlData{tutor-start=10,tutor-end=11}{+}\htmlData{tutor-start=11,tutor-end=17}{\beta }\htmlData{tutor-start=17,tutor-end=18}{y}\htmlData{tutor-start=18,tutor-end=19}{<}\htmlData{tutor-start=19,tutor-end=20}{M} m < α x + β y < M ( x , y ) ∈ S \htmlData{tutor-start=0,tutor-end=1}{(}\htmlData{tutor-start=1,tutor-end=2}{x}\htmlData{tutor-start=2,tutor-end=3}{,} \htmlData{tutor-start=4,tutor-end=5}{y}\htmlData{tutor-start=5,tutor-end=6}{)} \htmlData{tutor-start=7,tutor-end=11}{\in }\htmlData{tutor-start=11,tutor-end=12}{S} ( x , y ) ∈ S 题解状态： 标准答案与规范题解待补充
题目标签：2006 IMO Shortlist A3
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2006-A4 2006 · Algebra · IMO/代数
Prove the inequality
∑ i < j a i a j a i + a j ≤ n 2 ( a 1 + a 2 + ⋯ + a n ) ∑ i < j a i a j \htmlData{tutor-start=0,tutor-end=7}{\sum_{i}\htmlData{tutor-start=7,tutor-end=8}{<}\htmlData{tutor-start=8,tutor-end=9}{j}} \frac{\htmlData{tutor-start=17,tutor-end=18}{a}_{\htmlData{tutor-start=20,tutor-end=21}{i}} \htmlData{tutor-start=23,tutor-end=24}{a}_{\htmlData{tutor-start=26,tutor-end=27}{j}}}{\htmlData{tutor-start=30,tutor-end=31}{a}_{\htmlData{tutor-start=33,tutor-end=34}{i}}\htmlData{tutor-start=35,tutor-end=36}{+}\htmlData{tutor-start=36,tutor-end=37}{a}_{\htmlData{tutor-start=39,tutor-end=40}{j}}} \htmlData{tutor-start=43,tutor-end=48}{\leq }\frac{\htmlData{tutor-start=54,tutor-end=55}{n}}{\htmlData{tutor-start=57,tutor-end=58}{2}\left(\htmlData{tutor-start=64,tutor-end=65}{a}_{\htmlData{tutor-start=67,tutor-end=68}{1}}\htmlData{tutor-start=69,tutor-end=70}{+}\htmlData{tutor-start=70,tutor-end=71}{a}_{\htmlData{tutor-start=73,tutor-end=74}{2}}\htmlData{tutor-start=75,tutor-end=76}{+}\cdots\htmlData{tutor-start=82,tutor-end=83}{+}\htmlData{tutor-start=83,tutor-end=84}{a}_{\htmlData{tutor-start=86,tutor-end=87}{n}}\right)} \sum_{\htmlData{tutor-start=103,tutor-end=104}{i}\htmlData{tutor-start=104,tutor-end=105}{<}\htmlData{tutor-start=105,tutor-end=106}{j}} \htmlData{tutor-start=108,tutor-end=109}{a}_{\htmlData{tutor-start=111,tutor-end=112}{i}} \htmlData{tutor-start=114,tutor-end=115}{a}_{\htmlData{tutor-start=117,tutor-end=118}{j}} ∑ i < j a i + a j a i a j ≤ 2 ( a 1 + a 2 + ⋯ + a n ) n ∑ i < j a i a j a 1 , a 2 , … , a n \htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{a}_{\htmlData{tutor-start=10,tutor-end=11}{2}}\htmlData{tutor-start=12,tutor-end=13}{,} \htmlData{tutor-start=14,tutor-end=20}{\ldots}\htmlData{tutor-start=20,tutor-end=21}{,} \htmlData{tutor-start=22,tutor-end=23}{a}_{\htmlData{tutor-start=25,tutor-end=26}{n}} a 1 , a 2 , … , a n 题解状态： 标准答案与规范题解待补充
题目标签：2006 IMO Shortlist A4
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2006-A5 2006 · Algebra · IMO/代数
Let a , b , c \htmlData{tutor-start=0,tutor-end=1}{a}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{b}\htmlData{tutor-start=4,tutor-end=5}{,} \htmlData{tutor-start=6,tutor-end=7}{c} a , b , c b + c − a b + c − a + c + a − b c + a − b + a + b − c a + b − c ≤ 3. \frac{\sqrt{\htmlData{tutor-start=12,tutor-end=13}{b}\htmlData{tutor-start=13,tutor-end=14}{+}\htmlData{tutor-start=14,tutor-end=15}{c}\htmlData{tutor-start=15,tutor-end=16}{-}\htmlData{tutor-start=16,tutor-end=17}{a}}}{\sqrt{\htmlData{tutor-start=26,tutor-end=27}{b}}\htmlData{tutor-start=28,tutor-end=29}{+}\sqrt{\htmlData{tutor-start=35,tutor-end=36}{c}}\htmlData{tutor-start=37,tutor-end=38}{-}\sqrt{\htmlData{tutor-start=44,tutor-end=45}{a}}}\htmlData{tutor-start=47,tutor-end=48}{+}\frac{\sqrt{\htmlData{tutor-start=60,tutor-end=61}{c}\htmlData{tutor-start=61,tutor-end=62}{+}\htmlData{tutor-start=62,tutor-end=63}{a}\htmlData{tutor-start=63,tutor-end=64}{-}\htmlData{tutor-start=64,tutor-end=65}{b}}}{\sqrt{\htmlData{tutor-start=74,tutor-end=75}{c}}\htmlData{tutor-start=76,tutor-end=77}{+}\sqrt{\htmlData{tutor-start=83,tutor-end=84}{a}}\htmlData{tutor-start=85,tutor-end=86}{-}\sqrt{\htmlData{tutor-start=92,tutor-end=93}{b}}}\htmlData{tutor-start=95,tutor-end=96}{+}\frac{\sqrt{\htmlData{tutor-start=108,tutor-end=109}{a}\htmlData{tutor-start=109,tutor-end=110}{+}\htmlData{tutor-start=110,tutor-end=111}{b}\htmlData{tutor-start=111,tutor-end=112}{-}\htmlData{tutor-start=112,tutor-end=113}{c}}}{\sqrt{\htmlData{tutor-start=122,tutor-end=123}{a}}\htmlData{tutor-start=124,tutor-end=125}{+}\sqrt{\htmlData{tutor-start=131,tutor-end=132}{b}}\htmlData{tutor-start=133,tutor-end=134}{-}\sqrt{\htmlData{tutor-start=140,tutor-end=141}{c}}} \htmlData{tutor-start=144,tutor-end=149}{\leq }\htmlData{tutor-start=149,tutor-end=150}{3} \htmlData{tutor-start=151,tutor-end=152}{.} b + c − a b + c − a + c + a − b c + a − b + a + b − c a + b − c ≤ 3 . 题解状态： 标准答案与规范题解待补充
题目标签：2006 IMO Shortlist A5
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2006-A6 2006 · Algebra · IMO/代数
Determine the smallest number M \htmlData{tutor-start=0,tutor-end=1}{M} M ∣ a b ( a 2 − b 2 ) + b c ( b 2 − c 2 ) + c a ( c 2 − a 2 ) ∣ ≤ M ( a 2 + b 2 + c 2 ) 2 \left|\htmlData{tutor-start=6,tutor-end=7}{a} \htmlData{tutor-start=8,tutor-end=9}{b}\left(\htmlData{tutor-start=15,tutor-end=16}{a}^{\htmlData{tutor-start=18,tutor-end=19}{2}}\htmlData{tutor-start=20,tutor-end=21}{-}\htmlData{tutor-start=21,tutor-end=22}{b}^{\htmlData{tutor-start=24,tutor-end=25}{2}}\right)\htmlData{tutor-start=33,tutor-end=34}{+}\htmlData{tutor-start=34,tutor-end=35}{b} \htmlData{tutor-start=36,tutor-end=37}{c}\left(\htmlData{tutor-start=43,tutor-end=44}{b}^{\htmlData{tutor-start=46,tutor-end=47}{2}}\htmlData{tutor-start=48,tutor-end=49}{-}\htmlData{tutor-start=49,tutor-end=50}{c}^{\htmlData{tutor-start=52,tutor-end=53}{2}}\right)\htmlData{tutor-start=61,tutor-end=62}{+}\htmlData{tutor-start=62,tutor-end=63}{c} \htmlData{tutor-start=64,tutor-end=65}{a}\left(\htmlData{tutor-start=71,tutor-end=72}{c}^{\htmlData{tutor-start=74,tutor-end=75}{2}}\htmlData{tutor-start=76,tutor-end=77}{-}\htmlData{tutor-start=77,tutor-end=78}{a}^{\htmlData{tutor-start=80,tutor-end=81}{2}}\right)\right| \htmlData{tutor-start=97,tutor-end=102}{\leq }\htmlData{tutor-start=102,tutor-end=103}{M}\left(\htmlData{tutor-start=109,tutor-end=110}{a}^{\htmlData{tutor-start=112,tutor-end=113}{2}}\htmlData{tutor-start=114,tutor-end=115}{+}\htmlData{tutor-start=115,tutor-end=116}{b}^{\htmlData{tutor-start=118,tutor-end=119}{2}}\htmlData{tutor-start=120,tutor-end=121}{+}\htmlData{tutor-start=121,tutor-end=122}{c}^{\htmlData{tutor-start=124,tutor-end=125}{2}}\right)^{\htmlData{tutor-start=135,tutor-end=136}{2}} a b ( a 2 − b 2 ) + b c ( b 2 − c 2 ) + c a ( c 2 − a 2 ) ≤ M ( a 2 + b 2 + c 2 ) 2 a , b , c \htmlData{tutor-start=0,tutor-end=1}{a}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{b}\htmlData{tutor-start=4,tutor-end=5}{,} \htmlData{tutor-start=6,tutor-end=7}{c} a , b , c 题解状态： 标准答案与规范题解待补充
题目标签：2006 IMO 正式题 A6
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2007-A1 2007 · Algebra · IMO/代数
Given a sequence a 1 , a 2 , … , a n \htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{a}_{\htmlData{tutor-start=10,tutor-end=11}{2}}\htmlData{tutor-start=12,tutor-end=13}{,} \htmlData{tutor-start=14,tutor-end=20}{\ldots}\htmlData{tutor-start=20,tutor-end=21}{,} \htmlData{tutor-start=22,tutor-end=23}{a}_{\htmlData{tutor-start=25,tutor-end=26}{n}} a 1 , a 2 , … , a n i ( 1 ≤ i ≤ n ) \htmlData{tutor-start=0,tutor-end=1}{i}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{1} \htmlData{tutor-start=4,tutor-end=9}{\leq }\htmlData{tutor-start=9,tutor-end=10}{i} \htmlData{tutor-start=11,tutor-end=16}{\leq }\htmlData{tutor-start=16,tutor-end=17}{n}\htmlData{tutor-start=17,tutor-end=18}{)} i ( 1 ≤ i ≤ n ) d i = max { a j : 1 ≤ j ≤ i } − min { a j : i ≤ j ≤ n } \htmlData{tutor-start=0,tutor-end=1}{d}_{\htmlData{tutor-start=3,tutor-end=4}{i}}\htmlData{tutor-start=5,tutor-end=6}{=}\max \left\{\htmlData{tutor-start=18,tutor-end=19}{a}_{\htmlData{tutor-start=21,tutor-end=22}{j}}\htmlData{tutor-start=23,tutor-end=24}{:} \htmlData{tutor-start=25,tutor-end=26}{1} \htmlData{tutor-start=27,tutor-end=32}{\leq }\htmlData{tutor-start=32,tutor-end=33}{j} \htmlData{tutor-start=34,tutor-end=39}{\leq }\htmlData{tutor-start=39,tutor-end=40}{i}\right\}\htmlData{tutor-start=48,tutor-end=49}{-}\min \left\{\htmlData{tutor-start=61,tutor-end=62}{a}_{\htmlData{tutor-start=64,tutor-end=65}{j}}\htmlData{tutor-start=66,tutor-end=67}{:} \htmlData{tutor-start=68,tutor-end=69}{i} \htmlData{tutor-start=70,tutor-end=75}{\leq }\htmlData{tutor-start=75,tutor-end=76}{j} \htmlData{tutor-start=77,tutor-end=82}{\leq }\htmlData{tutor-start=82,tutor-end=83}{n}\right\} d i = max { a j : 1 ≤ j ≤ i } − min { a j : i ≤ j ≤ n } d = max { d i : 1 ≤ i ≤ n } \htmlData{tutor-start=0,tutor-end=1}{d}\htmlData{tutor-start=1,tutor-end=2}{=}\max \left\{\htmlData{tutor-start=14,tutor-end=15}{d}_{\htmlData{tutor-start=17,tutor-end=18}{i}}\htmlData{tutor-start=19,tutor-end=20}{:} \htmlData{tutor-start=21,tutor-end=22}{1} \htmlData{tutor-start=23,tutor-end=28}{\leq }\htmlData{tutor-start=28,tutor-end=29}{i} \htmlData{tutor-start=30,tutor-end=35}{\leq }\htmlData{tutor-start=35,tutor-end=36}{n}\right\} d = max { d i : 1 ≤ i ≤ n } x 1 ≤ x 2 ≤ … ≤ x n \htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{1}} \htmlData{tutor-start=6,tutor-end=11}{\leq }\htmlData{tutor-start=11,tutor-end=12}{x}_{\htmlData{tutor-start=14,tutor-end=15}{2}} \htmlData{tutor-start=17,tutor-end=22}{\leq }\htmlData{tutor-start=22,tutor-end=29}{\ldots }\htmlData{tutor-start=29,tutor-end=34}{\leq }\htmlData{tutor-start=34,tutor-end=35}{x}_{\htmlData{tutor-start=37,tutor-end=38}{n}} x 1 ≤ x 2 ≤ … ≤ x n max { ∣ x i − a i ∣ : 1 ≤ i ≤ n } ≥ d 2 \max \left\{\left|\htmlData{tutor-start=18,tutor-end=19}{x}_{\htmlData{tutor-start=21,tutor-end=22}{i}}\htmlData{tutor-start=23,tutor-end=24}{-}\htmlData{tutor-start=24,tutor-end=25}{a}_{\htmlData{tutor-start=27,tutor-end=28}{i}}\right|\htmlData{tutor-start=36,tutor-end=37}{:} \htmlData{tutor-start=38,tutor-end=39}{1} \htmlData{tutor-start=40,tutor-end=45}{\leq }\htmlData{tutor-start=45,tutor-end=46}{i} \htmlData{tutor-start=47,tutor-end=52}{\leq }\htmlData{tutor-start=52,tutor-end=53}{n}\right\} \htmlData{tutor-start=62,tutor-end=67}{\geq }\frac{\htmlData{tutor-start=73,tutor-end=74}{d}}{\htmlData{tutor-start=76,tutor-end=77}{2}} max { ∣ x i − a i ∣ : 1 ≤ i ≤ n } ≥ 2 d x 1 ≤ x 2 ≤ … ≤ x n \htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{1}} \htmlData{tutor-start=6,tutor-end=11}{\leq }\htmlData{tutor-start=11,tutor-end=12}{x}_{\htmlData{tutor-start=14,tutor-end=15}{2}} \htmlData{tutor-start=17,tutor-end=22}{\leq }\htmlData{tutor-start=22,tutor-end=29}{\ldots }\htmlData{tutor-start=29,tutor-end=34}{\leq }\htmlData{tutor-start=34,tutor-end=35}{x}_{\htmlData{tutor-start=37,tutor-end=38}{n}} x 1 ≤ x 2 ≤ … ≤ x n 题解状态： 标准答案与规范题解待补充
题目标签：2007 IMO 正式题 A1
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2007-A2 2007 · Algebra · IMO/代数
Consider those functions f : N → N \htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{:} \mathbb{\htmlData{tutor-start=11,tutor-end=12}{N}} \htmlData{tutor-start=14,tutor-end=26}{\rightarrow }\mathbb{\htmlData{tutor-start=34,tutor-end=35}{N}} f : N → N f ( m + n ) ≥ f ( m ) + f ( f ( n ) ) − 1 \htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{m}\htmlData{tutor-start=3,tutor-end=4}{+}\htmlData{tutor-start=4,tutor-end=5}{n}\htmlData{tutor-start=5,tutor-end=6}{)} \htmlData{tutor-start=7,tutor-end=12}{\geq }\htmlData{tutor-start=12,tutor-end=13}{f}\htmlData{tutor-start=13,tutor-end=14}{(}\htmlData{tutor-start=14,tutor-end=15}{m}\htmlData{tutor-start=15,tutor-end=16}{)}\htmlData{tutor-start=16,tutor-end=17}{+}\htmlData{tutor-start=17,tutor-end=18}{f}\htmlData{tutor-start=18,tutor-end=19}{(}\htmlData{tutor-start=19,tutor-end=20}{f}\htmlData{tutor-start=20,tutor-end=21}{(}\htmlData{tutor-start=21,tutor-end=22}{n}\htmlData{tutor-start=22,tutor-end=23}{)}\htmlData{tutor-start=23,tutor-end=24}{)}\htmlData{tutor-start=24,tutor-end=25}{-}\htmlData{tutor-start=25,tutor-end=26}{1} f ( m + n ) ≥ f ( m ) + f ( f ( n ) ) − 1 m , n ∈ N \htmlData{tutor-start=0,tutor-end=1}{m}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{n} \htmlData{tutor-start=5,tutor-end=9}{\in }\mathbb{\htmlData{tutor-start=17,tutor-end=18}{N}} m , n ∈ N f ( 2007 ) \htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{2}\htmlData{tutor-start=3,tutor-end=4}{0}\htmlData{tutor-start=4,tutor-end=5}{0}\htmlData{tutor-start=5,tutor-end=6}{7}\htmlData{tutor-start=6,tutor-end=7}{)} f ( 2 0 0 7 ) ( N \htmlData{tutor-start=0,tutor-end=1}{(}\mathbb{\htmlData{tutor-start=9,tutor-end=10}{N}} ( N 题解状态： 标准答案与规范题解待补充
题目标签：2007 IMO Shortlist A2
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2007-A3 2007 · Algebra · IMO/代数
Let n \htmlData{tutor-start=0,tutor-end=1}{n} n x \htmlData{tutor-start=0,tutor-end=1}{x} x y \htmlData{tutor-start=0,tutor-end=1}{y} y x n + y n = 1 \htmlData{tutor-start=0,tutor-end=1}{x}^{\htmlData{tutor-start=3,tutor-end=4}{n}}\htmlData{tutor-start=5,tutor-end=6}{+}\htmlData{tutor-start=6,tutor-end=7}{y}^{\htmlData{tutor-start=9,tutor-end=10}{n}}\htmlData{tutor-start=11,tutor-end=12}{=}\htmlData{tutor-start=12,tutor-end=13}{1} x n + y n = 1 ( ∑ k = 1 n 1 + x 2 k 1 + x 4 k ) ( ∑ k = 1 n 1 + y 2 k 1 + y 4 k ) < 1 ( 1 − x ) ( 1 − y ) \left(\sum_{\htmlData{tutor-start=12,tutor-end=13}{k}\htmlData{tutor-start=13,tutor-end=14}{=}\htmlData{tutor-start=14,tutor-end=15}{1}}^{\htmlData{tutor-start=18,tutor-end=19}{n}} \frac{\htmlData{tutor-start=27,tutor-end=28}{1}\htmlData{tutor-start=28,tutor-end=29}{+}\htmlData{tutor-start=29,tutor-end=30}{x}^{\htmlData{tutor-start=32,tutor-end=33}{2} \htmlData{tutor-start=34,tutor-end=35}{k}}}{\htmlData{tutor-start=38,tutor-end=39}{1}\htmlData{tutor-start=39,tutor-end=40}{+}\htmlData{tutor-start=40,tutor-end=41}{x}^{\htmlData{tutor-start=43,tutor-end=44}{4} \htmlData{tutor-start=45,tutor-end=46}{k}}}\right)\left(\sum_{\htmlData{tutor-start=67,tutor-end=68}{k}\htmlData{tutor-start=68,tutor-end=69}{=}\htmlData{tutor-start=69,tutor-end=70}{1}}^{\htmlData{tutor-start=73,tutor-end=74}{n}} \frac{\htmlData{tutor-start=82,tutor-end=83}{1}\htmlData{tutor-start=83,tutor-end=84}{+}\htmlData{tutor-start=84,tutor-end=85}{y}^{\htmlData{tutor-start=87,tutor-end=88}{2} \htmlData{tutor-start=89,tutor-end=90}{k}}}{\htmlData{tutor-start=93,tutor-end=94}{1}\htmlData{tutor-start=94,tutor-end=95}{+}\htmlData{tutor-start=95,tutor-end=96}{y}^{\htmlData{tutor-start=98,tutor-end=99}{4} \htmlData{tutor-start=100,tutor-end=101}{k}}}\right)\htmlData{tutor-start=110,tutor-end=111}{<}\frac{\htmlData{tutor-start=117,tutor-end=118}{1}}{\htmlData{tutor-start=120,tutor-end=121}{(}\htmlData{tutor-start=121,tutor-end=122}{1}\htmlData{tutor-start=122,tutor-end=123}{-}\htmlData{tutor-start=123,tutor-end=124}{x}\htmlData{tutor-start=124,tutor-end=125}{)}\htmlData{tutor-start=125,tutor-end=126}{(}\htmlData{tutor-start=126,tutor-end=127}{1}\htmlData{tutor-start=127,tutor-end=128}{-}\htmlData{tutor-start=128,tutor-end=129}{y}\htmlData{tutor-start=129,tutor-end=130}{)}} ( ∑ k = 1 n 1 + x 4 k 1 + x 2 k ) ( ∑ k = 1 n 1 + y 4 k 1 + y 2 k ) < ( 1 − x ) ( 1 − y ) 1 题解状态： 标准答案与规范题解待补充
题目标签：2007 IMO Shortlist A3
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2007-A4 2007 · Algebra · IMO/代数
Find all functions f : R + → R + \htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{:} \mathbb{\htmlData{tutor-start=11,tutor-end=12}{R}}^{\htmlData{tutor-start=15,tutor-end=16}{+}} \htmlData{tutor-start=18,tutor-end=30}{\rightarrow }\mathbb{\htmlData{tutor-start=38,tutor-end=39}{R}}^{\htmlData{tutor-start=42,tutor-end=43}{+}} f : R + → R + f ( x + f ( y ) ) = f ( x + y ) + f ( y ) \htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{x}\htmlData{tutor-start=3,tutor-end=4}{+}\htmlData{tutor-start=4,tutor-end=5}{f}\htmlData{tutor-start=5,tutor-end=6}{(}\htmlData{tutor-start=6,tutor-end=7}{y}\htmlData{tutor-start=7,tutor-end=8}{)}\htmlData{tutor-start=8,tutor-end=9}{)}\htmlData{tutor-start=9,tutor-end=10}{=}\htmlData{tutor-start=10,tutor-end=11}{f}\htmlData{tutor-start=11,tutor-end=12}{(}\htmlData{tutor-start=12,tutor-end=13}{x}\htmlData{tutor-start=13,tutor-end=14}{+}\htmlData{tutor-start=14,tutor-end=15}{y}\htmlData{tutor-start=15,tutor-end=16}{)}\htmlData{tutor-start=16,tutor-end=17}{+}\htmlData{tutor-start=17,tutor-end=18}{f}\htmlData{tutor-start=18,tutor-end=19}{(}\htmlData{tutor-start=19,tutor-end=20}{y}\htmlData{tutor-start=20,tutor-end=21}{)} f ( x + f ( y ) ) = f ( x + y ) + f ( y ) x , y ∈ R + \htmlData{tutor-start=0,tutor-end=1}{x}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{y} \htmlData{tutor-start=5,tutor-end=9}{\in }\mathbb{\htmlData{tutor-start=17,tutor-end=18}{R}}^{\htmlData{tutor-start=21,tutor-end=22}{+}} x , y ∈ R + R + \mathbb{\htmlData{tutor-start=8,tutor-end=9}{R}}^{\htmlData{tutor-start=12,tutor-end=13}{+}} R + f ( x ) = 2 x \htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{x}\htmlData{tutor-start=3,tutor-end=4}{)}\htmlData{tutor-start=4,tutor-end=5}{=}\htmlData{tutor-start=5,tutor-end=6}{2} \htmlData{tutor-start=7,tutor-end=8}{x} f ( x ) = 2 x 题解状态： 标准答案与规范题解待补充
题目标签：2007 IMO Shortlist A4
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2007-A5 2007 · Algebra · IMO/代数
Let c > 2 \htmlData{tutor-start=0,tutor-end=1}{c}\htmlData{tutor-start=1,tutor-end=2}{>}\htmlData{tutor-start=2,tutor-end=3}{2} c > 2 a ( 1 ) , a ( 2 ) , … \htmlData{tutor-start=0,tutor-end=1}{a}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{1}\htmlData{tutor-start=3,tutor-end=4}{)}\htmlData{tutor-start=4,tutor-end=5}{,} \htmlData{tutor-start=6,tutor-end=7}{a}\htmlData{tutor-start=7,tutor-end=8}{(}\htmlData{tutor-start=8,tutor-end=9}{2}\htmlData{tutor-start=9,tutor-end=10}{)}\htmlData{tutor-start=10,tutor-end=11}{,} \htmlData{tutor-start=12,tutor-end=18}{\ldots} a ( 1 ) , a ( 2 ) , … a ( m + n ) ≤ 2 a ( m ) + 2 a ( n ) for all m , n ≥ 1 , \htmlData{tutor-start=0,tutor-end=1}{a}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{m}\htmlData{tutor-start=3,tutor-end=4}{+}\htmlData{tutor-start=4,tutor-end=5}{n}\htmlData{tutor-start=5,tutor-end=6}{)} \htmlData{tutor-start=7,tutor-end=12}{\leq }\htmlData{tutor-start=12,tutor-end=13}{2} \htmlData{tutor-start=14,tutor-end=15}{a}\htmlData{tutor-start=15,tutor-end=16}{(}\htmlData{tutor-start=16,tutor-end=17}{m}\htmlData{tutor-start=17,tutor-end=18}{)}\htmlData{tutor-start=18,tutor-end=19}{+}\htmlData{tutor-start=19,tutor-end=20}{2} \htmlData{tutor-start=21,tutor-end=22}{a}\htmlData{tutor-start=22,tutor-end=23}{(}\htmlData{tutor-start=23,tutor-end=24}{n}\htmlData{tutor-start=24,tutor-end=25}{)} \text { \htmlData{tutor-start=34,tutor-end=35}{f}\htmlData{tutor-start=35,tutor-end=36}{o}\htmlData{tutor-start=36,tutor-end=37}{r} \htmlData{tutor-start=38,tutor-end=39}{a}\htmlData{tutor-start=39,tutor-end=40}{l}\htmlData{tutor-start=40,tutor-end=41}{l} } \htmlData{tutor-start=44,tutor-end=45}{m}\htmlData{tutor-start=45,tutor-end=46}{,} \htmlData{tutor-start=47,tutor-end=48}{n} \htmlData{tutor-start=49,tutor-end=54}{\geq }\htmlData{tutor-start=54,tutor-end=55}{1} \text {\htmlData{tutor-start=63,tutor-end=64}{,} } a ( m + n ) ≤ 2 a ( m ) + 2 a ( n ) f o r a l l m , n ≥ 1 , a ( 2 k ) ≤ 1 ( k + 1 ) c for all k ≥ 0 \htmlData{tutor-start=0,tutor-end=1}{a}\left(\htmlData{tutor-start=7,tutor-end=8}{2}^{\htmlData{tutor-start=10,tutor-end=11}{k}}\right) \htmlData{tutor-start=20,tutor-end=25}{\leq }\frac{\htmlData{tutor-start=31,tutor-end=32}{1}}{\htmlData{tutor-start=34,tutor-end=35}{(}\htmlData{tutor-start=35,tutor-end=36}{k}\htmlData{tutor-start=36,tutor-end=37}{+}\htmlData{tutor-start=37,tutor-end=38}{1}\htmlData{tutor-start=38,tutor-end=39}{)}^{\htmlData{tutor-start=41,tutor-end=42}{c}}} \quad \text { \htmlData{tutor-start=59,tutor-end=60}{f}\htmlData{tutor-start=60,tutor-end=61}{o}\htmlData{tutor-start=61,tutor-end=62}{r} \htmlData{tutor-start=63,tutor-end=64}{a}\htmlData{tutor-start=64,tutor-end=65}{l}\htmlData{tutor-start=65,tutor-end=66}{l} } \htmlData{tutor-start=69,tutor-end=70}{k} \htmlData{tutor-start=71,tutor-end=76}{\geq }\htmlData{tutor-start=76,tutor-end=77}{0} a ( 2 k ) ≤ ( k + 1 ) c 1 f o r a l l k ≥ 0 a ( n ) \htmlData{tutor-start=0,tutor-end=1}{a}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{n}\htmlData{tutor-start=3,tutor-end=4}{)} a ( n ) 题解状态： 标准答案与规范题解待补充
题目标签：2007 IMO Shortlist A5
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2007-A6 2007 · Algebra · IMO/代数
Let a 1 , a 2 , … , a 100 \htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{a}_{\htmlData{tutor-start=10,tutor-end=11}{2}}\htmlData{tutor-start=12,tutor-end=13}{,} \htmlData{tutor-start=14,tutor-end=20}{\ldots}\htmlData{tutor-start=20,tutor-end=21}{,} \htmlData{tutor-start=22,tutor-end=23}{a}_{\htmlData{tutor-start=25,tutor-end=26}{1}\htmlData{tutor-start=26,tutor-end=27}{0}\htmlData{tutor-start=27,tutor-end=28}{0}} a 1 , a 2 , … , a 1 0 0 a 1 2 + a 2 2 + … + a 100 2 = 1 \htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{1}}^{\htmlData{tutor-start=7,tutor-end=8}{2}}\htmlData{tutor-start=9,tutor-end=10}{+}\htmlData{tutor-start=10,tutor-end=11}{a}_{\htmlData{tutor-start=13,tutor-end=14}{2}}^{\htmlData{tutor-start=17,tutor-end=18}{2}}\htmlData{tutor-start=19,tutor-end=20}{+}\htmlData{tutor-start=20,tutor-end=26}{\ldots}\htmlData{tutor-start=26,tutor-end=27}{+}\htmlData{tutor-start=27,tutor-end=28}{a}_{\htmlData{tutor-start=30,tutor-end=31}{1}\htmlData{tutor-start=31,tutor-end=32}{0}\htmlData{tutor-start=32,tutor-end=33}{0}}^{\htmlData{tutor-start=36,tutor-end=37}{2}}\htmlData{tutor-start=38,tutor-end=39}{=}\htmlData{tutor-start=39,tutor-end=40}{1} a 1 2 + a 2 2 + … + a 1 0 0 2 = 1 a 1 2 a 2 + a 2 2 a 3 + … + a 100 2 a 1 < 12 25 \htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{1}}^{\htmlData{tutor-start=7,tutor-end=8}{2}} \htmlData{tutor-start=10,tutor-end=11}{a}_{\htmlData{tutor-start=13,tutor-end=14}{2}}\htmlData{tutor-start=15,tutor-end=16}{+}\htmlData{tutor-start=16,tutor-end=17}{a}_{\htmlData{tutor-start=19,tutor-end=20}{2}}^{\htmlData{tutor-start=23,tutor-end=24}{2}} \htmlData{tutor-start=26,tutor-end=27}{a}_{\htmlData{tutor-start=29,tutor-end=30}{3}}\htmlData{tutor-start=31,tutor-end=32}{+}\htmlData{tutor-start=32,tutor-end=38}{\ldots}\htmlData{tutor-start=38,tutor-end=39}{+}\htmlData{tutor-start=39,tutor-end=40}{a}_{\htmlData{tutor-start=42,tutor-end=43}{1}\htmlData{tutor-start=43,tutor-end=44}{0}\htmlData{tutor-start=44,tutor-end=45}{0}}^{\htmlData{tutor-start=48,tutor-end=49}{2}} \htmlData{tutor-start=51,tutor-end=52}{a}_{\htmlData{tutor-start=54,tutor-end=55}{1}}\htmlData{tutor-start=56,tutor-end=57}{<}\frac{\htmlData{tutor-start=63,tutor-end=64}{1}\htmlData{tutor-start=64,tutor-end=65}{2}}{\htmlData{tutor-start=67,tutor-end=68}{2}\htmlData{tutor-start=68,tutor-end=69}{5}} a 1 2 a 2 + a 2 2 a 3 + … + a 1 0 0 2 a 1 < 2 5 1 2 题解状态： 标准答案与规范题解待补充
题目标签：2007 IMO Shortlist A6
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2007-A7 2007 · Algebra · IMO/代数
Let n > 1 \htmlData{tutor-start=0,tutor-end=1}{n}\htmlData{tutor-start=1,tutor-end=2}{>}\htmlData{tutor-start=2,tutor-end=3}{1} n > 1 S = { ( x , y , z ) ∣ x , y , z ∈ { 0 , 1 , … , n } , x + y + z > 0 } \htmlData{tutor-start=0,tutor-end=1}{S}\htmlData{tutor-start=1,tutor-end=2}{=}\htmlData{tutor-start=2,tutor-end=4}{\{}\htmlData{tutor-start=4,tutor-end=5}{(}\htmlData{tutor-start=5,tutor-end=6}{x}\htmlData{tutor-start=6,tutor-end=7}{,} \htmlData{tutor-start=8,tutor-end=9}{y}\htmlData{tutor-start=9,tutor-end=10}{,} \htmlData{tutor-start=11,tutor-end=12}{z}\htmlData{tutor-start=12,tutor-end=13}{)} \htmlData{tutor-start=14,tutor-end=19}{\mid }\htmlData{tutor-start=19,tutor-end=20}{x}\htmlData{tutor-start=20,tutor-end=21}{,} \htmlData{tutor-start=22,tutor-end=23}{y}\htmlData{tutor-start=23,tutor-end=24}{,} \htmlData{tutor-start=25,tutor-end=26}{z} \htmlData{tutor-start=27,tutor-end=30}{\in}\htmlData{tutor-start=30,tutor-end=32}{\{}\htmlData{tutor-start=32,tutor-end=33}{0}\htmlData{tutor-start=33,tutor-end=34}{,}\htmlData{tutor-start=34,tutor-end=35}{1}\htmlData{tutor-start=35,tutor-end=36}{,} \htmlData{tutor-start=37,tutor-end=43}{\ldots}\htmlData{tutor-start=43,tutor-end=44}{,} \htmlData{tutor-start=45,tutor-end=46}{n}\htmlData{tutor-start=46,tutor-end=48}{\}}\htmlData{tutor-start=48,tutor-end=49}{,} \htmlData{tutor-start=50,tutor-end=51}{x}\htmlData{tutor-start=51,tutor-end=52}{+}\htmlData{tutor-start=52,tutor-end=53}{y}\htmlData{tutor-start=53,tutor-end=54}{+}\htmlData{tutor-start=54,tutor-end=55}{z}\htmlData{tutor-start=55,tutor-end=56}{>}\htmlData{tutor-start=56,tutor-end=57}{0}\htmlData{tutor-start=57,tutor-end=59}{\}} S = { ( x , y , z ) ∣ x , y , z ∈ { 0 , 1 , … , n } , x + y + z > 0 } ( n + 1 ) 3 − 1 \htmlData{tutor-start=0,tutor-end=1}{(}\htmlData{tutor-start=1,tutor-end=2}{n}\htmlData{tutor-start=2,tutor-end=3}{+}\htmlData{tutor-start=3,tutor-end=4}{1}\htmlData{tutor-start=4,tutor-end=5}{)}^{\htmlData{tutor-start=7,tutor-end=8}{3}}\htmlData{tutor-start=9,tutor-end=10}{-}\htmlData{tutor-start=10,tutor-end=11}{1} ( n + 1 ) 3 − 1 S \htmlData{tutor-start=0,tutor-end=1}{S} S 3 n \htmlData{tutor-start=0,tutor-end=1}{3} \htmlData{tutor-start=2,tutor-end=3}{n} 3 n 题解状态： 标准答案与规范题解待补充
题目标签：2007 IMO 正式题 A7
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2008-A1 2008 · Algebra · IMO/代数
Find all functions f : ( 0 , ∞ ) → ( 0 , ∞ ) \htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{:}\htmlData{tutor-start=2,tutor-end=3}{(}\htmlData{tutor-start=3,tutor-end=4}{0}\htmlData{tutor-start=4,tutor-end=5}{,} \htmlData{tutor-start=6,tutor-end=12}{\infty}\htmlData{tutor-start=12,tutor-end=13}{)} \htmlData{tutor-start=14,tutor-end=25}{\rightarrow}\htmlData{tutor-start=25,tutor-end=26}{(}\htmlData{tutor-start=26,tutor-end=27}{0}\htmlData{tutor-start=27,tutor-end=28}{,} \htmlData{tutor-start=29,tutor-end=35}{\infty}\htmlData{tutor-start=35,tutor-end=36}{)} f : ( 0 , ∞ ) → ( 0 , ∞ ) f ( p ) 2 + f ( q ) 2 f ( r 2 ) + f ( s 2 ) = p 2 + q 2 r 2 + s 2 \frac{\htmlData{tutor-start=6,tutor-end=7}{f}\htmlData{tutor-start=7,tutor-end=8}{(}\htmlData{tutor-start=8,tutor-end=9}{p}\htmlData{tutor-start=9,tutor-end=10}{)}^{\htmlData{tutor-start=12,tutor-end=13}{2}}\htmlData{tutor-start=14,tutor-end=15}{+}\htmlData{tutor-start=15,tutor-end=16}{f}\htmlData{tutor-start=16,tutor-end=17}{(}\htmlData{tutor-start=17,tutor-end=18}{q}\htmlData{tutor-start=18,tutor-end=19}{)}^{\htmlData{tutor-start=21,tutor-end=22}{2}}}{\htmlData{tutor-start=25,tutor-end=26}{f}\left(\htmlData{tutor-start=32,tutor-end=33}{r}^{\htmlData{tutor-start=35,tutor-end=36}{2}}\right)\htmlData{tutor-start=44,tutor-end=45}{+}\htmlData{tutor-start=45,tutor-end=46}{f}\left(\htmlData{tutor-start=52,tutor-end=53}{s}^{\htmlData{tutor-start=55,tutor-end=56}{2}}\right)}\htmlData{tutor-start=65,tutor-end=66}{=}\frac{\htmlData{tutor-start=72,tutor-end=73}{p}^{\htmlData{tutor-start=75,tutor-end=76}{2}}\htmlData{tutor-start=77,tutor-end=78}{+}\htmlData{tutor-start=78,tutor-end=79}{q}^{\htmlData{tutor-start=81,tutor-end=82}{2}}}{\htmlData{tutor-start=85,tutor-end=86}{r}^{\htmlData{tutor-start=88,tutor-end=89}{2}}\htmlData{tutor-start=90,tutor-end=91}{+}\htmlData{tutor-start=91,tutor-end=92}{s}^{\htmlData{tutor-start=94,tutor-end=95}{2}}} f ( r 2 ) + f ( s 2 ) f ( p ) 2 + f ( q ) 2 = r 2 + s 2 p 2 + q 2 p , q , r , s > 0 \htmlData{tutor-start=0,tutor-end=1}{p}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{q}\htmlData{tutor-start=4,tutor-end=5}{,} \htmlData{tutor-start=6,tutor-end=7}{r}\htmlData{tutor-start=7,tutor-end=8}{,} \htmlData{tutor-start=9,tutor-end=10}{s}\htmlData{tutor-start=10,tutor-end=11}{>}\htmlData{tutor-start=11,tutor-end=12}{0} p , q , r , s > 0 p q = r s \htmlData{tutor-start=0,tutor-end=1}{p} \htmlData{tutor-start=2,tutor-end=3}{q}\htmlData{tutor-start=3,tutor-end=4}{=}\htmlData{tutor-start=4,tutor-end=5}{r} \htmlData{tutor-start=6,tutor-end=7}{s} p q = r s 题解状态： 标准答案与规范题解待补充
题目标签：2008 IMO 正式题 A1
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2008-A2 2008 · Algebra · IMO/代数
(a) Prove the inequality
x 2 ( x − 1 ) 2 + y 2 ( y − 1 ) 2 + z 2 ( z − 1 ) 2 ≥ 1 \frac{\htmlData{tutor-start=6,tutor-end=7}{x}^{\htmlData{tutor-start=9,tutor-end=10}{2}}}{\htmlData{tutor-start=13,tutor-end=14}{(}\htmlData{tutor-start=14,tutor-end=15}{x}\htmlData{tutor-start=15,tutor-end=16}{-}\htmlData{tutor-start=16,tutor-end=17}{1}\htmlData{tutor-start=17,tutor-end=18}{)}^{\htmlData{tutor-start=20,tutor-end=21}{2}}}\htmlData{tutor-start=23,tutor-end=24}{+}\frac{\htmlData{tutor-start=30,tutor-end=31}{y}^{\htmlData{tutor-start=33,tutor-end=34}{2}}}{\htmlData{tutor-start=37,tutor-end=38}{(}\htmlData{tutor-start=38,tutor-end=39}{y}\htmlData{tutor-start=39,tutor-end=40}{-}\htmlData{tutor-start=40,tutor-end=41}{1}\htmlData{tutor-start=41,tutor-end=42}{)}^{\htmlData{tutor-start=44,tutor-end=45}{2}}}\htmlData{tutor-start=47,tutor-end=48}{+}\frac{\htmlData{tutor-start=54,tutor-end=55}{z}^{\htmlData{tutor-start=57,tutor-end=58}{2}}}{\htmlData{tutor-start=61,tutor-end=62}{(}\htmlData{tutor-start=62,tutor-end=63}{z}\htmlData{tutor-start=63,tutor-end=64}{-}\htmlData{tutor-start=64,tutor-end=65}{1}\htmlData{tutor-start=65,tutor-end=66}{)}^{\htmlData{tutor-start=68,tutor-end=69}{2}}} \htmlData{tutor-start=72,tutor-end=77}{\geq }\htmlData{tutor-start=77,tutor-end=78}{1} ( x − 1 ) 2 x 2 + ( y − 1 ) 2 y 2 + ( z − 1 ) 2 z 2 ≥ 1 x , y , z ≠ 1 \htmlData{tutor-start=0,tutor-end=1}{x}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{y}\htmlData{tutor-start=4,tutor-end=5}{,} \htmlData{tutor-start=6,tutor-end=7}{z} \neq \htmlData{tutor-start=13,tutor-end=14}{1} x , y , z = 1 x y z = 1 \htmlData{tutor-start=0,tutor-end=1}{x} \htmlData{tutor-start=2,tutor-end=3}{y} \htmlData{tutor-start=4,tutor-end=5}{z}\htmlData{tutor-start=5,tutor-end=6}{=}\htmlData{tutor-start=6,tutor-end=7}{1} x y z = 1 x , y , z \htmlData{tutor-start=0,tutor-end=1}{x}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{y}\htmlData{tutor-start=4,tutor-end=5}{,} \htmlData{tutor-start=6,tutor-end=7}{z} x , y , z 题解状态： 标准答案与规范题解待补充
题目标签：2008 IMO 正式题 A2
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2008-A3 2008 · Algebra · IMO/代数
Let S ⊆ R \htmlData{tutor-start=0,tutor-end=1}{S} \htmlData{tutor-start=2,tutor-end=12}{\subseteq }\mathbb{\htmlData{tutor-start=20,tutor-end=21}{R}} S ⊆ R ( f , g ) \htmlData{tutor-start=0,tutor-end=1}{(}\htmlData{tutor-start=1,tutor-end=2}{f}\htmlData{tutor-start=2,tutor-end=3}{,} \htmlData{tutor-start=4,tutor-end=5}{g}\htmlData{tutor-start=5,tutor-end=6}{)} ( f , g ) S \htmlData{tutor-start=0,tutor-end=1}{S} S S \htmlData{tutor-start=0,tutor-end=1}{S} S S \htmlData{tutor-start=0,tutor-end=1}{S} S f ( x ) < f ( y ) \htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{x}\htmlData{tutor-start=3,tutor-end=4}{)}\htmlData{tutor-start=4,tutor-end=5}{<}\htmlData{tutor-start=5,tutor-end=6}{f}\htmlData{tutor-start=6,tutor-end=7}{(}\htmlData{tutor-start=7,tutor-end=8}{y}\htmlData{tutor-start=8,tutor-end=9}{)} f ( x ) < f ( y ) g ( x ) < g ( y ) \htmlData{tutor-start=0,tutor-end=1}{g}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{x}\htmlData{tutor-start=3,tutor-end=4}{)}\htmlData{tutor-start=4,tutor-end=5}{<}\htmlData{tutor-start=5,tutor-end=6}{g}\htmlData{tutor-start=6,tutor-end=7}{(}\htmlData{tutor-start=7,tutor-end=8}{y}\htmlData{tutor-start=8,tutor-end=9}{)} g ( x ) < g ( y ) x , y ∈ S \htmlData{tutor-start=0,tutor-end=1}{x}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{y} \htmlData{tutor-start=5,tutor-end=9}{\in }\htmlData{tutor-start=9,tutor-end=10}{S} x , y ∈ S x < y \htmlData{tutor-start=0,tutor-end=1}{x}\htmlData{tutor-start=1,tutor-end=2}{<}\htmlData{tutor-start=2,tutor-end=3}{y} x < y f ( g ( g ( x ) ) ) < g ( f ( x ) ) \htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{g}\htmlData{tutor-start=3,tutor-end=4}{(}\htmlData{tutor-start=4,tutor-end=5}{g}\htmlData{tutor-start=5,tutor-end=6}{(}\htmlData{tutor-start=6,tutor-end=7}{x}\htmlData{tutor-start=7,tutor-end=8}{)}\htmlData{tutor-start=8,tutor-end=9}{)}\htmlData{tutor-start=9,tutor-end=10}{)}\htmlData{tutor-start=10,tutor-end=11}{<}\htmlData{tutor-start=11,tutor-end=12}{g}\htmlData{tutor-start=12,tutor-end=13}{(}\htmlData{tutor-start=13,tutor-end=14}{f}\htmlData{tutor-start=14,tutor-end=15}{(}\htmlData{tutor-start=15,tutor-end=16}{x}\htmlData{tutor-start=16,tutor-end=17}{)}\htmlData{tutor-start=17,tutor-end=18}{)} f ( g ( g ( x ) ) ) < g ( f ( x ) ) x ∈ S \htmlData{tutor-start=0,tutor-end=1}{x} \htmlData{tutor-start=2,tutor-end=6}{\in }\htmlData{tutor-start=6,tutor-end=7}{S} x ∈ S S = N \htmlData{tutor-start=0,tutor-end=1}{S}\htmlData{tutor-start=1,tutor-end=2}{=}\mathbb{\htmlData{tutor-start=10,tutor-end=11}{N}} S = N S = { a − 1 / b : a , b ∈ N } \htmlData{tutor-start=0,tutor-end=1}{S}\htmlData{tutor-start=1,tutor-end=2}{=}\htmlData{tutor-start=2,tutor-end=4}{\{}\htmlData{tutor-start=4,tutor-end=5}{a}\htmlData{tutor-start=5,tutor-end=6}{-}\htmlData{tutor-start=6,tutor-end=7}{1} \htmlData{tutor-start=8,tutor-end=9}{/} \htmlData{tutor-start=10,tutor-end=11}{b}\htmlData{tutor-start=11,tutor-end=12}{:} \htmlData{tutor-start=13,tutor-end=14}{a}\htmlData{tutor-start=14,tutor-end=15}{,} \htmlData{tutor-start=16,tutor-end=17}{b} \htmlData{tutor-start=18,tutor-end=22}{\in }\mathbb{\htmlData{tutor-start=30,tutor-end=31}{N}}\htmlData{tutor-start=32,tutor-end=34}{\}} S = { a − 1 / b : a , b ∈ N } 题解状态： 标准答案与规范题解待补充
题目标签：2008 IMO Shortlist A3
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2008-A4 2008 · Algebra · IMO/代数
For an integer m \htmlData{tutor-start=0,tutor-end=1}{m} m t ( m ) \htmlData{tutor-start=0,tutor-end=1}{t}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{m}\htmlData{tutor-start=3,tutor-end=4}{)} t ( m ) { 1 , 2 , 3 } \htmlData{tutor-start=0,tutor-end=2}{\{}\htmlData{tutor-start=2,tutor-end=3}{1}\htmlData{tutor-start=3,tutor-end=4}{,}\htmlData{tutor-start=4,tutor-end=5}{2}\htmlData{tutor-start=5,tutor-end=6}{,}\htmlData{tutor-start=6,tutor-end=7}{3}\htmlData{tutor-start=7,tutor-end=9}{\}} { 1 , 2 , 3 } m + t ( m ) \htmlData{tutor-start=0,tutor-end=1}{m}\htmlData{tutor-start=1,tutor-end=2}{+}\htmlData{tutor-start=2,tutor-end=3}{t}\htmlData{tutor-start=3,tutor-end=4}{(}\htmlData{tutor-start=4,tutor-end=5}{m}\htmlData{tutor-start=5,tutor-end=6}{)} m + t ( m ) f : Z → Z \htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{:} \mathbb{\htmlData{tutor-start=11,tutor-end=12}{Z}} \htmlData{tutor-start=14,tutor-end=26}{\rightarrow }\mathbb{\htmlData{tutor-start=34,tutor-end=35}{Z}} f : Z → Z f ( − 1 ) = 0 , f ( 0 ) = 1 , f ( 1 ) = − 1 \htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{-}\htmlData{tutor-start=3,tutor-end=4}{1}\htmlData{tutor-start=4,tutor-end=5}{)}\htmlData{tutor-start=5,tutor-end=6}{=}\htmlData{tutor-start=6,tutor-end=7}{0}\htmlData{tutor-start=7,tutor-end=8}{,} \htmlData{tutor-start=9,tutor-end=10}{f}\htmlData{tutor-start=10,tutor-end=11}{(}\htmlData{tutor-start=11,tutor-end=12}{0}\htmlData{tutor-start=12,tutor-end=13}{)}\htmlData{tutor-start=13,tutor-end=14}{=}\htmlData{tutor-start=14,tutor-end=15}{1}\htmlData{tutor-start=15,tutor-end=16}{,} \htmlData{tutor-start=17,tutor-end=18}{f}\htmlData{tutor-start=18,tutor-end=19}{(}\htmlData{tutor-start=19,tutor-end=20}{1}\htmlData{tutor-start=20,tutor-end=21}{)}\htmlData{tutor-start=21,tutor-end=22}{=}\htmlData{tutor-start=22,tutor-end=23}{-}\htmlData{tutor-start=23,tutor-end=24}{1} f ( − 1 ) = 0 , f ( 0 ) = 1 , f ( 1 ) = − 1 f ( 2 n + m ) = f ( 2 n − t ( m ) ) − f ( m ) for all integers m , n ≥ 0 with 2 n > m . \htmlData{tutor-start=0,tutor-end=1}{f}\left(\htmlData{tutor-start=7,tutor-end=8}{2}^{\htmlData{tutor-start=10,tutor-end=11}{n}}\htmlData{tutor-start=12,tutor-end=13}{+}\htmlData{tutor-start=13,tutor-end=14}{m}\right)\htmlData{tutor-start=21,tutor-end=22}{=}\htmlData{tutor-start=22,tutor-end=23}{f}\left(\htmlData{tutor-start=29,tutor-end=30}{2}^{\htmlData{tutor-start=32,tutor-end=33}{n}}\htmlData{tutor-start=34,tutor-end=35}{-}\htmlData{tutor-start=35,tutor-end=36}{t}\htmlData{tutor-start=36,tutor-end=37}{(}\htmlData{tutor-start=37,tutor-end=38}{m}\htmlData{tutor-start=38,tutor-end=39}{)}\right)\htmlData{tutor-start=46,tutor-end=47}{-}\htmlData{tutor-start=47,tutor-end=48}{f}\htmlData{tutor-start=48,tutor-end=49}{(}\htmlData{tutor-start=49,tutor-end=50}{m}\htmlData{tutor-start=50,tutor-end=51}{)} \quad \text { \htmlData{tutor-start=66,tutor-end=67}{f}\htmlData{tutor-start=67,tutor-end=68}{o}\htmlData{tutor-start=68,tutor-end=69}{r} \htmlData{tutor-start=70,tutor-end=71}{a}\htmlData{tutor-start=71,tutor-end=72}{l}\htmlData{tutor-start=72,tutor-end=73}{l} \htmlData{tutor-start=74,tutor-end=75}{i}\htmlData{tutor-start=75,tutor-end=76}{n}\htmlData{tutor-start=76,tutor-end=77}{t}\htmlData{tutor-start=77,tutor-end=78}{e}\htmlData{tutor-start=78,tutor-end=79}{g}\htmlData{tutor-start=79,tutor-end=80}{e}\htmlData{tutor-start=80,tutor-end=81}{r}\htmlData{tutor-start=81,tutor-end=82}{s} } \htmlData{tutor-start=85,tutor-end=86}{m}\htmlData{tutor-start=86,tutor-end=87}{,} \htmlData{tutor-start=88,tutor-end=89}{n} \htmlData{tutor-start=90,tutor-end=95}{\geq }\htmlData{tutor-start=95,tutor-end=96}{0} \text { \htmlData{tutor-start=105,tutor-end=106}{w}\htmlData{tutor-start=106,tutor-end=107}{i}\htmlData{tutor-start=107,tutor-end=108}{t}\htmlData{tutor-start=108,tutor-end=109}{h} } \htmlData{tutor-start=112,tutor-end=113}{2}^{\htmlData{tutor-start=115,tutor-end=116}{n}}\htmlData{tutor-start=117,tutor-end=118}{>}\htmlData{tutor-start=118,tutor-end=119}{m} \text {\htmlData{tutor-start=127,tutor-end=128}{.} } f ( 2 n + m ) = f ( 2 n − t ( m ) ) − f ( m ) f o r a l l i n t e g e r s m , n ≥ 0 w i t h 2 n > m . f ( 3 p ) ≥ 0 \htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{3} \htmlData{tutor-start=4,tutor-end=5}{p}\htmlData{tutor-start=5,tutor-end=6}{)} \htmlData{tutor-start=7,tutor-end=12}{\geq }\htmlData{tutor-start=12,tutor-end=13}{0} f ( 3 p ) ≥ 0 p ≥ 0 \htmlData{tutor-start=0,tutor-end=1}{p} \htmlData{tutor-start=2,tutor-end=7}{\geq }\htmlData{tutor-start=7,tutor-end=8}{0} p ≥ 0 题解状态： 标准答案与规范题解待补充
题目标签：2008 IMO Shortlist A4
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2008-A5 2008 · Algebra · IMO/代数
Let a , b , c , d \htmlData{tutor-start=0,tutor-end=1}{a}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{b}\htmlData{tutor-start=4,tutor-end=5}{,} \htmlData{tutor-start=6,tutor-end=7}{c}\htmlData{tutor-start=7,tutor-end=8}{,} \htmlData{tutor-start=9,tutor-end=10}{d} a , b , c , d a b c d = 1 and a + b + c + d > a b + b c + c d + d a \htmlData{tutor-start=0,tutor-end=1}{a} \htmlData{tutor-start=2,tutor-end=3}{b} \htmlData{tutor-start=4,tutor-end=5}{c} \htmlData{tutor-start=6,tutor-end=7}{d}\htmlData{tutor-start=7,tutor-end=8}{=}\htmlData{tutor-start=8,tutor-end=9}{1} \quad \text { \htmlData{tutor-start=24,tutor-end=25}{a}\htmlData{tutor-start=25,tutor-end=26}{n}\htmlData{tutor-start=26,tutor-end=27}{d} } \quad \htmlData{tutor-start=36,tutor-end=37}{a}\htmlData{tutor-start=37,tutor-end=38}{+}\htmlData{tutor-start=38,tutor-end=39}{b}\htmlData{tutor-start=39,tutor-end=40}{+}\htmlData{tutor-start=40,tutor-end=41}{c}\htmlData{tutor-start=41,tutor-end=42}{+}\htmlData{tutor-start=42,tutor-end=43}{d}\htmlData{tutor-start=43,tutor-end=44}{>}\frac{\htmlData{tutor-start=50,tutor-end=51}{a}}{\htmlData{tutor-start=53,tutor-end=54}{b}}\htmlData{tutor-start=55,tutor-end=56}{+}\frac{\htmlData{tutor-start=62,tutor-end=63}{b}}{\htmlData{tutor-start=65,tutor-end=66}{c}}\htmlData{tutor-start=67,tutor-end=68}{+}\frac{\htmlData{tutor-start=74,tutor-end=75}{c}}{\htmlData{tutor-start=77,tutor-end=78}{d}}\htmlData{tutor-start=79,tutor-end=80}{+}\frac{\htmlData{tutor-start=86,tutor-end=87}{d}}{\htmlData{tutor-start=89,tutor-end=90}{a}} a b c d = 1 a n d a + b + c + d > b a + c b + d c + a d a + b + c + d < b a + c b + d c + a d \htmlData{tutor-start=0,tutor-end=1}{a}\htmlData{tutor-start=1,tutor-end=2}{+}\htmlData{tutor-start=2,tutor-end=3}{b}\htmlData{tutor-start=3,tutor-end=4}{+}\htmlData{tutor-start=4,tutor-end=5}{c}\htmlData{tutor-start=5,tutor-end=6}{+}\htmlData{tutor-start=6,tutor-end=7}{d}\htmlData{tutor-start=7,tutor-end=8}{<}\frac{\htmlData{tutor-start=14,tutor-end=15}{b}}{\htmlData{tutor-start=17,tutor-end=18}{a}}\htmlData{tutor-start=19,tutor-end=20}{+}\frac{\htmlData{tutor-start=26,tutor-end=27}{c}}{\htmlData{tutor-start=29,tutor-end=30}{b}}\htmlData{tutor-start=31,tutor-end=32}{+}\frac{\htmlData{tutor-start=38,tutor-end=39}{d}}{\htmlData{tutor-start=41,tutor-end=42}{c}}\htmlData{tutor-start=43,tutor-end=44}{+}\frac{\htmlData{tutor-start=50,tutor-end=51}{a}}{\htmlData{tutor-start=53,tutor-end=54}{d}} a + b + c + d < a b + b c + c d + d a 题解状态： 标准答案与规范题解待补充
题目标签：2008 IMO Shortlist A5
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2008-A6 2008 · Algebra · IMO/代数
Let f : R → N \htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{:} \mathbb{\htmlData{tutor-start=11,tutor-end=12}{R}} \htmlData{tutor-start=14,tutor-end=26}{\rightarrow }\mathbb{\htmlData{tutor-start=34,tutor-end=35}{N}} f : R → N f ( x + 1 f ( y ) ) = f ( y + 1 f ( x ) ) for all x , y ∈ R . \htmlData{tutor-start=0,tutor-end=1}{f}\left(\htmlData{tutor-start=7,tutor-end=8}{x}\htmlData{tutor-start=8,tutor-end=9}{+}\frac{\htmlData{tutor-start=15,tutor-end=16}{1}}{\htmlData{tutor-start=18,tutor-end=19}{f}\htmlData{tutor-start=19,tutor-end=20}{(}\htmlData{tutor-start=20,tutor-end=21}{y}\htmlData{tutor-start=21,tutor-end=22}{)}}\right)\htmlData{tutor-start=30,tutor-end=31}{=}\htmlData{tutor-start=31,tutor-end=32}{f}\left(\htmlData{tutor-start=38,tutor-end=39}{y}\htmlData{tutor-start=39,tutor-end=40}{+}\frac{\htmlData{tutor-start=46,tutor-end=47}{1}}{\htmlData{tutor-start=49,tutor-end=50}{f}\htmlData{tutor-start=50,tutor-end=51}{(}\htmlData{tutor-start=51,tutor-end=52}{x}\htmlData{tutor-start=52,tutor-end=53}{)}}\right) \quad \text { \htmlData{tutor-start=76,tutor-end=77}{f}\htmlData{tutor-start=77,tutor-end=78}{o}\htmlData{tutor-start=78,tutor-end=79}{r} \htmlData{tutor-start=80,tutor-end=81}{a}\htmlData{tutor-start=81,tutor-end=82}{l}\htmlData{tutor-start=82,tutor-end=83}{l} } \htmlData{tutor-start=86,tutor-end=87}{x}\htmlData{tutor-start=87,tutor-end=88}{,} \htmlData{tutor-start=89,tutor-end=90}{y} \htmlData{tutor-start=91,tutor-end=95}{\in }\mathbb{\htmlData{tutor-start=103,tutor-end=104}{R}} \htmlData{tutor-start=106,tutor-end=107}{.} f ( x + f ( y ) 1 ) = f ( y + f ( x ) 1 ) f o r a l l x , y ∈ R . f \htmlData{tutor-start=0,tutor-end=1}{f} f 题解状态： 标准答案与规范题解待补充
题目标签：2008 IMO Shortlist A6
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2008-A7 2008 · Algebra · IMO/代数
Prove that for any four positive real numbers a , b , c , d \htmlData{tutor-start=0,tutor-end=1}{a}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{b}\htmlData{tutor-start=4,tutor-end=5}{,} \htmlData{tutor-start=6,tutor-end=7}{c}\htmlData{tutor-start=7,tutor-end=8}{,} \htmlData{tutor-start=9,tutor-end=10}{d} a , b , c , d ( a − b ) ( a − c ) a + b + c + ( b − c ) ( b − d ) b + c + d + ( c − d ) ( c − a ) c + d + a + ( d − a ) ( d − b ) d + a + b ≥ 0 \frac{\htmlData{tutor-start=6,tutor-end=7}{(}\htmlData{tutor-start=7,tutor-end=8}{a}\htmlData{tutor-start=8,tutor-end=9}{-}\htmlData{tutor-start=9,tutor-end=10}{b}\htmlData{tutor-start=10,tutor-end=11}{)}\htmlData{tutor-start=11,tutor-end=12}{(}\htmlData{tutor-start=12,tutor-end=13}{a}\htmlData{tutor-start=13,tutor-end=14}{-}\htmlData{tutor-start=14,tutor-end=15}{c}\htmlData{tutor-start=15,tutor-end=16}{)}}{\htmlData{tutor-start=18,tutor-end=19}{a}\htmlData{tutor-start=19,tutor-end=20}{+}\htmlData{tutor-start=20,tutor-end=21}{b}\htmlData{tutor-start=21,tutor-end=22}{+}\htmlData{tutor-start=22,tutor-end=23}{c}}\htmlData{tutor-start=24,tutor-end=25}{+}\frac{\htmlData{tutor-start=31,tutor-end=32}{(}\htmlData{tutor-start=32,tutor-end=33}{b}\htmlData{tutor-start=33,tutor-end=34}{-}\htmlData{tutor-start=34,tutor-end=35}{c}\htmlData{tutor-start=35,tutor-end=36}{)}\htmlData{tutor-start=36,tutor-end=37}{(}\htmlData{tutor-start=37,tutor-end=38}{b}\htmlData{tutor-start=38,tutor-end=39}{-}\htmlData{tutor-start=39,tutor-end=40}{d}\htmlData{tutor-start=40,tutor-end=41}{)}}{\htmlData{tutor-start=43,tutor-end=44}{b}\htmlData{tutor-start=44,tutor-end=45}{+}\htmlData{tutor-start=45,tutor-end=46}{c}\htmlData{tutor-start=46,tutor-end=47}{+}\htmlData{tutor-start=47,tutor-end=48}{d}}\htmlData{tutor-start=49,tutor-end=50}{+}\frac{\htmlData{tutor-start=56,tutor-end=57}{(}\htmlData{tutor-start=57,tutor-end=58}{c}\htmlData{tutor-start=58,tutor-end=59}{-}\htmlData{tutor-start=59,tutor-end=60}{d}\htmlData{tutor-start=60,tutor-end=61}{)}\htmlData{tutor-start=61,tutor-end=62}{(}\htmlData{tutor-start=62,tutor-end=63}{c}\htmlData{tutor-start=63,tutor-end=64}{-}\htmlData{tutor-start=64,tutor-end=65}{a}\htmlData{tutor-start=65,tutor-end=66}{)}}{\htmlData{tutor-start=68,tutor-end=69}{c}\htmlData{tutor-start=69,tutor-end=70}{+}\htmlData{tutor-start=70,tutor-end=71}{d}\htmlData{tutor-start=71,tutor-end=72}{+}\htmlData{tutor-start=72,tutor-end=73}{a}}\htmlData{tutor-start=74,tutor-end=75}{+}\frac{\htmlData{tutor-start=81,tutor-end=82}{(}\htmlData{tutor-start=82,tutor-end=83}{d}\htmlData{tutor-start=83,tutor-end=84}{-}\htmlData{tutor-start=84,tutor-end=85}{a}\htmlData{tutor-start=85,tutor-end=86}{)}\htmlData{tutor-start=86,tutor-end=87}{(}\htmlData{tutor-start=87,tutor-end=88}{d}\htmlData{tutor-start=88,tutor-end=89}{-}\htmlData{tutor-start=89,tutor-end=90}{b}\htmlData{tutor-start=90,tutor-end=91}{)}}{\htmlData{tutor-start=93,tutor-end=94}{d}\htmlData{tutor-start=94,tutor-end=95}{+}\htmlData{tutor-start=95,tutor-end=96}{a}\htmlData{tutor-start=96,tutor-end=97}{+}\htmlData{tutor-start=97,tutor-end=98}{b}} \htmlData{tutor-start=100,tutor-end=105}{\geq }\htmlData{tutor-start=105,tutor-end=106}{0} a + b + c ( a − b ) ( a − c ) + b + c + d ( b − c ) ( b − d ) + c + d + a ( c − d ) ( c − a ) + d + a + b ( d − a ) ( d − b ) ≥ 0 题解状态： 标准答案与规范题解待补充
题目标签：2008 IMO Shortlist A7
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2009-A1 2009 · Algebra · IMO/代数
Find the largest possible integer k \htmlData{tutor-start=0,tutor-end=1}{k} k b 1 ≤ b 2 ≤ … ≤ b 2009 the lengths of the blue sides, r 1 ≤ r 2 ≤ … ≤ r 2009 the lengths of the red sides, and w 1 ≤ w 2 ≤ … ≤ w 2009 the lengths of the white sides. \begin{aligned}
& \htmlData{tutor-start=18,tutor-end=19}{b}_{\htmlData{tutor-start=21,tutor-end=22}{1}} \htmlData{tutor-start=24,tutor-end=29}{\leq }\htmlData{tutor-start=29,tutor-end=30}{b}_{\htmlData{tutor-start=32,tutor-end=33}{2}} \\
& \htmlData{tutor-start=40,tutor-end=45}{\leq }\htmlData{tutor-start=45,tutor-end=52}{\ldots }\htmlData{tutor-start=52,tutor-end=57}{\leq }\htmlData{tutor-start=57,tutor-end=58}{b}_{\htmlData{tutor-start=60,tutor-end=61}{2}\htmlData{tutor-start=61,tutor-end=62}{0}\htmlData{tutor-start=62,tutor-end=63}{0}\htmlData{tutor-start=63,tutor-end=64}{9}} \quad \text { \htmlData{tutor-start=80,tutor-end=81}{t}\htmlData{tutor-start=81,tutor-end=82}{h}\htmlData{tutor-start=82,tutor-end=83}{e} \htmlData{tutor-start=84,tutor-end=85}{l}\htmlData{tutor-start=85,tutor-end=86}{e}\htmlData{tutor-start=86,tutor-end=87}{n}\htmlData{tutor-start=87,tutor-end=88}{g}\htmlData{tutor-start=88,tutor-end=89}{t}\htmlData{tutor-start=89,tutor-end=90}{h}\htmlData{tutor-start=90,tutor-end=91}{s} \htmlData{tutor-start=92,tutor-end=93}{o}\htmlData{tutor-start=93,tutor-end=94}{f} \htmlData{tutor-start=95,tutor-end=96}{t}\htmlData{tutor-start=96,tutor-end=97}{h}\htmlData{tutor-start=97,tutor-end=98}{e} \htmlData{tutor-start=99,tutor-end=100}{b}\htmlData{tutor-start=100,tutor-end=101}{l}\htmlData{tutor-start=101,tutor-end=102}{u}\htmlData{tutor-start=102,tutor-end=103}{e} \htmlData{tutor-start=104,tutor-end=105}{s}\htmlData{tutor-start=105,tutor-end=106}{i}\htmlData{tutor-start=106,tutor-end=107}{d}\htmlData{tutor-start=107,tutor-end=108}{e}\htmlData{tutor-start=108,tutor-end=109}{s}\htmlData{tutor-start=109,tutor-end=110}{,} } \\
\htmlData{tutor-start=116,tutor-end=117}{r}_{\htmlData{tutor-start=119,tutor-end=120}{1}} & \htmlData{tutor-start=124,tutor-end=129}{\leq }\htmlData{tutor-start=129,tutor-end=130}{r}_{\htmlData{tutor-start=132,tutor-end=133}{2}} \htmlData{tutor-start=135,tutor-end=140}{\leq }\htmlData{tutor-start=140,tutor-end=147}{\ldots }\htmlData{tutor-start=147,tutor-end=152}{\leq }\htmlData{tutor-start=152,tutor-end=153}{r}_{\htmlData{tutor-start=155,tutor-end=156}{2}\htmlData{tutor-start=156,tutor-end=157}{0}\htmlData{tutor-start=157,tutor-end=158}{0}\htmlData{tutor-start=158,tutor-end=159}{9}} \quad \text { \htmlData{tutor-start=175,tutor-end=176}{t}\htmlData{tutor-start=176,tutor-end=177}{h}\htmlData{tutor-start=177,tutor-end=178}{e} \htmlData{tutor-start=179,tutor-end=180}{l}\htmlData{tutor-start=180,tutor-end=181}{e}\htmlData{tutor-start=181,tutor-end=182}{n}\htmlData{tutor-start=182,tutor-end=183}{g}\htmlData{tutor-start=183,tutor-end=184}{t}\htmlData{tutor-start=184,tutor-end=185}{h}\htmlData{tutor-start=185,tutor-end=186}{s} \htmlData{tutor-start=187,tutor-end=188}{o}\htmlData{tutor-start=188,tutor-end=189}{f} \htmlData{tutor-start=190,tutor-end=191}{t}\htmlData{tutor-start=191,tutor-end=192}{h}\htmlData{tutor-start=192,tutor-end=193}{e} \htmlData{tutor-start=194,tutor-end=195}{r}\htmlData{tutor-start=195,tutor-end=196}{e}\htmlData{tutor-start=196,tutor-end=197}{d} \htmlData{tutor-start=198,tutor-end=199}{s}\htmlData{tutor-start=199,tutor-end=200}{i}\htmlData{tutor-start=200,tutor-end=201}{d}\htmlData{tutor-start=201,tutor-end=202}{e}\htmlData{tutor-start=202,tutor-end=203}{s}\htmlData{tutor-start=203,tutor-end=204}{,} } \\
\text { \htmlData{tutor-start=218,tutor-end=219}{a}\htmlData{tutor-start=219,tutor-end=220}{n}\htmlData{tutor-start=220,tutor-end=221}{d} } \quad \htmlData{tutor-start=230,tutor-end=231}{w}_{\htmlData{tutor-start=233,tutor-end=234}{1}} & \htmlData{tutor-start=238,tutor-end=243}{\leq }\htmlData{tutor-start=243,tutor-end=244}{w}_{\htmlData{tutor-start=246,tutor-end=247}{2}} \htmlData{tutor-start=249,tutor-end=254}{\leq }\htmlData{tutor-start=254,tutor-end=261}{\ldots }\htmlData{tutor-start=261,tutor-end=266}{\leq }\htmlData{tutor-start=266,tutor-end=267}{w}_{\htmlData{tutor-start=269,tutor-end=270}{2}\htmlData{tutor-start=270,tutor-end=271}{0}\htmlData{tutor-start=271,tutor-end=272}{0}\htmlData{tutor-start=272,tutor-end=273}{9}} \quad \text { \htmlData{tutor-start=289,tutor-end=290}{t}\htmlData{tutor-start=290,tutor-end=291}{h}\htmlData{tutor-start=291,tutor-end=292}{e} \htmlData{tutor-start=293,tutor-end=294}{l}\htmlData{tutor-start=294,tutor-end=295}{e}\htmlData{tutor-start=295,tutor-end=296}{n}\htmlData{tutor-start=296,tutor-end=297}{g}\htmlData{tutor-start=297,tutor-end=298}{t}\htmlData{tutor-start=298,tutor-end=299}{h}\htmlData{tutor-start=299,tutor-end=300}{s} \htmlData{tutor-start=301,tutor-end=302}{o}\htmlData{tutor-start=302,tutor-end=303}{f} \htmlData{tutor-start=304,tutor-end=305}{t}\htmlData{tutor-start=305,tutor-end=306}{h}\htmlData{tutor-start=306,tutor-end=307}{e} \htmlData{tutor-start=308,tutor-end=309}{w}\htmlData{tutor-start=309,tutor-end=310}{h}\htmlData{tutor-start=310,tutor-end=311}{i}\htmlData{tutor-start=311,tutor-end=312}{t}\htmlData{tutor-start=312,tutor-end=313}{e} \htmlData{tutor-start=314,tutor-end=315}{s}\htmlData{tutor-start=315,tutor-end=316}{i}\htmlData{tutor-start=316,tutor-end=317}{d}\htmlData{tutor-start=317,tutor-end=318}{e}\htmlData{tutor-start=318,tutor-end=319}{s}\htmlData{tutor-start=319,tutor-end=320}{.} }
\end{aligned} r 1 a n d w 1 b 1 ≤ b 2 ≤ … ≤ b 2 0 0 9 t h e l e n g t h s o f t h e b l u e s i d e s , ≤ r 2 ≤ … ≤ r 2 0 0 9 t h e l e n g t h s o f t h e r e d s i d e s , ≤ w 2 ≤ … ≤ w 2 0 0 9 t h e l e n g t h s o f t h e w h i t e s i d e s . k \htmlData{tutor-start=0,tutor-end=1}{k} k j \htmlData{tutor-start=0,tutor-end=1}{j} j b j , r j , w j \htmlData{tutor-start=0,tutor-end=1}{b}_{\htmlData{tutor-start=3,tutor-end=4}{j}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{r}_{\htmlData{tutor-start=10,tutor-end=11}{j}}\htmlData{tutor-start=12,tutor-end=13}{,} \htmlData{tutor-start=14,tutor-end=15}{w}_{\htmlData{tutor-start=17,tutor-end=18}{j}} b j , r j , w j 题解状态： 标准答案与规范题解待补充
题目标签：2009 IMO Shortlist A1
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2009-A2 2009 · Algebra · IMO/代数
Let a , b , c \htmlData{tutor-start=0,tutor-end=1}{a}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{b}\htmlData{tutor-start=4,tutor-end=5}{,} \htmlData{tutor-start=6,tutor-end=7}{c} a , b , c 1 a + 1 b + 1 c = a + b + c \frac{\htmlData{tutor-start=6,tutor-end=7}{1}}{\htmlData{tutor-start=9,tutor-end=10}{a}}\htmlData{tutor-start=11,tutor-end=12}{+}\frac{\htmlData{tutor-start=18,tutor-end=19}{1}}{\htmlData{tutor-start=21,tutor-end=22}{b}}\htmlData{tutor-start=23,tutor-end=24}{+}\frac{\htmlData{tutor-start=30,tutor-end=31}{1}}{\htmlData{tutor-start=33,tutor-end=34}{c}}\htmlData{tutor-start=35,tutor-end=36}{=}\htmlData{tutor-start=36,tutor-end=37}{a}\htmlData{tutor-start=37,tutor-end=38}{+}\htmlData{tutor-start=38,tutor-end=39}{b}\htmlData{tutor-start=39,tutor-end=40}{+}\htmlData{tutor-start=40,tutor-end=41}{c} a 1 + b 1 + c 1 = a + b + c 1 ( 2 a + b + c ) 2 + 1 ( 2 b + c + a ) 2 + 1 ( 2 c + a + b ) 2 ≤ 3 16 \frac{\htmlData{tutor-start=6,tutor-end=7}{1}}{\htmlData{tutor-start=9,tutor-end=10}{(}\htmlData{tutor-start=10,tutor-end=11}{2} \htmlData{tutor-start=12,tutor-end=13}{a}\htmlData{tutor-start=13,tutor-end=14}{+}\htmlData{tutor-start=14,tutor-end=15}{b}\htmlData{tutor-start=15,tutor-end=16}{+}\htmlData{tutor-start=16,tutor-end=17}{c}\htmlData{tutor-start=17,tutor-end=18}{)}^{\htmlData{tutor-start=20,tutor-end=21}{2}}}\htmlData{tutor-start=23,tutor-end=24}{+}\frac{\htmlData{tutor-start=30,tutor-end=31}{1}}{\htmlData{tutor-start=33,tutor-end=34}{(}\htmlData{tutor-start=34,tutor-end=35}{2} \htmlData{tutor-start=36,tutor-end=37}{b}\htmlData{tutor-start=37,tutor-end=38}{+}\htmlData{tutor-start=38,tutor-end=39}{c}\htmlData{tutor-start=39,tutor-end=40}{+}\htmlData{tutor-start=40,tutor-end=41}{a}\htmlData{tutor-start=41,tutor-end=42}{)}^{\htmlData{tutor-start=44,tutor-end=45}{2}}}\htmlData{tutor-start=47,tutor-end=48}{+}\frac{\htmlData{tutor-start=54,tutor-end=55}{1}}{\htmlData{tutor-start=57,tutor-end=58}{(}\htmlData{tutor-start=58,tutor-end=59}{2} \htmlData{tutor-start=60,tutor-end=61}{c}\htmlData{tutor-start=61,tutor-end=62}{+}\htmlData{tutor-start=62,tutor-end=63}{a}\htmlData{tutor-start=63,tutor-end=64}{+}\htmlData{tutor-start=64,tutor-end=65}{b}\htmlData{tutor-start=65,tutor-end=66}{)}^{\htmlData{tutor-start=68,tutor-end=69}{2}}} \htmlData{tutor-start=72,tutor-end=77}{\leq }\frac{\htmlData{tutor-start=83,tutor-end=84}{3}}{\htmlData{tutor-start=86,tutor-end=87}{1}\htmlData{tutor-start=87,tutor-end=88}{6}} ( 2 a + b + c ) 2 1 + ( 2 b + c + a ) 2 1 + ( 2 c + a + b ) 2 1 ≤ 1 6 3 题解状态： 标准答案与规范题解待补充
题目标签：2009 IMO Shortlist A2
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2009-A3 2009 · Algebra · IMO/代数
Determine all functions f \htmlData{tutor-start=0,tutor-end=1}{f} f x \htmlData{tutor-start=0,tutor-end=1}{x} x y \htmlData{tutor-start=0,tutor-end=1}{y} y x , f ( y ) and f ( y + f ( x ) − 1 ) \htmlData{tutor-start=0,tutor-end=1}{x}\htmlData{tutor-start=1,tutor-end=2}{,} \quad \htmlData{tutor-start=9,tutor-end=10}{f}\htmlData{tutor-start=10,tutor-end=11}{(}\htmlData{tutor-start=11,tutor-end=12}{y}\htmlData{tutor-start=12,tutor-end=13}{)} \quad \text { \htmlData{tutor-start=28,tutor-end=29}{a}\htmlData{tutor-start=29,tutor-end=30}{n}\htmlData{tutor-start=30,tutor-end=31}{d} } \quad \htmlData{tutor-start=40,tutor-end=41}{f}\htmlData{tutor-start=41,tutor-end=42}{(}\htmlData{tutor-start=42,tutor-end=43}{y}\htmlData{tutor-start=43,tutor-end=44}{+}\htmlData{tutor-start=44,tutor-end=45}{f}\htmlData{tutor-start=45,tutor-end=46}{(}\htmlData{tutor-start=46,tutor-end=47}{x}\htmlData{tutor-start=47,tutor-end=48}{)}\htmlData{tutor-start=48,tutor-end=49}{-}\htmlData{tutor-start=49,tutor-end=50}{1}\htmlData{tutor-start=50,tutor-end=51}{)} x , f ( y ) a n d f ( y + f ( x ) − 1 ) 题解状态： 标准答案与规范题解待补充
题目标签：2009 IMO 正式题 A3
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2009-A4 2009 · Algebra · IMO/代数
Let a , b , c \htmlData{tutor-start=0,tutor-end=1}{a}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{b}\htmlData{tutor-start=4,tutor-end=5}{,} \htmlData{tutor-start=6,tutor-end=7}{c} a , b , c a b + b c + c a ≤ 3 a b c \htmlData{tutor-start=0,tutor-end=1}{a} \htmlData{tutor-start=2,tutor-end=3}{b}\htmlData{tutor-start=3,tutor-end=4}{+}\htmlData{tutor-start=4,tutor-end=5}{b} \htmlData{tutor-start=6,tutor-end=7}{c}\htmlData{tutor-start=7,tutor-end=8}{+}\htmlData{tutor-start=8,tutor-end=9}{c} \htmlData{tutor-start=10,tutor-end=11}{a} \htmlData{tutor-start=12,tutor-end=17}{\leq }\htmlData{tutor-start=17,tutor-end=18}{3} \htmlData{tutor-start=19,tutor-end=20}{a} \htmlData{tutor-start=21,tutor-end=22}{b} \htmlData{tutor-start=23,tutor-end=24}{c} a b + b c + c a ≤ 3 a b c a 2 + b 2 a + b + b 2 + c 2 b + c + c 2 + a 2 c + a + 3 ≤ 2 ( a + b + b + c + c + a ) . \sqrt{\frac{\htmlData{tutor-start=12,tutor-end=13}{a}^{\htmlData{tutor-start=15,tutor-end=16}{2}}\htmlData{tutor-start=17,tutor-end=18}{+}\htmlData{tutor-start=18,tutor-end=19}{b}^{\htmlData{tutor-start=21,tutor-end=22}{2}}}{\htmlData{tutor-start=25,tutor-end=26}{a}\htmlData{tutor-start=26,tutor-end=27}{+}\htmlData{tutor-start=27,tutor-end=28}{b}}}\htmlData{tutor-start=30,tutor-end=31}{+}\sqrt{\frac{\htmlData{tutor-start=43,tutor-end=44}{b}^{\htmlData{tutor-start=46,tutor-end=47}{2}}\htmlData{tutor-start=48,tutor-end=49}{+}\htmlData{tutor-start=49,tutor-end=50}{c}^{\htmlData{tutor-start=52,tutor-end=53}{2}}}{\htmlData{tutor-start=56,tutor-end=57}{b}\htmlData{tutor-start=57,tutor-end=58}{+}\htmlData{tutor-start=58,tutor-end=59}{c}}}\htmlData{tutor-start=61,tutor-end=62}{+}\sqrt{\frac{\htmlData{tutor-start=74,tutor-end=75}{c}^{\htmlData{tutor-start=77,tutor-end=78}{2}}\htmlData{tutor-start=79,tutor-end=80}{+}\htmlData{tutor-start=80,tutor-end=81}{a}^{\htmlData{tutor-start=83,tutor-end=84}{2}}}{\htmlData{tutor-start=87,tutor-end=88}{c}\htmlData{tutor-start=88,tutor-end=89}{+}\htmlData{tutor-start=89,tutor-end=90}{a}}}\htmlData{tutor-start=92,tutor-end=93}{+}\htmlData{tutor-start=93,tutor-end=94}{3} \htmlData{tutor-start=95,tutor-end=100}{\leq }\sqrt{\htmlData{tutor-start=106,tutor-end=107}{2}}\htmlData{tutor-start=108,tutor-end=109}{(}\sqrt{\htmlData{tutor-start=115,tutor-end=116}{a}\htmlData{tutor-start=116,tutor-end=117}{+}\htmlData{tutor-start=117,tutor-end=118}{b}}\htmlData{tutor-start=119,tutor-end=120}{+}\sqrt{\htmlData{tutor-start=126,tutor-end=127}{b}\htmlData{tutor-start=127,tutor-end=128}{+}\htmlData{tutor-start=128,tutor-end=129}{c}}\htmlData{tutor-start=130,tutor-end=131}{+}\sqrt{\htmlData{tutor-start=137,tutor-end=138}{c}\htmlData{tutor-start=138,tutor-end=139}{+}\htmlData{tutor-start=139,tutor-end=140}{a}}\htmlData{tutor-start=141,tutor-end=142}{)} \htmlData{tutor-start=143,tutor-end=144}{.} a + b a 2 + b 2 + b + c b 2 + c 2 + c + a c 2 + a 2 + 3 ≤ 2 ( a + b + b + c + c + a ) . 题解状态： 标准答案与规范题解待补充
题目标签：2009 IMO Shortlist A4
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2009-A5 2009 · Algebra · IMO/代数
Let f \htmlData{tutor-start=0,tutor-end=1}{f} f x \htmlData{tutor-start=0,tutor-end=1}{x} x y \htmlData{tutor-start=0,tutor-end=1}{y} y f ( x − f ( y ) ) > y f ( x ) + x \htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{x}\htmlData{tutor-start=3,tutor-end=4}{-}\htmlData{tutor-start=4,tutor-end=5}{f}\htmlData{tutor-start=5,tutor-end=6}{(}\htmlData{tutor-start=6,tutor-end=7}{y}\htmlData{tutor-start=7,tutor-end=8}{)}\htmlData{tutor-start=8,tutor-end=9}{)}\htmlData{tutor-start=9,tutor-end=10}{>}\htmlData{tutor-start=10,tutor-end=11}{y} \htmlData{tutor-start=12,tutor-end=13}{f}\htmlData{tutor-start=13,tutor-end=14}{(}\htmlData{tutor-start=14,tutor-end=15}{x}\htmlData{tutor-start=15,tutor-end=16}{)}\htmlData{tutor-start=16,tutor-end=17}{+}\htmlData{tutor-start=17,tutor-end=18}{x} f ( x − f ( y ) ) > y f ( x ) + x 题解状态： 标准答案与规范题解待补充
题目标签：2009 IMO Shortlist A5
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2009-A6 2009 · Algebra · IMO/代数
Suppose that s 1 , s 2 , s 3 , … \htmlData{tutor-start=0,tutor-end=1}{s}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{s}_{\htmlData{tutor-start=10,tutor-end=11}{2}}\htmlData{tutor-start=12,tutor-end=13}{,} \htmlData{tutor-start=14,tutor-end=15}{s}_{\htmlData{tutor-start=17,tutor-end=18}{3}}\htmlData{tutor-start=19,tutor-end=20}{,} \htmlData{tutor-start=21,tutor-end=27}{\ldots} s 1 , s 2 , s 3 , … s s 1 , s s 2 , s s 3 , … and s s 1 + 1 , s s 2 + 1 , s s 3 + 1 , … \htmlData{tutor-start=0,tutor-end=1}{s}_{\htmlData{tutor-start=3,tutor-end=4}{s}_{\htmlData{tutor-start=6,tutor-end=7}{1}}}\htmlData{tutor-start=9,tutor-end=10}{,} \htmlData{tutor-start=11,tutor-end=12}{s}_{\htmlData{tutor-start=14,tutor-end=15}{s}_{\htmlData{tutor-start=17,tutor-end=18}{2}}}\htmlData{tutor-start=20,tutor-end=21}{,} \htmlData{tutor-start=22,tutor-end=23}{s}_{\htmlData{tutor-start=25,tutor-end=26}{s}_{\htmlData{tutor-start=28,tutor-end=29}{3}}}\htmlData{tutor-start=31,tutor-end=32}{,} \htmlData{tutor-start=33,tutor-end=40}{\ldots }\quad \text { \htmlData{tutor-start=54,tutor-end=55}{a}\htmlData{tutor-start=55,tutor-end=56}{n}\htmlData{tutor-start=56,tutor-end=57}{d} } \quad \htmlData{tutor-start=66,tutor-end=67}{s}_{\htmlData{tutor-start=69,tutor-end=70}{s}_{\htmlData{tutor-start=72,tutor-end=73}{1}}\htmlData{tutor-start=74,tutor-end=75}{+}\htmlData{tutor-start=75,tutor-end=76}{1}}\htmlData{tutor-start=77,tutor-end=78}{,} \htmlData{tutor-start=79,tutor-end=80}{s}_{\htmlData{tutor-start=82,tutor-end=83}{s}_{\htmlData{tutor-start=85,tutor-end=86}{2}}\htmlData{tutor-start=87,tutor-end=88}{+}\htmlData{tutor-start=88,tutor-end=89}{1}}\htmlData{tutor-start=90,tutor-end=91}{,} \htmlData{tutor-start=92,tutor-end=93}{s}_{\htmlData{tutor-start=95,tutor-end=96}{s}_{\htmlData{tutor-start=98,tutor-end=99}{3}}\htmlData{tutor-start=100,tutor-end=101}{+}\htmlData{tutor-start=101,tutor-end=102}{1}}\htmlData{tutor-start=103,tutor-end=104}{,} \htmlData{tutor-start=105,tutor-end=111}{\ldots} s s 1 , s s 2 , s s 3 , … a n d s s 1 + 1 , s s 2 + 1 , s s 3 + 1 , … s 1 , s 2 , s 3 , … \htmlData{tutor-start=0,tutor-end=1}{s}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{s}_{\htmlData{tutor-start=10,tutor-end=11}{2}}\htmlData{tutor-start=12,tutor-end=13}{,} \htmlData{tutor-start=14,tutor-end=15}{s}_{\htmlData{tutor-start=17,tutor-end=18}{3}}\htmlData{tutor-start=19,tutor-end=20}{,} \htmlData{tutor-start=21,tutor-end=27}{\ldots} s 1 , s 2 , s 3 , … 题解状态： 标准答案与规范题解待补充
题目标签：2009 IMO 正式题 A6
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2009-A7 2009 · Algebra · IMO/代数
Find all functions f \htmlData{tutor-start=0,tutor-end=1}{f} f x , y \htmlData{tutor-start=0,tutor-end=1}{x}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{y} x , y f ( x f ( x + y ) ) = f ( y f ( x ) ) + x 2 \htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{x} \htmlData{tutor-start=4,tutor-end=5}{f}\htmlData{tutor-start=5,tutor-end=6}{(}\htmlData{tutor-start=6,tutor-end=7}{x}\htmlData{tutor-start=7,tutor-end=8}{+}\htmlData{tutor-start=8,tutor-end=9}{y}\htmlData{tutor-start=9,tutor-end=10}{)}\htmlData{tutor-start=10,tutor-end=11}{)}\htmlData{tutor-start=11,tutor-end=12}{=}\htmlData{tutor-start=12,tutor-end=13}{f}\htmlData{tutor-start=13,tutor-end=14}{(}\htmlData{tutor-start=14,tutor-end=15}{y} \htmlData{tutor-start=16,tutor-end=17}{f}\htmlData{tutor-start=17,tutor-end=18}{(}\htmlData{tutor-start=18,tutor-end=19}{x}\htmlData{tutor-start=19,tutor-end=20}{)}\htmlData{tutor-start=20,tutor-end=21}{)}\htmlData{tutor-start=21,tutor-end=22}{+}\htmlData{tutor-start=22,tutor-end=23}{x}^{\htmlData{tutor-start=25,tutor-end=26}{2}} f ( x f ( x + y ) ) = f ( y f ( x ) ) + x 2 题解状态： 标准答案与规范题解待补充
题目标签：2009 IMO Shortlist A7
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2010-A1 2010 · Algebra · IMO/代数
Determine all functions f : R → R \htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{:} \mathbb{\htmlData{tutor-start=11,tutor-end=12}{R}} \htmlData{tutor-start=14,tutor-end=26}{\rightarrow }\mathbb{\htmlData{tutor-start=34,tutor-end=35}{R}} f : R → R f ( [ x ] y ) = f ( x ) [ f ( y ) ] . \htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{[}\htmlData{tutor-start=3,tutor-end=4}{x}\htmlData{tutor-start=4,tutor-end=5}{]} \htmlData{tutor-start=6,tutor-end=7}{y}\htmlData{tutor-start=7,tutor-end=8}{)}\htmlData{tutor-start=8,tutor-end=9}{=}\htmlData{tutor-start=9,tutor-end=10}{f}\htmlData{tutor-start=10,tutor-end=11}{(}\htmlData{tutor-start=11,tutor-end=12}{x}\htmlData{tutor-start=12,tutor-end=13}{)}\htmlData{tutor-start=13,tutor-end=14}{[}\htmlData{tutor-start=14,tutor-end=15}{f}\htmlData{tutor-start=15,tutor-end=16}{(}\htmlData{tutor-start=16,tutor-end=17}{y}\htmlData{tutor-start=17,tutor-end=18}{)}\htmlData{tutor-start=18,tutor-end=19}{]} \htmlData{tutor-start=20,tutor-end=21}{.} f ( [ x ] y ) = f ( x ) [ f ( y ) ] . x , y ∈ R \htmlData{tutor-start=0,tutor-end=1}{x}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{y} \htmlData{tutor-start=5,tutor-end=9}{\in }\mathbb{\htmlData{tutor-start=17,tutor-end=18}{R}} x , y ∈ R [ x ] \htmlData{tutor-start=0,tutor-end=1}{[}\htmlData{tutor-start=1,tutor-end=2}{x}\htmlData{tutor-start=2,tutor-end=3}{]} [ x ] x \htmlData{tutor-start=0,tutor-end=1}{x} x f ( x ) = \htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{x}\htmlData{tutor-start=3,tutor-end=4}{)}\htmlData{tutor-start=4,tutor-end=5}{=} f ( x ) = = C \htmlData{tutor-start=0,tutor-end=1}{=}\htmlData{tutor-start=1,tutor-end=2}{C} = C C = 0 \htmlData{tutor-start=0,tutor-end=1}{C}\htmlData{tutor-start=1,tutor-end=2}{=}\htmlData{tutor-start=2,tutor-end=3}{0} C = 0 1 ≤ C < 2 \htmlData{tutor-start=0,tutor-end=1}{1} \htmlData{tutor-start=2,tutor-end=7}{\leq }\htmlData{tutor-start=7,tutor-end=8}{C}\htmlData{tutor-start=8,tutor-end=9}{<}\htmlData{tutor-start=9,tutor-end=10}{2} 1 ≤ C < 2 题解状态： 标准答案与规范题解待补充
题目标签：2010 IMO 正式题 A1
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2010-A2 2010 · Algebra · IMO/代数
Let the real numbers a , b , c , d \htmlData{tutor-start=0,tutor-end=1}{a}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{b}\htmlData{tutor-start=4,tutor-end=5}{,} \htmlData{tutor-start=6,tutor-end=7}{c}\htmlData{tutor-start=7,tutor-end=8}{,} \htmlData{tutor-start=9,tutor-end=10}{d} a , b , c , d a + b + c + d = 6 \htmlData{tutor-start=0,tutor-end=1}{a}\htmlData{tutor-start=1,tutor-end=2}{+}\htmlData{tutor-start=2,tutor-end=3}{b}\htmlData{tutor-start=3,tutor-end=4}{+}\htmlData{tutor-start=4,tutor-end=5}{c}\htmlData{tutor-start=5,tutor-end=6}{+}\htmlData{tutor-start=6,tutor-end=7}{d}\htmlData{tutor-start=7,tutor-end=8}{=}\htmlData{tutor-start=8,tutor-end=9}{6} a + b + c + d = 6 a 2 + b 2 + c 2 + d 2 = 12 \htmlData{tutor-start=0,tutor-end=1}{a}^{\htmlData{tutor-start=3,tutor-end=4}{2}}\htmlData{tutor-start=5,tutor-end=6}{+}\htmlData{tutor-start=6,tutor-end=7}{b}^{\htmlData{tutor-start=9,tutor-end=10}{2}}\htmlData{tutor-start=11,tutor-end=12}{+}\htmlData{tutor-start=12,tutor-end=13}{c}^{\htmlData{tutor-start=15,tutor-end=16}{2}}\htmlData{tutor-start=17,tutor-end=18}{+}\htmlData{tutor-start=18,tutor-end=19}{d}^{\htmlData{tutor-start=21,tutor-end=22}{2}}\htmlData{tutor-start=23,tutor-end=24}{=}\htmlData{tutor-start=24,tutor-end=25}{1}\htmlData{tutor-start=25,tutor-end=26}{2} a 2 + b 2 + c 2 + d 2 = 1 2 36 ≤ 4 ( a 3 + b 3 + c 3 + d 3 ) − ( a 4 + b 4 + c 4 + d 4 ) ≤ 48 \htmlData{tutor-start=0,tutor-end=1}{3}\htmlData{tutor-start=1,tutor-end=2}{6} \htmlData{tutor-start=3,tutor-end=8}{\leq }\htmlData{tutor-start=8,tutor-end=9}{4}\left(\htmlData{tutor-start=15,tutor-end=16}{a}^{\htmlData{tutor-start=18,tutor-end=19}{3}}\htmlData{tutor-start=20,tutor-end=21}{+}\htmlData{tutor-start=21,tutor-end=22}{b}^{\htmlData{tutor-start=24,tutor-end=25}{3}}\htmlData{tutor-start=26,tutor-end=27}{+}\htmlData{tutor-start=27,tutor-end=28}{c}^{\htmlData{tutor-start=30,tutor-end=31}{3}}\htmlData{tutor-start=32,tutor-end=33}{+}\htmlData{tutor-start=33,tutor-end=34}{d}^{\htmlData{tutor-start=36,tutor-end=37}{3}}\right)\htmlData{tutor-start=45,tutor-end=46}{-}\left(\htmlData{tutor-start=52,tutor-end=53}{a}^{\htmlData{tutor-start=55,tutor-end=56}{4}}\htmlData{tutor-start=57,tutor-end=58}{+}\htmlData{tutor-start=58,tutor-end=59}{b}^{\htmlData{tutor-start=61,tutor-end=62}{4}}\htmlData{tutor-start=63,tutor-end=64}{+}\htmlData{tutor-start=64,tutor-end=65}{c}^{\htmlData{tutor-start=67,tutor-end=68}{4}}\htmlData{tutor-start=69,tutor-end=70}{+}\htmlData{tutor-start=70,tutor-end=71}{d}^{\htmlData{tutor-start=73,tutor-end=74}{4}}\right) \htmlData{tutor-start=83,tutor-end=88}{\leq }\htmlData{tutor-start=88,tutor-end=89}{4}\htmlData{tutor-start=89,tutor-end=90}{8} 3 6 ≤ 4 ( a 3 + b 3 + c 3 + d 3 ) − ( a 4 + b 4 + c 4 + d 4 ) ≤ 4 8 题解状态： 标准答案与规范题解待补充
题目标签：2010 IMO Shortlist A2
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2010-A3 2010 · Algebra · IMO/代数
Let x 1 , … , x 100 \htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=13}{\ldots}\htmlData{tutor-start=13,tutor-end=14}{,} \htmlData{tutor-start=15,tutor-end=16}{x}_{\htmlData{tutor-start=18,tutor-end=19}{1}\htmlData{tutor-start=19,tutor-end=20}{0}\htmlData{tutor-start=20,tutor-end=21}{0}} x 1 , … , x 1 0 0 x i + x i + 1 + x i + 2 ≤ 1 \htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{i}}\htmlData{tutor-start=5,tutor-end=6}{+}\htmlData{tutor-start=6,tutor-end=7}{x}_{\htmlData{tutor-start=9,tutor-end=10}{i}\htmlData{tutor-start=10,tutor-end=11}{+}\htmlData{tutor-start=11,tutor-end=12}{1}}\htmlData{tutor-start=13,tutor-end=14}{+}\htmlData{tutor-start=14,tutor-end=15}{x}_{\htmlData{tutor-start=17,tutor-end=18}{i}\htmlData{tutor-start=18,tutor-end=19}{+}\htmlData{tutor-start=19,tutor-end=20}{2}} \htmlData{tutor-start=22,tutor-end=27}{\leq }\htmlData{tutor-start=27,tutor-end=28}{1} x i + x i + 1 + x i + 2 ≤ 1 i = 1 , … , 100 \htmlData{tutor-start=0,tutor-end=1}{i}\htmlData{tutor-start=1,tutor-end=2}{=}\htmlData{tutor-start=2,tutor-end=3}{1}\htmlData{tutor-start=3,tutor-end=4}{,} \htmlData{tutor-start=5,tutor-end=11}{\ldots}\htmlData{tutor-start=11,tutor-end=12}{,} \htmlData{tutor-start=13,tutor-end=14}{1}\htmlData{tutor-start=14,tutor-end=15}{0}\htmlData{tutor-start=15,tutor-end=16}{0} i = 1 , … , 1 0 0 x 101 = x 1 , x 102 = x 2 \htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{1}\htmlData{tutor-start=4,tutor-end=5}{0}\htmlData{tutor-start=5,tutor-end=6}{1}}\htmlData{tutor-start=7,tutor-end=8}{=}\htmlData{tutor-start=8,tutor-end=9}{x}_{\htmlData{tutor-start=11,tutor-end=12}{1}}\htmlData{tutor-start=13,tutor-end=14}{,} \htmlData{tutor-start=15,tutor-end=16}{x}_{\htmlData{tutor-start=18,tutor-end=19}{1}\htmlData{tutor-start=19,tutor-end=20}{0}\htmlData{tutor-start=20,tutor-end=21}{2}}\htmlData{tutor-start=22,tutor-end=23}{=}\htmlData{tutor-start=23,tutor-end=24}{x}_{\htmlData{tutor-start=26,tutor-end=27}{2}} x 1 0 1 = x 1 , x 1 0 2 = x 2 S = ∑ i = 1 100 x i x i + 2 \htmlData{tutor-start=0,tutor-end=1}{S}\htmlData{tutor-start=1,tutor-end=2}{=}\sum_{\htmlData{tutor-start=8,tutor-end=9}{i}\htmlData{tutor-start=9,tutor-end=10}{=}\htmlData{tutor-start=10,tutor-end=11}{1}}^{\htmlData{tutor-start=14,tutor-end=15}{1}\htmlData{tutor-start=15,tutor-end=16}{0}\htmlData{tutor-start=16,tutor-end=17}{0}} \htmlData{tutor-start=19,tutor-end=20}{x}_{\htmlData{tutor-start=22,tutor-end=23}{i}} \htmlData{tutor-start=25,tutor-end=26}{x}_{\htmlData{tutor-start=28,tutor-end=29}{i}\htmlData{tutor-start=29,tutor-end=30}{+}\htmlData{tutor-start=30,tutor-end=31}{2}} S = ∑ i = 1 1 0 0 x i x i + 2 25 2 \frac{\htmlData{tutor-start=6,tutor-end=7}{2}\htmlData{tutor-start=7,tutor-end=8}{5}}{\htmlData{tutor-start=10,tutor-end=11}{2}} 2 2 5 题解状态： 标准答案与规范题解待补充
题目标签：2010 IMO Shortlist A3
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2010-A4 2010 · Algebra · IMO/代数
A sequence x 1 , x 2 , … \htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{x}_{\htmlData{tutor-start=10,tutor-end=11}{2}}\htmlData{tutor-start=12,tutor-end=13}{,} \htmlData{tutor-start=14,tutor-end=20}{\ldots} x 1 , x 2 , … x 1 = 1 \htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{=}\htmlData{tutor-start=6,tutor-end=7}{1} x 1 = 1 x 2 k = − x k , x 2 k − 1 = ( − 1 ) k + 1 x k \htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{2} \htmlData{tutor-start=5,tutor-end=6}{k}}\htmlData{tutor-start=7,tutor-end=8}{=}\htmlData{tutor-start=8,tutor-end=9}{-}\htmlData{tutor-start=9,tutor-end=10}{x}_{\htmlData{tutor-start=12,tutor-end=13}{k}}\htmlData{tutor-start=14,tutor-end=15}{,} \htmlData{tutor-start=16,tutor-end=17}{x}_{\htmlData{tutor-start=19,tutor-end=20}{2} \htmlData{tutor-start=21,tutor-end=22}{k}\htmlData{tutor-start=22,tutor-end=23}{-}\htmlData{tutor-start=23,tutor-end=24}{1}}\htmlData{tutor-start=25,tutor-end=26}{=}\htmlData{tutor-start=26,tutor-end=27}{(}\htmlData{tutor-start=27,tutor-end=28}{-}\htmlData{tutor-start=28,tutor-end=29}{1}\htmlData{tutor-start=29,tutor-end=30}{)}^{\htmlData{tutor-start=32,tutor-end=33}{k}\htmlData{tutor-start=33,tutor-end=34}{+}\htmlData{tutor-start=34,tutor-end=35}{1}} \htmlData{tutor-start=37,tutor-end=38}{x}_{\htmlData{tutor-start=40,tutor-end=41}{k}} x 2 k = − x k , x 2 k − 1 = ( − 1 ) k + 1 x k k ≥ 1 \htmlData{tutor-start=0,tutor-end=1}{k} \htmlData{tutor-start=2,tutor-end=7}{\geq }\htmlData{tutor-start=7,tutor-end=8}{1} k ≥ 1 x 1 + x 2 + ⋯ + x n ≥ 0 \htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{+}\htmlData{tutor-start=6,tutor-end=7}{x}_{\htmlData{tutor-start=9,tutor-end=10}{2}}\htmlData{tutor-start=11,tutor-end=12}{+}\cdots\htmlData{tutor-start=18,tutor-end=19}{+}\htmlData{tutor-start=19,tutor-end=20}{x}_{\htmlData{tutor-start=22,tutor-end=23}{n}} \htmlData{tutor-start=25,tutor-end=30}{\geq }\htmlData{tutor-start=30,tutor-end=31}{0} x 1 + x 2 + ⋯ + x n ≥ 0 n ≥ 1 \htmlData{tutor-start=0,tutor-end=1}{n} \htmlData{tutor-start=2,tutor-end=7}{\geq }\htmlData{tutor-start=7,tutor-end=8}{1} n ≥ 1 题解状态： 标准答案与规范题解待补充
题目标签：2010 IMO Shortlist A4
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2010-A5 2010 · Algebra · IMO/代数
Denote by Q + \mathbb{\htmlData{tutor-start=8,tutor-end=9}{Q}}^{\htmlData{tutor-start=12,tutor-end=13}{+}} Q + f : Q + → Q + \htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{:} \mathbb{\htmlData{tutor-start=11,tutor-end=12}{Q}}^{\htmlData{tutor-start=15,tutor-end=16}{+}} \htmlData{tutor-start=18,tutor-end=30}{\rightarrow }\mathbb{\htmlData{tutor-start=38,tutor-end=39}{Q}}^{\htmlData{tutor-start=42,tutor-end=43}{+}} f : Q + → Q + x , y ∈ Q + \htmlData{tutor-start=0,tutor-end=1}{x}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{y} \htmlData{tutor-start=5,tutor-end=9}{\in }\mathbb{\htmlData{tutor-start=17,tutor-end=18}{Q}}^{\htmlData{tutor-start=21,tutor-end=22}{+}} x , y ∈ Q + f ( f ( x ) 2 y ) = x 3 f ( x y ) \htmlData{tutor-start=0,tutor-end=1}{f}\left(\htmlData{tutor-start=7,tutor-end=8}{f}\htmlData{tutor-start=8,tutor-end=9}{(}\htmlData{tutor-start=9,tutor-end=10}{x}\htmlData{tutor-start=10,tutor-end=11}{)}^{\htmlData{tutor-start=13,tutor-end=14}{2}} \htmlData{tutor-start=16,tutor-end=17}{y}\right)\htmlData{tutor-start=24,tutor-end=25}{=}\htmlData{tutor-start=25,tutor-end=26}{x}^{\htmlData{tutor-start=28,tutor-end=29}{3}} \htmlData{tutor-start=31,tutor-end=32}{f}\htmlData{tutor-start=32,tutor-end=33}{(}\htmlData{tutor-start=33,tutor-end=34}{x} \htmlData{tutor-start=35,tutor-end=36}{y}\htmlData{tutor-start=36,tutor-end=37}{)} f ( f ( x ) 2 y ) = x 3 f ( x y ) f ( x ) = 1 x \htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{x}\htmlData{tutor-start=3,tutor-end=4}{)}\htmlData{tutor-start=4,tutor-end=5}{=}\frac{\htmlData{tutor-start=11,tutor-end=12}{1}}{\htmlData{tutor-start=14,tutor-end=15}{x}} f ( x ) = x 1 题解状态： 标准答案与规范题解待补充
题目标签：2010 IMO Shortlist A5
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2010-A6 2010 · Algebra · IMO/代数
Suppose that f \htmlData{tutor-start=0,tutor-end=1}{f} f g \htmlData{tutor-start=0,tutor-end=1}{g} g f ( g ( n ) ) = f ( n ) + 1 \htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{g}\htmlData{tutor-start=3,tutor-end=4}{(}\htmlData{tutor-start=4,tutor-end=5}{n}\htmlData{tutor-start=5,tutor-end=6}{)}\htmlData{tutor-start=6,tutor-end=7}{)}\htmlData{tutor-start=7,tutor-end=8}{=}\htmlData{tutor-start=8,tutor-end=9}{f}\htmlData{tutor-start=9,tutor-end=10}{(}\htmlData{tutor-start=10,tutor-end=11}{n}\htmlData{tutor-start=11,tutor-end=12}{)}\htmlData{tutor-start=12,tutor-end=13}{+}\htmlData{tutor-start=13,tutor-end=14}{1} f ( g ( n ) ) = f ( n ) + 1 g ( f ( n ) ) = \htmlData{tutor-start=0,tutor-end=1}{g}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{f}\htmlData{tutor-start=3,tutor-end=4}{(}\htmlData{tutor-start=4,tutor-end=5}{n}\htmlData{tutor-start=5,tutor-end=6}{)}\htmlData{tutor-start=6,tutor-end=7}{)}\htmlData{tutor-start=7,tutor-end=8}{=} g ( f ( n ) ) = g ( n ) + 1 \htmlData{tutor-start=0,tutor-end=1}{g}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{n}\htmlData{tutor-start=3,tutor-end=4}{)}\htmlData{tutor-start=4,tutor-end=5}{+}\htmlData{tutor-start=5,tutor-end=6}{1} g ( n ) + 1 f ( n ) = g ( n ) \htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{n}\htmlData{tutor-start=3,tutor-end=4}{)}\htmlData{tutor-start=4,tutor-end=5}{=}\htmlData{tutor-start=5,tutor-end=6}{g}\htmlData{tutor-start=6,tutor-end=7}{(}\htmlData{tutor-start=7,tutor-end=8}{n}\htmlData{tutor-start=8,tutor-end=9}{)} f ( n ) = g ( n ) n \htmlData{tutor-start=0,tutor-end=1}{n} n 题解状态： 标准答案与规范题解待补充
题目标签：2010 IMO Shortlist A6
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2010-A7 2010 · Algebra · IMO/代数
Let a 1 , … , a r \htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=13}{\ldots}\htmlData{tutor-start=13,tutor-end=14}{,} \htmlData{tutor-start=15,tutor-end=16}{a}_{\htmlData{tutor-start=18,tutor-end=19}{r}} a 1 , … , a r n > r \htmlData{tutor-start=0,tutor-end=1}{n}\htmlData{tutor-start=1,tutor-end=2}{>}\htmlData{tutor-start=2,tutor-end=3}{r} n > r a n = max 1 ≤ k ≤ n − 1 ( a k + a n − k ) \htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{n}}\htmlData{tutor-start=5,tutor-end=6}{=}\max _{\htmlData{tutor-start=13,tutor-end=14}{1} \htmlData{tutor-start=15,tutor-end=20}{\leq }\htmlData{tutor-start=20,tutor-end=21}{k} \htmlData{tutor-start=22,tutor-end=27}{\leq }\htmlData{tutor-start=27,tutor-end=28}{n}\htmlData{tutor-start=28,tutor-end=29}{-}\htmlData{tutor-start=29,tutor-end=30}{1}}\left(\htmlData{tutor-start=37,tutor-end=38}{a}_{\htmlData{tutor-start=40,tutor-end=41}{k}}\htmlData{tutor-start=42,tutor-end=43}{+}\htmlData{tutor-start=43,tutor-end=44}{a}_{\htmlData{tutor-start=46,tutor-end=47}{n}\htmlData{tutor-start=47,tutor-end=48}{-}\htmlData{tutor-start=48,tutor-end=49}{k}}\right) a n = max 1 ≤ k ≤ n − 1 ( a k + a n − k ) ℓ ≤ r \htmlData{tutor-start=0,tutor-end=5}{\ell }\htmlData{tutor-start=5,tutor-end=10}{\leq }\htmlData{tutor-start=10,tutor-end=11}{r} ℓ ≤ r N \htmlData{tutor-start=0,tutor-end=1}{N} N a n = a n − ℓ + a ℓ \htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{n}}\htmlData{tutor-start=5,tutor-end=6}{=}\htmlData{tutor-start=6,tutor-end=7}{a}_{\htmlData{tutor-start=9,tutor-end=10}{n}\htmlData{tutor-start=10,tutor-end=11}{-}\htmlData{tutor-start=11,tutor-end=15}{\ell}}\htmlData{tutor-start=16,tutor-end=17}{+}\htmlData{tutor-start=17,tutor-end=18}{a}_{\htmlData{tutor-start=20,tutor-end=24}{\ell}} a n = a n − ℓ + a ℓ n ≥ N \htmlData{tutor-start=0,tutor-end=1}{n} \htmlData{tutor-start=2,tutor-end=7}{\geq }\htmlData{tutor-start=7,tutor-end=8}{N} n ≥ N 题解状态： 标准答案与规范题解待补充
题目标签：2010 IMO 正式题 A7
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2010-A8 2010 · Algebra · IMO/代数
Given six positive numbers a , b , c , d , e , f \htmlData{tutor-start=0,tutor-end=1}{a}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{b}\htmlData{tutor-start=4,tutor-end=5}{,} \htmlData{tutor-start=6,tutor-end=7}{c}\htmlData{tutor-start=7,tutor-end=8}{,} \htmlData{tutor-start=9,tutor-end=10}{d}\htmlData{tutor-start=10,tutor-end=11}{,} \htmlData{tutor-start=12,tutor-end=13}{e}\htmlData{tutor-start=13,tutor-end=14}{,} \htmlData{tutor-start=15,tutor-end=16}{f} a , b , c , d , e , f a < b < c < d < e < f \htmlData{tutor-start=0,tutor-end=1}{a}\htmlData{tutor-start=1,tutor-end=2}{<}\htmlData{tutor-start=2,tutor-end=3}{b}\htmlData{tutor-start=3,tutor-end=4}{<}\htmlData{tutor-start=4,tutor-end=5}{c}\htmlData{tutor-start=5,tutor-end=6}{<}\htmlData{tutor-start=6,tutor-end=7}{d}\htmlData{tutor-start=7,tutor-end=8}{<}\htmlData{tutor-start=8,tutor-end=9}{e}\htmlData{tutor-start=9,tutor-end=10}{<}\htmlData{tutor-start=10,tutor-end=11}{f} a < b < c < d < e < f a + c + e = S \htmlData{tutor-start=0,tutor-end=1}{a}\htmlData{tutor-start=1,tutor-end=2}{+}\htmlData{tutor-start=2,tutor-end=3}{c}\htmlData{tutor-start=3,tutor-end=4}{+}\htmlData{tutor-start=4,tutor-end=5}{e}\htmlData{tutor-start=5,tutor-end=6}{=}\htmlData{tutor-start=6,tutor-end=7}{S} a + c + e = S b + d + f = T \htmlData{tutor-start=0,tutor-end=1}{b}\htmlData{tutor-start=1,tutor-end=2}{+}\htmlData{tutor-start=2,tutor-end=3}{d}\htmlData{tutor-start=3,tutor-end=4}{+}\htmlData{tutor-start=4,tutor-end=5}{f}\htmlData{tutor-start=5,tutor-end=6}{=}\htmlData{tutor-start=6,tutor-end=7}{T} b + d + f = T 2 S T > 3 ( S + T ) ( S ( b d + b f + d f ) + T ( a c + a e + c e ) ) \htmlData{tutor-start=0,tutor-end=1}{2} \htmlData{tutor-start=2,tutor-end=3}{S} \htmlData{tutor-start=4,tutor-end=5}{T}\htmlData{tutor-start=5,tutor-end=6}{>}\sqrt{\htmlData{tutor-start=12,tutor-end=13}{3}\htmlData{tutor-start=13,tutor-end=14}{(}\htmlData{tutor-start=14,tutor-end=15}{S}\htmlData{tutor-start=15,tutor-end=16}{+}\htmlData{tutor-start=16,tutor-end=17}{T}\htmlData{tutor-start=17,tutor-end=18}{)}\htmlData{tutor-start=18,tutor-end=19}{(}\htmlData{tutor-start=19,tutor-end=20}{S}\htmlData{tutor-start=20,tutor-end=21}{(}\htmlData{tutor-start=21,tutor-end=22}{b} \htmlData{tutor-start=23,tutor-end=24}{d}\htmlData{tutor-start=24,tutor-end=25}{+}\htmlData{tutor-start=25,tutor-end=26}{b} \htmlData{tutor-start=27,tutor-end=28}{f}\htmlData{tutor-start=28,tutor-end=29}{+}\htmlData{tutor-start=29,tutor-end=30}{d} \htmlData{tutor-start=31,tutor-end=32}{f}\htmlData{tutor-start=32,tutor-end=33}{)}\htmlData{tutor-start=33,tutor-end=34}{+}\htmlData{tutor-start=34,tutor-end=35}{T}\htmlData{tutor-start=35,tutor-end=36}{(}\htmlData{tutor-start=36,tutor-end=37}{a} \htmlData{tutor-start=38,tutor-end=39}{c}\htmlData{tutor-start=39,tutor-end=40}{+}\htmlData{tutor-start=40,tutor-end=41}{a} \htmlData{tutor-start=42,tutor-end=43}{e}\htmlData{tutor-start=43,tutor-end=44}{+}\htmlData{tutor-start=44,tutor-end=45}{c} \htmlData{tutor-start=46,tutor-end=47}{e}\htmlData{tutor-start=47,tutor-end=48}{)}\htmlData{tutor-start=48,tutor-end=49}{)}} 2 S T > 3 ( S + T ) ( S ( b d + b f + d f ) + T ( a c + a e + c e ) ) 题解状态： 标准答案与规范题解待补充
题目标签：2010 IMO Shortlist A8
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2011-A1 2011 · Algebra · IMO/代数
For any set A = { a 1 , a 2 , a 3 , a 4 } \htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{=}\left\{\htmlData{tutor-start=9,tutor-end=10}{a}_{\htmlData{tutor-start=12,tutor-end=13}{1}}\htmlData{tutor-start=14,tutor-end=15}{,} \htmlData{tutor-start=16,tutor-end=17}{a}_{\htmlData{tutor-start=19,tutor-end=20}{2}}\htmlData{tutor-start=21,tutor-end=22}{,} \htmlData{tutor-start=23,tutor-end=24}{a}_{\htmlData{tutor-start=26,tutor-end=27}{3}}\htmlData{tutor-start=28,tutor-end=29}{,} \htmlData{tutor-start=30,tutor-end=31}{a}_{\htmlData{tutor-start=33,tutor-end=34}{4}}\right\} A = { a 1 , a 2 , a 3 , a 4 } s A = a 1 + a 2 + a 3 + a 4 \htmlData{tutor-start=0,tutor-end=1}{s}_{\htmlData{tutor-start=3,tutor-end=4}{A}}\htmlData{tutor-start=5,tutor-end=6}{=}\htmlData{tutor-start=6,tutor-end=7}{a}_{\htmlData{tutor-start=9,tutor-end=10}{1}}\htmlData{tutor-start=11,tutor-end=12}{+}\htmlData{tutor-start=12,tutor-end=13}{a}_{\htmlData{tutor-start=15,tutor-end=16}{2}}\htmlData{tutor-start=17,tutor-end=18}{+}\htmlData{tutor-start=18,tutor-end=19}{a}_{\htmlData{tutor-start=21,tutor-end=22}{3}}\htmlData{tutor-start=23,tutor-end=24}{+}\htmlData{tutor-start=24,tutor-end=25}{a}_{\htmlData{tutor-start=27,tutor-end=28}{4}} s A = a 1 + a 2 + a 3 + a 4 p A \htmlData{tutor-start=0,tutor-end=1}{p}_{\htmlData{tutor-start=3,tutor-end=4}{A}} p A ( i , j ) \htmlData{tutor-start=0,tutor-end=1}{(}\htmlData{tutor-start=1,tutor-end=2}{i}\htmlData{tutor-start=2,tutor-end=3}{,} \htmlData{tutor-start=4,tutor-end=5}{j}\htmlData{tutor-start=5,tutor-end=6}{)} ( i , j ) 1 ≤ i < j ≤ 4 \htmlData{tutor-start=0,tutor-end=1}{1} \htmlData{tutor-start=2,tutor-end=7}{\leq }\htmlData{tutor-start=7,tutor-end=8}{i}\htmlData{tutor-start=8,tutor-end=9}{<}\htmlData{tutor-start=9,tutor-end=10}{j} \htmlData{tutor-start=11,tutor-end=16}{\leq }\htmlData{tutor-start=16,tutor-end=17}{4} 1 ≤ i < j ≤ 4 a i + a j \htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{i}}\htmlData{tutor-start=5,tutor-end=6}{+}\htmlData{tutor-start=6,tutor-end=7}{a}_{\htmlData{tutor-start=9,tutor-end=10}{j}} a i + a j s A \htmlData{tutor-start=0,tutor-end=1}{s}_{\htmlData{tutor-start=3,tutor-end=4}{A}} s A A \htmlData{tutor-start=0,tutor-end=1}{A} A p A \htmlData{tutor-start=0,tutor-end=1}{p}_{\htmlData{tutor-start=3,tutor-end=4}{A}} p A 题解状态： 标准答案与规范题解待补充
题目标签：2011 IMO 正式题 A1
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2011-A2 2011 · Algebra · IMO/代数
Determine all sequences ( x 1 , x 2 , … , x 2011 ) \left(\htmlData{tutor-start=6,tutor-end=7}{x}_{\htmlData{tutor-start=9,tutor-end=10}{1}}\htmlData{tutor-start=11,tutor-end=12}{,} \htmlData{tutor-start=13,tutor-end=14}{x}_{\htmlData{tutor-start=16,tutor-end=17}{2}}\htmlData{tutor-start=18,tutor-end=19}{,} \htmlData{tutor-start=20,tutor-end=26}{\ldots}\htmlData{tutor-start=26,tutor-end=27}{,} \htmlData{tutor-start=28,tutor-end=29}{x}_{\htmlData{tutor-start=31,tutor-end=32}{2}\htmlData{tutor-start=32,tutor-end=33}{0}\htmlData{tutor-start=33,tutor-end=34}{1}\htmlData{tutor-start=34,tutor-end=35}{1}}\right) ( x 1 , x 2 , … , x 2 0 1 1 ) n \htmlData{tutor-start=0,tutor-end=1}{n} n a \htmlData{tutor-start=0,tutor-end=1}{a} a x 1 n + 2 x 2 n + ⋯ + 2011 x 2011 n = a n + 1 + 1. \htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{1}}^{\htmlData{tutor-start=7,tutor-end=8}{n}}\htmlData{tutor-start=9,tutor-end=10}{+}\htmlData{tutor-start=10,tutor-end=11}{2} \htmlData{tutor-start=12,tutor-end=13}{x}_{\htmlData{tutor-start=15,tutor-end=16}{2}}^{\htmlData{tutor-start=19,tutor-end=20}{n}}\htmlData{tutor-start=21,tutor-end=22}{+}\cdots\htmlData{tutor-start=28,tutor-end=29}{+}\htmlData{tutor-start=29,tutor-end=30}{2}\htmlData{tutor-start=30,tutor-end=31}{0}\htmlData{tutor-start=31,tutor-end=32}{1}\htmlData{tutor-start=32,tutor-end=33}{1} \htmlData{tutor-start=34,tutor-end=35}{x}_{\htmlData{tutor-start=37,tutor-end=38}{2}\htmlData{tutor-start=38,tutor-end=39}{0}\htmlData{tutor-start=39,tutor-end=40}{1}\htmlData{tutor-start=40,tutor-end=41}{1}}^{\htmlData{tutor-start=44,tutor-end=45}{n}}\htmlData{tutor-start=46,tutor-end=47}{=}\htmlData{tutor-start=47,tutor-end=48}{a}^{\htmlData{tutor-start=50,tutor-end=51}{n}\htmlData{tutor-start=51,tutor-end=52}{+}\htmlData{tutor-start=52,tutor-end=53}{1}}\htmlData{tutor-start=54,tutor-end=55}{+}\htmlData{tutor-start=55,tutor-end=56}{1} \htmlData{tutor-start=57,tutor-end=58}{.} x 1 n + 2 x 2 n + ⋯ + 2 0 1 1 x 2 0 1 1 n = a n + 1 + 1 . 题解状态： 标准答案与规范题解待补充
题目标签：2011 IMO Shortlist A2
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2011-A3 2011 · Algebra · IMO/代数
Determine all pairs ( f , g ) \htmlData{tutor-start=0,tutor-end=1}{(}\htmlData{tutor-start=1,tutor-end=2}{f}\htmlData{tutor-start=2,tutor-end=3}{,} \htmlData{tutor-start=4,tutor-end=5}{g}\htmlData{tutor-start=5,tutor-end=6}{)} ( f , g ) g ( f ( x + y ) ) = f ( x ) + ( 2 x + y ) g ( y ) \htmlData{tutor-start=0,tutor-end=1}{g}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{f}\htmlData{tutor-start=3,tutor-end=4}{(}\htmlData{tutor-start=4,tutor-end=5}{x}\htmlData{tutor-start=5,tutor-end=6}{+}\htmlData{tutor-start=6,tutor-end=7}{y}\htmlData{tutor-start=7,tutor-end=8}{)}\htmlData{tutor-start=8,tutor-end=9}{)}\htmlData{tutor-start=9,tutor-end=10}{=}\htmlData{tutor-start=10,tutor-end=11}{f}\htmlData{tutor-start=11,tutor-end=12}{(}\htmlData{tutor-start=12,tutor-end=13}{x}\htmlData{tutor-start=13,tutor-end=14}{)}\htmlData{tutor-start=14,tutor-end=15}{+}\htmlData{tutor-start=15,tutor-end=16}{(}\htmlData{tutor-start=16,tutor-end=17}{2} \htmlData{tutor-start=18,tutor-end=19}{x}\htmlData{tutor-start=19,tutor-end=20}{+}\htmlData{tutor-start=20,tutor-end=21}{y}\htmlData{tutor-start=21,tutor-end=22}{)} \htmlData{tutor-start=23,tutor-end=24}{g}\htmlData{tutor-start=24,tutor-end=25}{(}\htmlData{tutor-start=25,tutor-end=26}{y}\htmlData{tutor-start=26,tutor-end=27}{)} g ( f ( x + y ) ) = f ( x ) + ( 2 x + y ) g ( y ) x \htmlData{tutor-start=0,tutor-end=1}{x} x y \htmlData{tutor-start=0,tutor-end=1}{y} y 题解状态： 标准答案与规范题解待补充
题目标签：2011 IMO Shortlist A3
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2011-A4 2011 · Algebra · IMO/代数
Determine all pairs ( f , g ) \htmlData{tutor-start=0,tutor-end=1}{(}\htmlData{tutor-start=1,tutor-end=2}{f}\htmlData{tutor-start=2,tutor-end=3}{,} \htmlData{tutor-start=4,tutor-end=5}{g}\htmlData{tutor-start=5,tutor-end=6}{)} ( f , g ) f g ( n ) + 1 ( n ) + g f ( n ) ( n ) = f ( n + 1 ) − g ( n + 1 ) + 1 \htmlData{tutor-start=0,tutor-end=1}{f}^{\htmlData{tutor-start=3,tutor-end=4}{g}\htmlData{tutor-start=4,tutor-end=5}{(}\htmlData{tutor-start=5,tutor-end=6}{n}\htmlData{tutor-start=6,tutor-end=7}{)}\htmlData{tutor-start=7,tutor-end=8}{+}\htmlData{tutor-start=8,tutor-end=9}{1}}\htmlData{tutor-start=10,tutor-end=11}{(}\htmlData{tutor-start=11,tutor-end=12}{n}\htmlData{tutor-start=12,tutor-end=13}{)}\htmlData{tutor-start=13,tutor-end=14}{+}\htmlData{tutor-start=14,tutor-end=15}{g}^{\htmlData{tutor-start=17,tutor-end=18}{f}\htmlData{tutor-start=18,tutor-end=19}{(}\htmlData{tutor-start=19,tutor-end=20}{n}\htmlData{tutor-start=20,tutor-end=21}{)}}\htmlData{tutor-start=22,tutor-end=23}{(}\htmlData{tutor-start=23,tutor-end=24}{n}\htmlData{tutor-start=24,tutor-end=25}{)}\htmlData{tutor-start=25,tutor-end=26}{=}\htmlData{tutor-start=26,tutor-end=27}{f}\htmlData{tutor-start=27,tutor-end=28}{(}\htmlData{tutor-start=28,tutor-end=29}{n}\htmlData{tutor-start=29,tutor-end=30}{+}\htmlData{tutor-start=30,tutor-end=31}{1}\htmlData{tutor-start=31,tutor-end=32}{)}\htmlData{tutor-start=32,tutor-end=33}{-}\htmlData{tutor-start=33,tutor-end=34}{g}\htmlData{tutor-start=34,tutor-end=35}{(}\htmlData{tutor-start=35,tutor-end=36}{n}\htmlData{tutor-start=36,tutor-end=37}{+}\htmlData{tutor-start=37,tutor-end=38}{1}\htmlData{tutor-start=38,tutor-end=39}{)}\htmlData{tutor-start=39,tutor-end=40}{+}\htmlData{tutor-start=40,tutor-end=41}{1} f g ( n ) + 1 ( n ) + g f ( n ) ( n ) = f ( n + 1 ) − g ( n + 1 ) + 1 n \htmlData{tutor-start=0,tutor-end=1}{n} n f k ( n ) \htmlData{tutor-start=0,tutor-end=1}{f}^{\htmlData{tutor-start=3,tutor-end=4}{k}}\htmlData{tutor-start=5,tutor-end=6}{(}\htmlData{tutor-start=6,tutor-end=7}{n}\htmlData{tutor-start=7,tutor-end=8}{)} f k ( n ) f ( f ( … f ⏟ k ( n ) … ) ) \underbrace{\htmlData{tutor-start=12,tutor-end=13}{f}\htmlData{tutor-start=13,tutor-end=14}{(}\htmlData{tutor-start=14,tutor-end=15}{f}\htmlData{tutor-start=15,tutor-end=16}{(}\htmlData{tutor-start=16,tutor-end=23}{\ldots }\htmlData{tutor-start=23,tutor-end=24}{f}}_{\htmlData{tutor-start=27,tutor-end=28}{k}}\htmlData{tutor-start=29,tutor-end=30}{(}\htmlData{tutor-start=30,tutor-end=31}{n}\htmlData{tutor-start=31,tutor-end=32}{)} \htmlData{tutor-start=33,tutor-end=39}{\ldots}\htmlData{tutor-start=39,tutor-end=40}{)}\htmlData{tutor-start=40,tutor-end=41}{)} k f ( f ( … f ( n ) … ) ) 题解状态： 标准答案与规范题解待补充
题目标签：2011 IMO Shortlist A4
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2011-A5 2011 · Algebra · IMO/代数
Prove that for every positive integer n \htmlData{tutor-start=0,tutor-end=1}{n} n { 2 , 3 , 4 , … , 3 n + 1 } \htmlData{tutor-start=0,tutor-end=2}{\{}\htmlData{tutor-start=2,tutor-end=3}{2}\htmlData{tutor-start=3,tutor-end=4}{,}\htmlData{tutor-start=4,tutor-end=5}{3}\htmlData{tutor-start=5,tutor-end=6}{,}\htmlData{tutor-start=6,tutor-end=7}{4}\htmlData{tutor-start=7,tutor-end=8}{,} \htmlData{tutor-start=9,tutor-end=15}{\ldots}\htmlData{tutor-start=15,tutor-end=16}{,} \htmlData{tutor-start=17,tutor-end=18}{3} \htmlData{tutor-start=19,tutor-end=20}{n}\htmlData{tutor-start=20,tutor-end=21}{+}\htmlData{tutor-start=21,tutor-end=22}{1}\htmlData{tutor-start=22,tutor-end=24}{\}} { 2 , 3 , 4 , … , 3 n + 1 } n \htmlData{tutor-start=0,tutor-end=1}{n} n 题解状态： 标准答案与规范题解待补充
题目标签：2011 IMO Shortlist A5
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2011-A6 2011 · Algebra · IMO/代数
Let f \htmlData{tutor-start=0,tutor-end=1}{f} f f ( x + y ) ≤ y f ( x ) + f ( f ( x ) ) \htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{x}\htmlData{tutor-start=3,tutor-end=4}{+}\htmlData{tutor-start=4,tutor-end=5}{y}\htmlData{tutor-start=5,tutor-end=6}{)} \htmlData{tutor-start=7,tutor-end=12}{\leq }\htmlData{tutor-start=12,tutor-end=13}{y} \htmlData{tutor-start=14,tutor-end=15}{f}\htmlData{tutor-start=15,tutor-end=16}{(}\htmlData{tutor-start=16,tutor-end=17}{x}\htmlData{tutor-start=17,tutor-end=18}{)}\htmlData{tutor-start=18,tutor-end=19}{+}\htmlData{tutor-start=19,tutor-end=20}{f}\htmlData{tutor-start=20,tutor-end=21}{(}\htmlData{tutor-start=21,tutor-end=22}{f}\htmlData{tutor-start=22,tutor-end=23}{(}\htmlData{tutor-start=23,tutor-end=24}{x}\htmlData{tutor-start=24,tutor-end=25}{)}\htmlData{tutor-start=25,tutor-end=26}{)} f ( x + y ) ≤ y f ( x ) + f ( f ( x ) ) x \htmlData{tutor-start=0,tutor-end=1}{x} x y \htmlData{tutor-start=0,tutor-end=1}{y} y f ( x ) = 0 \htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{x}\htmlData{tutor-start=3,tutor-end=4}{)}\htmlData{tutor-start=4,tutor-end=5}{=}\htmlData{tutor-start=5,tutor-end=6}{0} f ( x ) = 0 x ≤ 0 \htmlData{tutor-start=0,tutor-end=1}{x} \htmlData{tutor-start=2,tutor-end=7}{\leq }\htmlData{tutor-start=7,tutor-end=8}{0} x ≤ 0 题解状态： 标准答案与规范题解待补充
题目标签：2011 IMO 正式题 A6
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2011-A7 2011 · Algebra · IMO/代数
Let a , b \htmlData{tutor-start=0,tutor-end=1}{a}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{b} a , b c \htmlData{tutor-start=0,tutor-end=1}{c} c min ( a + b , b + c , c + a ) > 2 \min \htmlData{tutor-start=5,tutor-end=6}{(}\htmlData{tutor-start=6,tutor-end=7}{a}\htmlData{tutor-start=7,tutor-end=8}{+}\htmlData{tutor-start=8,tutor-end=9}{b}\htmlData{tutor-start=9,tutor-end=10}{,} \htmlData{tutor-start=11,tutor-end=12}{b}\htmlData{tutor-start=12,tutor-end=13}{+}\htmlData{tutor-start=13,tutor-end=14}{c}\htmlData{tutor-start=14,tutor-end=15}{,} \htmlData{tutor-start=16,tutor-end=17}{c}\htmlData{tutor-start=17,tutor-end=18}{+}\htmlData{tutor-start=18,tutor-end=19}{a}\htmlData{tutor-start=19,tutor-end=20}{)}\htmlData{tutor-start=20,tutor-end=21}{>}\sqrt{\htmlData{tutor-start=27,tutor-end=28}{2}} min ( a + b , b + c , c + a ) > 2 a 2 + b 2 + c 2 = 3 \htmlData{tutor-start=0,tutor-end=1}{a}^{\htmlData{tutor-start=3,tutor-end=4}{2}}\htmlData{tutor-start=5,tutor-end=6}{+}\htmlData{tutor-start=6,tutor-end=7}{b}^{\htmlData{tutor-start=9,tutor-end=10}{2}}\htmlData{tutor-start=11,tutor-end=12}{+}\htmlData{tutor-start=12,tutor-end=13}{c}^{\htmlData{tutor-start=15,tutor-end=16}{2}}\htmlData{tutor-start=17,tutor-end=18}{=}\htmlData{tutor-start=18,tutor-end=19}{3} a 2 + b 2 + c 2 = 3 a ( b + c − a ) 2 + b ( c + a − b ) 2 + c ( a + b − c ) 2 ≥ 3 ( a b c ) 2 \frac{\htmlData{tutor-start=6,tutor-end=7}{a}}{\htmlData{tutor-start=9,tutor-end=10}{(}\htmlData{tutor-start=10,tutor-end=11}{b}\htmlData{tutor-start=11,tutor-end=12}{+}\htmlData{tutor-start=12,tutor-end=13}{c}\htmlData{tutor-start=13,tutor-end=14}{-}\htmlData{tutor-start=14,tutor-end=15}{a}\htmlData{tutor-start=15,tutor-end=16}{)}^{\htmlData{tutor-start=18,tutor-end=19}{2}}}\htmlData{tutor-start=21,tutor-end=22}{+}\frac{\htmlData{tutor-start=28,tutor-end=29}{b}}{\htmlData{tutor-start=31,tutor-end=32}{(}\htmlData{tutor-start=32,tutor-end=33}{c}\htmlData{tutor-start=33,tutor-end=34}{+}\htmlData{tutor-start=34,tutor-end=35}{a}\htmlData{tutor-start=35,tutor-end=36}{-}\htmlData{tutor-start=36,tutor-end=37}{b}\htmlData{tutor-start=37,tutor-end=38}{)}^{\htmlData{tutor-start=40,tutor-end=41}{2}}}\htmlData{tutor-start=43,tutor-end=44}{+}\frac{\htmlData{tutor-start=50,tutor-end=51}{c}}{\htmlData{tutor-start=53,tutor-end=54}{(}\htmlData{tutor-start=54,tutor-end=55}{a}\htmlData{tutor-start=55,tutor-end=56}{+}\htmlData{tutor-start=56,tutor-end=57}{b}\htmlData{tutor-start=57,tutor-end=58}{-}\htmlData{tutor-start=58,tutor-end=59}{c}\htmlData{tutor-start=59,tutor-end=60}{)}^{\htmlData{tutor-start=62,tutor-end=63}{2}}} \htmlData{tutor-start=66,tutor-end=71}{\geq }\frac{\htmlData{tutor-start=77,tutor-end=78}{3}}{\htmlData{tutor-start=80,tutor-end=81}{(}\htmlData{tutor-start=81,tutor-end=82}{a} \htmlData{tutor-start=83,tutor-end=84}{b} \htmlData{tutor-start=85,tutor-end=86}{c}\htmlData{tutor-start=86,tutor-end=87}{)}^{\htmlData{tutor-start=89,tutor-end=90}{2}}} ( b + c − a ) 2 a + ( c + a − b ) 2 b + ( a + b − c ) 2 c ≥ ( a b c ) 2 3 题解状态： 标准答案与规范题解待补充
题目标签：2011 IMO Shortlist A7
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2012-A1 2012 · Algebra · IMO/代数
Find all the functions f : Z → Z \htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{:} \mathbb{\htmlData{tutor-start=11,tutor-end=12}{Z}} \htmlData{tutor-start=14,tutor-end=26}{\rightarrow }\mathbb{\htmlData{tutor-start=34,tutor-end=35}{Z}} f : Z → Z f ( a ) 2 + f ( b ) 2 + f ( c ) 2 = 2 f ( a ) f ( b ) + 2 f ( b ) f ( c ) + 2 f ( c ) f ( a ) \htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{a}\htmlData{tutor-start=3,tutor-end=4}{)}^{\htmlData{tutor-start=6,tutor-end=7}{2}}\htmlData{tutor-start=8,tutor-end=9}{+}\htmlData{tutor-start=9,tutor-end=10}{f}\htmlData{tutor-start=10,tutor-end=11}{(}\htmlData{tutor-start=11,tutor-end=12}{b}\htmlData{tutor-start=12,tutor-end=13}{)}^{\htmlData{tutor-start=15,tutor-end=16}{2}}\htmlData{tutor-start=17,tutor-end=18}{+}\htmlData{tutor-start=18,tutor-end=19}{f}\htmlData{tutor-start=19,tutor-end=20}{(}\htmlData{tutor-start=20,tutor-end=21}{c}\htmlData{tutor-start=21,tutor-end=22}{)}^{\htmlData{tutor-start=24,tutor-end=25}{2}}\htmlData{tutor-start=26,tutor-end=27}{=}\htmlData{tutor-start=27,tutor-end=28}{2} \htmlData{tutor-start=29,tutor-end=30}{f}\htmlData{tutor-start=30,tutor-end=31}{(}\htmlData{tutor-start=31,tutor-end=32}{a}\htmlData{tutor-start=32,tutor-end=33}{)} \htmlData{tutor-start=34,tutor-end=35}{f}\htmlData{tutor-start=35,tutor-end=36}{(}\htmlData{tutor-start=36,tutor-end=37}{b}\htmlData{tutor-start=37,tutor-end=38}{)}\htmlData{tutor-start=38,tutor-end=39}{+}\htmlData{tutor-start=39,tutor-end=40}{2} \htmlData{tutor-start=41,tutor-end=42}{f}\htmlData{tutor-start=42,tutor-end=43}{(}\htmlData{tutor-start=43,tutor-end=44}{b}\htmlData{tutor-start=44,tutor-end=45}{)} \htmlData{tutor-start=46,tutor-end=47}{f}\htmlData{tutor-start=47,tutor-end=48}{(}\htmlData{tutor-start=48,tutor-end=49}{c}\htmlData{tutor-start=49,tutor-end=50}{)}\htmlData{tutor-start=50,tutor-end=51}{+}\htmlData{tutor-start=51,tutor-end=52}{2} \htmlData{tutor-start=53,tutor-end=54}{f}\htmlData{tutor-start=54,tutor-end=55}{(}\htmlData{tutor-start=55,tutor-end=56}{c}\htmlData{tutor-start=56,tutor-end=57}{)} \htmlData{tutor-start=58,tutor-end=59}{f}\htmlData{tutor-start=59,tutor-end=60}{(}\htmlData{tutor-start=60,tutor-end=61}{a}\htmlData{tutor-start=61,tutor-end=62}{)} f ( a ) 2 + f ( b ) 2 + f ( c ) 2 = 2 f ( a ) f ( b ) + 2 f ( b ) f ( c ) + 2 f ( c ) f ( a ) a , b , c \htmlData{tutor-start=0,tutor-end=1}{a}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{b}\htmlData{tutor-start=4,tutor-end=5}{,} \htmlData{tutor-start=6,tutor-end=7}{c} a , b , c a + b + c = 0 \htmlData{tutor-start=0,tutor-end=1}{a}\htmlData{tutor-start=1,tutor-end=2}{+}\htmlData{tutor-start=2,tutor-end=3}{b}\htmlData{tutor-start=3,tutor-end=4}{+}\htmlData{tutor-start=4,tutor-end=5}{c}\htmlData{tutor-start=5,tutor-end=6}{=}\htmlData{tutor-start=6,tutor-end=7}{0} a + b + c = 0 题解状态： 标准答案与规范题解待补充
题目标签：2012 IMO 正式题 A1
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2012-A2 2012 · Algebra · IMO/代数
Let Z \mathbb{\htmlData{tutor-start=8,tutor-end=9}{Z}} Z Q \mathbb{\htmlData{tutor-start=8,tutor-end=9}{Q}} Q Z \mathbb{\htmlData{tutor-start=8,tutor-end=9}{Z}} Z A , B , C \htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{B}\htmlData{tutor-start=4,tutor-end=5}{,} \htmlData{tutor-start=6,tutor-end=7}{C} A , B , C A + B , B + C , C + A \htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{+}\htmlData{tutor-start=2,tutor-end=3}{B}\htmlData{tutor-start=3,tutor-end=4}{,} \htmlData{tutor-start=5,tutor-end=6}{B}\htmlData{tutor-start=6,tutor-end=7}{+}\htmlData{tutor-start=7,tutor-end=8}{C}\htmlData{tutor-start=8,tutor-end=9}{,} \htmlData{tutor-start=10,tutor-end=11}{C}\htmlData{tutor-start=11,tutor-end=12}{+}\htmlData{tutor-start=12,tutor-end=13}{A} A + B , B + C , C + A Q \mathbb{\htmlData{tutor-start=8,tutor-end=9}{Q}} Q A , B , C \htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{B}\htmlData{tutor-start=4,tutor-end=5}{,} \htmlData{tutor-start=6,tutor-end=7}{C} A , B , C A + B , B + C , C + A \htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{+}\htmlData{tutor-start=2,tutor-end=3}{B}\htmlData{tutor-start=3,tutor-end=4}{,} \htmlData{tutor-start=5,tutor-end=6}{B}\htmlData{tutor-start=6,tutor-end=7}{+}\htmlData{tutor-start=7,tutor-end=8}{C}\htmlData{tutor-start=8,tutor-end=9}{,} \htmlData{tutor-start=10,tutor-end=11}{C}\htmlData{tutor-start=11,tutor-end=12}{+}\htmlData{tutor-start=12,tutor-end=13}{A} A + B , B + C , C + A X + Y \htmlData{tutor-start=0,tutor-end=1}{X}\htmlData{tutor-start=1,tutor-end=2}{+}\htmlData{tutor-start=2,tutor-end=3}{Y} X + Y { x + y ∣ x ∈ X , y ∈ Y } \htmlData{tutor-start=0,tutor-end=2}{\{}\htmlData{tutor-start=2,tutor-end=3}{x}\htmlData{tutor-start=3,tutor-end=4}{+}\htmlData{tutor-start=4,tutor-end=5}{y} \htmlData{tutor-start=6,tutor-end=11}{\mid }\htmlData{tutor-start=11,tutor-end=12}{x} \htmlData{tutor-start=13,tutor-end=17}{\in }\htmlData{tutor-start=17,tutor-end=18}{X}\htmlData{tutor-start=18,tutor-end=19}{,} \htmlData{tutor-start=20,tutor-end=21}{y} \htmlData{tutor-start=22,tutor-end=26}{\in }\htmlData{tutor-start=26,tutor-end=27}{Y}\htmlData{tutor-start=27,tutor-end=29}{\}} { x + y ∣ x ∈ X , y ∈ Y } X , Y ⊆ Z \htmlData{tutor-start=0,tutor-end=1}{X}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{Y} \htmlData{tutor-start=5,tutor-end=15}{\subseteq }\mathbb{\htmlData{tutor-start=23,tutor-end=24}{Z}} X , Y ⊆ Z X , Y ⊆ Q \htmlData{tutor-start=0,tutor-end=1}{X}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{Y} \htmlData{tutor-start=5,tutor-end=15}{\subseteq }\mathbb{\htmlData{tutor-start=23,tutor-end=24}{Q}} X , Y ⊆ Q 题解状态： 标准答案与规范题解待补充
题目标签：2012 IMO Shortlist A2
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2012-A3 2012 · Algebra · IMO/代数
Let a 2 , … , a n \htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{2}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=13}{\ldots}\htmlData{tutor-start=13,tutor-end=14}{,} \htmlData{tutor-start=15,tutor-end=16}{a}_{\htmlData{tutor-start=18,tutor-end=19}{n}} a 2 , … , a n n − 1 \htmlData{tutor-start=0,tutor-end=1}{n}\htmlData{tutor-start=1,tutor-end=2}{-}\htmlData{tutor-start=2,tutor-end=3}{1} n − 1 n ≥ 3 \htmlData{tutor-start=0,tutor-end=1}{n} \htmlData{tutor-start=2,tutor-end=7}{\geq }\htmlData{tutor-start=7,tutor-end=8}{3} n ≥ 3 a 2 a 3 ⋯ a n = 1 \htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{2}} \htmlData{tutor-start=6,tutor-end=7}{a}_{\htmlData{tutor-start=9,tutor-end=10}{3}} \cdots \htmlData{tutor-start=19,tutor-end=20}{a}_{\htmlData{tutor-start=22,tutor-end=23}{n}}\htmlData{tutor-start=24,tutor-end=25}{=}\htmlData{tutor-start=25,tutor-end=26}{1} a 2 a 3 ⋯ a n = 1 ( 1 + a 2 ) 2 ( 1 + a 3 ) 3 ⋯ ( 1 + a n ) n > n n . \left(1+a_{2}\right)^{2}\left(1+a_{3}\right)^{3} \cdots\left(1+a_{n}\right)^{n}>n^{n} . ( 1 + a 2 ) 2 ( 1 + a 3 ) 3 ⋯ ( 1 + a n ) n > n n . 题解状态： 标准答案与规范题解待补充
题目标签：2012 IMO 正式题 A3
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2012-A4 2012 · Algebra · IMO/代数
Let f \htmlData{tutor-start=0,tutor-end=1}{f} f g \htmlData{tutor-start=0,tutor-end=1}{g} g deg f > deg g \operatorname{\htmlData{tutor-start=14,tutor-end=15}{d}\htmlData{tutor-start=15,tutor-end=16}{e}\htmlData{tutor-start=16,tutor-end=17}{g}} \htmlData{tutor-start=19,tutor-end=20}{f}\htmlData{tutor-start=20,tutor-end=21}{>}\operatorname{\htmlData{tutor-start=35,tutor-end=36}{d}\htmlData{tutor-start=36,tutor-end=37}{e}\htmlData{tutor-start=37,tutor-end=38}{g}} \htmlData{tutor-start=40,tutor-end=41}{g} d e g f > d e g g p \htmlData{tutor-start=0,tutor-end=1}{p} p p f + g \htmlData{tutor-start=0,tutor-end=1}{p} \htmlData{tutor-start=2,tutor-end=3}{f}\htmlData{tutor-start=3,tutor-end=4}{+}\htmlData{tutor-start=4,tutor-end=5}{g} p f + g f \htmlData{tutor-start=0,tutor-end=1}{f} f 题解状态： 标准答案与规范题解待补充
题目标签：2012 IMO Shortlist A4
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2012-A5 2012 · Algebra · IMO/代数
Find all functions f : R → R \htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{:} \mathbb{\htmlData{tutor-start=11,tutor-end=12}{R}} \htmlData{tutor-start=14,tutor-end=26}{\rightarrow }\mathbb{\htmlData{tutor-start=34,tutor-end=35}{R}} f : R → R f ( 1 + x y ) − f ( x + y ) = f ( x ) f ( y ) for all x , y ∈ R \htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{1}\htmlData{tutor-start=3,tutor-end=4}{+}\htmlData{tutor-start=4,tutor-end=5}{x} \htmlData{tutor-start=6,tutor-end=7}{y}\htmlData{tutor-start=7,tutor-end=8}{)}\htmlData{tutor-start=8,tutor-end=9}{-}\htmlData{tutor-start=9,tutor-end=10}{f}\htmlData{tutor-start=10,tutor-end=11}{(}\htmlData{tutor-start=11,tutor-end=12}{x}\htmlData{tutor-start=12,tutor-end=13}{+}\htmlData{tutor-start=13,tutor-end=14}{y}\htmlData{tutor-start=14,tutor-end=15}{)}\htmlData{tutor-start=15,tutor-end=16}{=}\htmlData{tutor-start=16,tutor-end=17}{f}\htmlData{tutor-start=17,tutor-end=18}{(}\htmlData{tutor-start=18,tutor-end=19}{x}\htmlData{tutor-start=19,tutor-end=20}{)} \htmlData{tutor-start=21,tutor-end=22}{f}\htmlData{tutor-start=22,tutor-end=23}{(}\htmlData{tutor-start=23,tutor-end=24}{y}\htmlData{tutor-start=24,tutor-end=25}{)} \quad \text { \htmlData{tutor-start=40,tutor-end=41}{f}\htmlData{tutor-start=41,tutor-end=42}{o}\htmlData{tutor-start=42,tutor-end=43}{r} \htmlData{tutor-start=44,tutor-end=45}{a}\htmlData{tutor-start=45,tutor-end=46}{l}\htmlData{tutor-start=46,tutor-end=47}{l} } \htmlData{tutor-start=50,tutor-end=51}{x}\htmlData{tutor-start=51,tutor-end=52}{,} \htmlData{tutor-start=53,tutor-end=54}{y} \htmlData{tutor-start=55,tutor-end=59}{\in }\mathbb{\htmlData{tutor-start=67,tutor-end=68}{R}} f ( 1 + x y ) − f ( x + y ) = f ( x ) f ( y ) f o r a l l x , y ∈ R f ( − 1 ) ≠ 0 \htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{-}\htmlData{tutor-start=3,tutor-end=4}{1}\htmlData{tutor-start=4,tutor-end=5}{)} \neq \htmlData{tutor-start=11,tutor-end=12}{0} f ( − 1 ) = 0 题解状态： 标准答案与规范题解待补充
题目标签：2012 IMO Shortlist A5
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2012-A6 2012 · Algebra · IMO/代数
Let f : N → N \htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{:} \mathbb{\htmlData{tutor-start=11,tutor-end=12}{N}} \htmlData{tutor-start=14,tutor-end=26}{\rightarrow }\mathbb{\htmlData{tutor-start=34,tutor-end=35}{N}} f : N → N f m \htmlData{tutor-start=0,tutor-end=1}{f}^{\htmlData{tutor-start=3,tutor-end=4}{m}} f m f \htmlData{tutor-start=0,tutor-end=1}{f} f m \htmlData{tutor-start=0,tutor-end=1}{m} m n ∈ N \htmlData{tutor-start=0,tutor-end=1}{n} \htmlData{tutor-start=2,tutor-end=6}{\in }\mathbb{\htmlData{tutor-start=14,tutor-end=15}{N}} n ∈ N k ∈ N \htmlData{tutor-start=0,tutor-end=1}{k} \htmlData{tutor-start=2,tutor-end=6}{\in }\mathbb{\htmlData{tutor-start=14,tutor-end=15}{N}} k ∈ N f 2 k ( n ) = n + k \htmlData{tutor-start=0,tutor-end=1}{f}^{\htmlData{tutor-start=3,tutor-end=4}{2} \htmlData{tutor-start=5,tutor-end=6}{k}}\htmlData{tutor-start=7,tutor-end=8}{(}\htmlData{tutor-start=8,tutor-end=9}{n}\htmlData{tutor-start=9,tutor-end=10}{)}\htmlData{tutor-start=10,tutor-end=11}{=}\htmlData{tutor-start=11,tutor-end=12}{n}\htmlData{tutor-start=12,tutor-end=13}{+}\htmlData{tutor-start=13,tutor-end=14}{k} f 2 k ( n ) = n + k k n \htmlData{tutor-start=0,tutor-end=1}{k}_{\htmlData{tutor-start=3,tutor-end=4}{n}} k n k \htmlData{tutor-start=0,tutor-end=1}{k} k k 1 , k 2 , … \htmlData{tutor-start=0,tutor-end=1}{k}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{k}_{\htmlData{tutor-start=10,tutor-end=11}{2}}\htmlData{tutor-start=12,tutor-end=13}{,} \htmlData{tutor-start=14,tutor-end=20}{\ldots} k 1 , k 2 , … 题解状态： 标准答案与规范题解待补充
题目标签：2012 IMO Shortlist A6
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2012-A7 2012 · Algebra · IMO/代数
We say that a function f : R k → R \htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{:} \mathbb{\htmlData{tutor-start=11,tutor-end=12}{R}}^{\htmlData{tutor-start=15,tutor-end=16}{k}} \htmlData{tutor-start=18,tutor-end=30}{\rightarrow }\mathbb{\htmlData{tutor-start=38,tutor-end=39}{R}} f : R k → R m \htmlData{tutor-start=0,tutor-end=1}{m} m n \htmlData{tutor-start=0,tutor-end=1}{n} n f ( x 1 , … , x k ) = max i = 1 , … , m min j = 1 , … , n P i , j ( x 1 , … , x k ) \htmlData{tutor-start=0,tutor-end=1}{f}\left(\htmlData{tutor-start=7,tutor-end=8}{x}_{\htmlData{tutor-start=10,tutor-end=11}{1}}\htmlData{tutor-start=12,tutor-end=13}{,} \htmlData{tutor-start=14,tutor-end=20}{\ldots}\htmlData{tutor-start=20,tutor-end=21}{,} \htmlData{tutor-start=22,tutor-end=23}{x}_{\htmlData{tutor-start=25,tutor-end=26}{k}}\right)\htmlData{tutor-start=34,tutor-end=35}{=}\max _{\htmlData{tutor-start=42,tutor-end=43}{i}\htmlData{tutor-start=43,tutor-end=44}{=}\htmlData{tutor-start=44,tutor-end=45}{1}\htmlData{tutor-start=45,tutor-end=46}{,} \htmlData{tutor-start=47,tutor-end=53}{\ldots}\htmlData{tutor-start=53,tutor-end=54}{,} \htmlData{tutor-start=55,tutor-end=56}{m}} \min _{\htmlData{tutor-start=65,tutor-end=66}{j}\htmlData{tutor-start=66,tutor-end=67}{=}\htmlData{tutor-start=67,tutor-end=68}{1}\htmlData{tutor-start=68,tutor-end=69}{,} \htmlData{tutor-start=70,tutor-end=76}{\ldots}\htmlData{tutor-start=76,tutor-end=77}{,} \htmlData{tutor-start=78,tutor-end=79}{n}} \htmlData{tutor-start=81,tutor-end=82}{P}_{\htmlData{tutor-start=84,tutor-end=85}{i}\htmlData{tutor-start=85,tutor-end=86}{,} \htmlData{tutor-start=87,tutor-end=88}{j}}\left(\htmlData{tutor-start=95,tutor-end=96}{x}_{\htmlData{tutor-start=98,tutor-end=99}{1}}\htmlData{tutor-start=100,tutor-end=101}{,} \htmlData{tutor-start=102,tutor-end=108}{\ldots}\htmlData{tutor-start=108,tutor-end=109}{,} \htmlData{tutor-start=110,tutor-end=111}{x}_{\htmlData{tutor-start=113,tutor-end=114}{k}}\right) f ( x 1 , … , x k ) = max i = 1 , … , m min j = 1 , … , n P i , j ( x 1 , … , x k ) P i , j \htmlData{tutor-start=0,tutor-end=1}{P}_{\htmlData{tutor-start=3,tutor-end=4}{i}\htmlData{tutor-start=4,tutor-end=5}{,} \htmlData{tutor-start=6,tutor-end=7}{j}} P i , j 题解状态： 标准答案与规范题解待补充
题目标签：2012 IMO Shortlist A7
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2013-A1 2013 · Algebra · IMO/代数
Let n \htmlData{tutor-start=0,tutor-end=1}{n} n a 1 , … , a n − 1 \htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=13}{\ldots}\htmlData{tutor-start=13,tutor-end=14}{,} \htmlData{tutor-start=15,tutor-end=16}{a}_{\htmlData{tutor-start=18,tutor-end=19}{n}\htmlData{tutor-start=19,tutor-end=20}{-}\htmlData{tutor-start=20,tutor-end=21}{1}} a 1 , … , a n − 1 u 0 , … , u n \htmlData{tutor-start=0,tutor-end=1}{u}_{\htmlData{tutor-start=3,tutor-end=4}{0}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=13}{\ldots}\htmlData{tutor-start=13,tutor-end=14}{,} \htmlData{tutor-start=15,tutor-end=16}{u}_{\htmlData{tutor-start=18,tutor-end=19}{n}} u 0 , … , u n v 0 , … , v n \htmlData{tutor-start=0,tutor-end=1}{v}_{\htmlData{tutor-start=3,tutor-end=4}{0}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=13}{\ldots}\htmlData{tutor-start=13,tutor-end=14}{,} \htmlData{tutor-start=15,tutor-end=16}{v}_{\htmlData{tutor-start=18,tutor-end=19}{n}} v 0 , … , v n u 0 = u 1 = v 0 = v 1 = 1 \htmlData{tutor-start=0,tutor-end=1}{u}_{\htmlData{tutor-start=3,tutor-end=4}{0}}\htmlData{tutor-start=5,tutor-end=6}{=}\htmlData{tutor-start=6,tutor-end=7}{u}_{\htmlData{tutor-start=9,tutor-end=10}{1}}\htmlData{tutor-start=11,tutor-end=12}{=}\htmlData{tutor-start=12,tutor-end=13}{v}_{\htmlData{tutor-start=15,tutor-end=16}{0}}\htmlData{tutor-start=17,tutor-end=18}{=}\htmlData{tutor-start=18,tutor-end=19}{v}_{\htmlData{tutor-start=21,tutor-end=22}{1}}\htmlData{tutor-start=23,tutor-end=24}{=}\htmlData{tutor-start=24,tutor-end=25}{1} u 0 = u 1 = v 0 = v 1 = 1 u k + 1 = u k + a k u k − 1 , v k + 1 = v k + a n − k v k − 1 for k = 1 , … , n − 1. \htmlData{tutor-start=0,tutor-end=1}{u}_{\htmlData{tutor-start=3,tutor-end=4}{k}\htmlData{tutor-start=4,tutor-end=5}{+}\htmlData{tutor-start=5,tutor-end=6}{1}}\htmlData{tutor-start=7,tutor-end=8}{=}\htmlData{tutor-start=8,tutor-end=9}{u}_{\htmlData{tutor-start=11,tutor-end=12}{k}}\htmlData{tutor-start=13,tutor-end=14}{+}\htmlData{tutor-start=14,tutor-end=15}{a}_{\htmlData{tutor-start=17,tutor-end=18}{k}} \htmlData{tutor-start=20,tutor-end=21}{u}_{\htmlData{tutor-start=23,tutor-end=24}{k}\htmlData{tutor-start=24,tutor-end=25}{-}\htmlData{tutor-start=25,tutor-end=26}{1}}\htmlData{tutor-start=27,tutor-end=28}{,} \quad \htmlData{tutor-start=35,tutor-end=36}{v}_{\htmlData{tutor-start=38,tutor-end=39}{k}\htmlData{tutor-start=39,tutor-end=40}{+}\htmlData{tutor-start=40,tutor-end=41}{1}}\htmlData{tutor-start=42,tutor-end=43}{=}\htmlData{tutor-start=43,tutor-end=44}{v}_{\htmlData{tutor-start=46,tutor-end=47}{k}}\htmlData{tutor-start=48,tutor-end=49}{+}\htmlData{tutor-start=49,tutor-end=50}{a}_{\htmlData{tutor-start=52,tutor-end=53}{n}\htmlData{tutor-start=53,tutor-end=54}{-}\htmlData{tutor-start=54,tutor-end=55}{k}} \htmlData{tutor-start=57,tutor-end=58}{v}_{\htmlData{tutor-start=60,tutor-end=61}{k}\htmlData{tutor-start=61,tutor-end=62}{-}\htmlData{tutor-start=62,tutor-end=63}{1}} \quad \text { \htmlData{tutor-start=79,tutor-end=80}{f}\htmlData{tutor-start=80,tutor-end=81}{o}\htmlData{tutor-start=81,tutor-end=82}{r} } \htmlData{tutor-start=85,tutor-end=86}{k}\htmlData{tutor-start=86,tutor-end=87}{=}\htmlData{tutor-start=87,tutor-end=88}{1}\htmlData{tutor-start=88,tutor-end=89}{,} \htmlData{tutor-start=90,tutor-end=96}{\ldots}\htmlData{tutor-start=96,tutor-end=97}{,} \htmlData{tutor-start=98,tutor-end=99}{n}\htmlData{tutor-start=99,tutor-end=100}{-}\htmlData{tutor-start=100,tutor-end=101}{1} \htmlData{tutor-start=102,tutor-end=103}{.} u k + 1 = u k + a k u k − 1 , v k + 1 = v k + a n − k v k − 1 f o r k = 1 , … , n − 1 . u n = v n \htmlData{tutor-start=0,tutor-end=1}{u}_{\htmlData{tutor-start=3,tutor-end=4}{n}}\htmlData{tutor-start=5,tutor-end=6}{=}\htmlData{tutor-start=6,tutor-end=7}{v}_{\htmlData{tutor-start=9,tutor-end=10}{n}} u n = v n 题解状态： 标准答案与规范题解待补充
题目标签：2013 IMO Shortlist A1
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2013-A2 2013 · Algebra · IMO/代数
Prove that in any set of 2000 distinct real numbers there exist two pairs a > b \htmlData{tutor-start=0,tutor-end=1}{a}\htmlData{tutor-start=1,tutor-end=2}{>}\htmlData{tutor-start=2,tutor-end=3}{b} a > b c > d \htmlData{tutor-start=0,tutor-end=1}{c}\htmlData{tutor-start=1,tutor-end=2}{>}\htmlData{tutor-start=2,tutor-end=3}{d} c > d a ≠ c \htmlData{tutor-start=0,tutor-end=1}{a} \neq \htmlData{tutor-start=7,tutor-end=8}{c} a = c b ≠ d \htmlData{tutor-start=0,tutor-end=1}{b} \neq \htmlData{tutor-start=7,tutor-end=8}{d} b = d ∣ a − b c − d − 1 ∣ < 1 100000 \left|\frac{\htmlData{tutor-start=12,tutor-end=13}{a}\htmlData{tutor-start=13,tutor-end=14}{-}\htmlData{tutor-start=14,tutor-end=15}{b}}{\htmlData{tutor-start=17,tutor-end=18}{c}\htmlData{tutor-start=18,tutor-end=19}{-}\htmlData{tutor-start=19,tutor-end=20}{d}}\htmlData{tutor-start=21,tutor-end=22}{-}\htmlData{tutor-start=22,tutor-end=23}{1}\right|\htmlData{tutor-start=30,tutor-end=31}{<}\frac{\htmlData{tutor-start=37,tutor-end=38}{1}}{\htmlData{tutor-start=40,tutor-end=41}{1}\htmlData{tutor-start=41,tutor-end=42}{0}\htmlData{tutor-start=42,tutor-end=43}{0}\htmlData{tutor-start=43,tutor-end=44}{0}\htmlData{tutor-start=44,tutor-end=45}{0}\htmlData{tutor-start=45,tutor-end=46}{0}} c − d a − b − 1 < 1 0 0 0 0 0 1 题解状态： 标准答案与规范题解待补充
题目标签：2013 IMO Shortlist A2
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2013-A3 2013 · Algebra · IMO/代数
Let Q > 0 \mathbb{\htmlData{tutor-start=8,tutor-end=9}{Q}}_{\htmlData{tutor-start=12,tutor-end=13}{>}\htmlData{tutor-start=13,tutor-end=14}{0}} Q > 0 f : Q > 0 → R \htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{:} \mathbb{\htmlData{tutor-start=11,tutor-end=12}{Q}}_{\htmlData{tutor-start=15,tutor-end=16}{>}\htmlData{tutor-start=16,tutor-end=17}{0}} \htmlData{tutor-start=19,tutor-end=31}{\rightarrow }\mathbb{\htmlData{tutor-start=39,tutor-end=40}{R}} f : Q > 0 → R f ( x ) f ( y ) ⩾ f ( x y ) and f ( x + y ) ⩾ f ( x ) + f ( y ) \htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{x}\htmlData{tutor-start=3,tutor-end=4}{)} \htmlData{tutor-start=5,tutor-end=6}{f}\htmlData{tutor-start=6,tutor-end=7}{(}\htmlData{tutor-start=7,tutor-end=8}{y}\htmlData{tutor-start=8,tutor-end=9}{)} \htmlData{tutor-start=10,tutor-end=20}{\geqslant }\htmlData{tutor-start=20,tutor-end=21}{f}\htmlData{tutor-start=21,tutor-end=22}{(}\htmlData{tutor-start=22,tutor-end=23}{x} \htmlData{tutor-start=24,tutor-end=25}{y}\htmlData{tutor-start=25,tutor-end=26}{)} \text { \htmlData{tutor-start=35,tutor-end=36}{a}\htmlData{tutor-start=36,tutor-end=37}{n}\htmlData{tutor-start=37,tutor-end=38}{d} } \htmlData{tutor-start=41,tutor-end=42}{f}\htmlData{tutor-start=42,tutor-end=43}{(}\htmlData{tutor-start=43,tutor-end=44}{x}\htmlData{tutor-start=44,tutor-end=45}{+}\htmlData{tutor-start=45,tutor-end=46}{y}\htmlData{tutor-start=46,tutor-end=47}{)} \htmlData{tutor-start=48,tutor-end=58}{\geqslant }\htmlData{tutor-start=58,tutor-end=59}{f}\htmlData{tutor-start=59,tutor-end=60}{(}\htmlData{tutor-start=60,tutor-end=61}{x}\htmlData{tutor-start=61,tutor-end=62}{)}\htmlData{tutor-start=62,tutor-end=63}{+}\htmlData{tutor-start=63,tutor-end=64}{f}\htmlData{tutor-start=64,tutor-end=65}{(}\htmlData{tutor-start=65,tutor-end=66}{y}\htmlData{tutor-start=66,tutor-end=67}{)} f ( x ) f ( y ) ⩾ f ( x y ) a n d f ( x + y ) ⩾ f ( x ) + f ( y ) x , y ∈ Q > 0 \htmlData{tutor-start=0,tutor-end=1}{x}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{y} \htmlData{tutor-start=5,tutor-end=9}{\in }\mathbb{\htmlData{tutor-start=17,tutor-end=18}{Q}}_{\htmlData{tutor-start=21,tutor-end=22}{>}\htmlData{tutor-start=22,tutor-end=23}{0}} x , y ∈ Q > 0 f ( a ) = a \htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{a}\htmlData{tutor-start=3,tutor-end=4}{)}\htmlData{tutor-start=4,tutor-end=5}{=}\htmlData{tutor-start=5,tutor-end=6}{a} f ( a ) = a a > 1 \htmlData{tutor-start=0,tutor-end=1}{a}\htmlData{tutor-start=1,tutor-end=2}{>}\htmlData{tutor-start=2,tutor-end=3}{1} a > 1 f ( x ) = x \htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{x}\htmlData{tutor-start=3,tutor-end=4}{)}\htmlData{tutor-start=4,tutor-end=5}{=}\htmlData{tutor-start=5,tutor-end=6}{x} f ( x ) = x x ∈ Q > 0 \htmlData{tutor-start=0,tutor-end=1}{x} \htmlData{tutor-start=2,tutor-end=6}{\in }\mathbb{\htmlData{tutor-start=14,tutor-end=15}{Q}}_{\htmlData{tutor-start=18,tutor-end=19}{>}\htmlData{tutor-start=19,tutor-end=20}{0}} x ∈ Q > 0 题解状态： 标准答案与规范题解待补充
题目标签：2013 IMO 正式题 A3
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2013-A4 2013 · Algebra · IMO/代数
Let n \htmlData{tutor-start=0,tutor-end=1}{n} n a 1 , a 2 , … , a n \htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{a}_{\htmlData{tutor-start=10,tutor-end=11}{2}}\htmlData{tutor-start=12,tutor-end=13}{,} \htmlData{tutor-start=14,tutor-end=20}{\ldots}\htmlData{tutor-start=20,tutor-end=21}{,} \htmlData{tutor-start=22,tutor-end=23}{a}_{\htmlData{tutor-start=25,tutor-end=26}{n}} a 1 , a 2 , … , a n a 1 , a 2 , … \htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{a}_{\htmlData{tutor-start=10,tutor-end=11}{2}}\htmlData{tutor-start=12,tutor-end=13}{,} \htmlData{tutor-start=14,tutor-end=20}{\ldots} a 1 , a 2 , … a n + i = a i \htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{n}\htmlData{tutor-start=4,tutor-end=5}{+}\htmlData{tutor-start=5,tutor-end=6}{i}}\htmlData{tutor-start=7,tutor-end=8}{=}\htmlData{tutor-start=8,tutor-end=9}{a}_{\htmlData{tutor-start=11,tutor-end=12}{i}} a n + i = a i i ⩾ 1 \htmlData{tutor-start=0,tutor-end=1}{i} \htmlData{tutor-start=2,tutor-end=12}{\geqslant }\htmlData{tutor-start=12,tutor-end=13}{1} i ⩾ 1 a 1 ⩽ a 2 ⩽ ⋯ ⩽ a n ⩽ a 1 + n \htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{1}} \htmlData{tutor-start=6,tutor-end=16}{\leqslant }\htmlData{tutor-start=16,tutor-end=17}{a}_{\htmlData{tutor-start=19,tutor-end=20}{2}} \htmlData{tutor-start=22,tutor-end=32}{\leqslant }\cdots \htmlData{tutor-start=39,tutor-end=49}{\leqslant }\htmlData{tutor-start=49,tutor-end=50}{a}_{\htmlData{tutor-start=52,tutor-end=53}{n}} \htmlData{tutor-start=55,tutor-end=65}{\leqslant }\htmlData{tutor-start=65,tutor-end=66}{a}_{\htmlData{tutor-start=68,tutor-end=69}{1}}\htmlData{tutor-start=70,tutor-end=71}{+}\htmlData{tutor-start=71,tutor-end=72}{n} a 1 ⩽ a 2 ⩽ ⋯ ⩽ a n ⩽ a 1 + n a a i ⩽ n + i − 1 for i = 1 , 2 , … , n \htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{a}_{\htmlData{tutor-start=6,tutor-end=7}{i}}} \htmlData{tutor-start=10,tutor-end=20}{\leqslant }\htmlData{tutor-start=20,tutor-end=21}{n}\htmlData{tutor-start=21,tutor-end=22}{+}\htmlData{tutor-start=22,tutor-end=23}{i}\htmlData{tutor-start=23,tutor-end=24}{-}\htmlData{tutor-start=24,tutor-end=25}{1} \quad \text { \htmlData{tutor-start=40,tutor-end=41}{f}\htmlData{tutor-start=41,tutor-end=42}{o}\htmlData{tutor-start=42,tutor-end=43}{r} } \htmlData{tutor-start=46,tutor-end=47}{i}\htmlData{tutor-start=47,tutor-end=48}{=}\htmlData{tutor-start=48,tutor-end=49}{1}\htmlData{tutor-start=49,tutor-end=50}{,}\htmlData{tutor-start=50,tutor-end=51}{2}\htmlData{tutor-start=51,tutor-end=52}{,} \htmlData{tutor-start=53,tutor-end=59}{\ldots}\htmlData{tutor-start=59,tutor-end=60}{,} \htmlData{tutor-start=61,tutor-end=62}{n} a a i ⩽ n + i − 1 f o r i = 1 , 2 , … , n a 1 + ⋯ + a n ⩽ n 2 . \htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{+}\cdots\htmlData{tutor-start=12,tutor-end=13}{+}\htmlData{tutor-start=13,tutor-end=14}{a}_{\htmlData{tutor-start=16,tutor-end=17}{n}} \htmlData{tutor-start=19,tutor-end=29}{\leqslant }\htmlData{tutor-start=29,tutor-end=30}{n}^{\htmlData{tutor-start=32,tutor-end=33}{2}} \htmlData{tutor-start=35,tutor-end=36}{.} a 1 + ⋯ + a n ⩽ n 2 . 题解状态： 标准答案与规范题解待补充
题目标签：2013 IMO Shortlist A4
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2013-A5 2013 · Algebra · IMO/代数
Let Z ⩾ 0 \mathbb{\htmlData{tutor-start=8,tutor-end=9}{Z}}_{\htmlData{tutor-start=12,tutor-end=22}{\geqslant }\htmlData{tutor-start=22,tutor-end=23}{0}} Z ⩾ 0 f : Z ⩾ 0 → Z ⩾ 0 \htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{:} \mathbb{\htmlData{tutor-start=11,tutor-end=12}{Z}}_{\htmlData{tutor-start=15,tutor-end=25}{\geqslant }\htmlData{tutor-start=25,tutor-end=26}{0}} \htmlData{tutor-start=28,tutor-end=40}{\rightarrow }\mathbb{\htmlData{tutor-start=48,tutor-end=49}{Z}}_{\htmlData{tutor-start=52,tutor-end=62}{\geqslant }\htmlData{tutor-start=62,tutor-end=63}{0}} f : Z ⩾ 0 → Z ⩾ 0 f ( f ( f ( n ) ) ) = f ( n + 1 ) + 1 \htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{f}\htmlData{tutor-start=3,tutor-end=4}{(}\htmlData{tutor-start=4,tutor-end=5}{f}\htmlData{tutor-start=5,tutor-end=6}{(}\htmlData{tutor-start=6,tutor-end=7}{n}\htmlData{tutor-start=7,tutor-end=8}{)}\htmlData{tutor-start=8,tutor-end=9}{)}\htmlData{tutor-start=9,tutor-end=10}{)}\htmlData{tutor-start=10,tutor-end=11}{=}\htmlData{tutor-start=11,tutor-end=12}{f}\htmlData{tutor-start=12,tutor-end=13}{(}\htmlData{tutor-start=13,tutor-end=14}{n}\htmlData{tutor-start=14,tutor-end=15}{+}\htmlData{tutor-start=15,tutor-end=16}{1}\htmlData{tutor-start=16,tutor-end=17}{)}\htmlData{tutor-start=17,tutor-end=18}{+}\htmlData{tutor-start=18,tutor-end=19}{1} f ( f ( f ( n ) ) ) = f ( n + 1 ) + 1 n ∈ Z ⩾ 0 \htmlData{tutor-start=0,tutor-end=1}{n} \htmlData{tutor-start=2,tutor-end=6}{\in }\mathbb{\htmlData{tutor-start=14,tutor-end=15}{Z}}_{\htmlData{tutor-start=18,tutor-end=28}{\geqslant }\htmlData{tutor-start=28,tutor-end=29}{0}} n ∈ Z ⩾ 0 题解状态： 标准答案与规范题解待补充
题目标签：2013 IMO Shortlist A5
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2013-A6 2013 · Algebra · IMO/代数
Let m ≠ 0 \htmlData{tutor-start=0,tutor-end=1}{m} \neq \htmlData{tutor-start=7,tutor-end=8}{0} m = 0 P ( x ) \htmlData{tutor-start=0,tutor-end=1}{P}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{x}\htmlData{tutor-start=3,tutor-end=4}{)} P ( x ) ( x 3 − m x 2 + 1 ) P ( x + 1 ) + ( x 3 + m x 2 + 1 ) P ( x − 1 ) = 2 ( x 3 − m x + 1 ) P ( x ) \left(\htmlData{tutor-start=6,tutor-end=7}{x}^{\htmlData{tutor-start=9,tutor-end=10}{3}}\htmlData{tutor-start=11,tutor-end=12}{-}\htmlData{tutor-start=12,tutor-end=13}{m} \htmlData{tutor-start=14,tutor-end=15}{x}^{\htmlData{tutor-start=17,tutor-end=18}{2}}\htmlData{tutor-start=19,tutor-end=20}{+}\htmlData{tutor-start=20,tutor-end=21}{1}\right) \htmlData{tutor-start=29,tutor-end=30}{P}\htmlData{tutor-start=30,tutor-end=31}{(}\htmlData{tutor-start=31,tutor-end=32}{x}\htmlData{tutor-start=32,tutor-end=33}{+}\htmlData{tutor-start=33,tutor-end=34}{1}\htmlData{tutor-start=34,tutor-end=35}{)}\htmlData{tutor-start=35,tutor-end=36}{+}\left(\htmlData{tutor-start=42,tutor-end=43}{x}^{\htmlData{tutor-start=45,tutor-end=46}{3}}\htmlData{tutor-start=47,tutor-end=48}{+}\htmlData{tutor-start=48,tutor-end=49}{m} \htmlData{tutor-start=50,tutor-end=51}{x}^{\htmlData{tutor-start=53,tutor-end=54}{2}}\htmlData{tutor-start=55,tutor-end=56}{+}\htmlData{tutor-start=56,tutor-end=57}{1}\right) \htmlData{tutor-start=65,tutor-end=66}{P}\htmlData{tutor-start=66,tutor-end=67}{(}\htmlData{tutor-start=67,tutor-end=68}{x}\htmlData{tutor-start=68,tutor-end=69}{-}\htmlData{tutor-start=69,tutor-end=70}{1}\htmlData{tutor-start=70,tutor-end=71}{)}\htmlData{tutor-start=71,tutor-end=72}{=}\htmlData{tutor-start=72,tutor-end=73}{2}\left(\htmlData{tutor-start=79,tutor-end=80}{x}^{\htmlData{tutor-start=82,tutor-end=83}{3}}\htmlData{tutor-start=84,tutor-end=85}{-}\htmlData{tutor-start=85,tutor-end=86}{m} \htmlData{tutor-start=87,tutor-end=88}{x}\htmlData{tutor-start=88,tutor-end=89}{+}\htmlData{tutor-start=89,tutor-end=90}{1}\right) \htmlData{tutor-start=98,tutor-end=99}{P}\htmlData{tutor-start=99,tutor-end=100}{(}\htmlData{tutor-start=100,tutor-end=101}{x}\htmlData{tutor-start=101,tutor-end=102}{)} ( x 3 − m x 2 + 1 ) P ( x + 1 ) + ( x 3 + m x 2 + 1 ) P ( x − 1 ) = 2 ( x 3 − m x + 1 ) P ( x ) x \htmlData{tutor-start=0,tutor-end=1}{x} x 题解状态： 标准答案与规范题解待补充
题目标签：2013 IMO Shortlist A6
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2014-A1 2014 · Algebra · IMO/代数
Let z 0 < z 1 < z 2 < ⋯ \htmlData{tutor-start=0,tutor-end=1}{z}_{\htmlData{tutor-start=3,tutor-end=4}{0}}\htmlData{tutor-start=5,tutor-end=6}{<}\htmlData{tutor-start=6,tutor-end=7}{z}_{\htmlData{tutor-start=9,tutor-end=10}{1}}\htmlData{tutor-start=11,tutor-end=12}{<}\htmlData{tutor-start=12,tutor-end=13}{z}_{\htmlData{tutor-start=15,tutor-end=16}{2}}\htmlData{tutor-start=17,tutor-end=18}{<}\cdots z 0 < z 1 < z 2 < ⋯ n ⩾ 1 \htmlData{tutor-start=0,tutor-end=1}{n} \htmlData{tutor-start=2,tutor-end=12}{\geqslant }\htmlData{tutor-start=12,tutor-end=13}{1} n ⩾ 1 z n < z 0 + z 1 + ⋯ + z n n ⩽ z n + 1 \htmlData{tutor-start=0,tutor-end=1}{z}_{\htmlData{tutor-start=3,tutor-end=4}{n}}\htmlData{tutor-start=5,tutor-end=6}{<}\frac{\htmlData{tutor-start=12,tutor-end=13}{z}_{\htmlData{tutor-start=15,tutor-end=16}{0}}\htmlData{tutor-start=17,tutor-end=18}{+}\htmlData{tutor-start=18,tutor-end=19}{z}_{\htmlData{tutor-start=21,tutor-end=22}{1}}\htmlData{tutor-start=23,tutor-end=24}{+}\cdots\htmlData{tutor-start=30,tutor-end=31}{+}\htmlData{tutor-start=31,tutor-end=32}{z}_{\htmlData{tutor-start=34,tutor-end=35}{n}}}{\htmlData{tutor-start=38,tutor-end=39}{n}} \htmlData{tutor-start=41,tutor-end=51}{\leqslant }\htmlData{tutor-start=51,tutor-end=52}{z}_{\htmlData{tutor-start=54,tutor-end=55}{n}\htmlData{tutor-start=55,tutor-end=56}{+}\htmlData{tutor-start=56,tutor-end=57}{1}} z n < n z 0 + z 1 + ⋯ + z n ⩽ z n + 1 题解状态： 标准答案与规范题解待补充
题目标签：2014 IMO 正式题 A1
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2014-A2 2014 · Algebra · IMO/代数
Define the function f : ( 0 , 1 ) → ( 0 , 1 ) \htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{:}\htmlData{tutor-start=2,tutor-end=3}{(}\htmlData{tutor-start=3,tutor-end=4}{0}\htmlData{tutor-start=4,tutor-end=5}{,}\htmlData{tutor-start=5,tutor-end=6}{1}\htmlData{tutor-start=6,tutor-end=7}{)} \htmlData{tutor-start=8,tutor-end=19}{\rightarrow}\htmlData{tutor-start=19,tutor-end=20}{(}\htmlData{tutor-start=20,tutor-end=21}{0}\htmlData{tutor-start=21,tutor-end=22}{,}\htmlData{tutor-start=22,tutor-end=23}{1}\htmlData{tutor-start=23,tutor-end=24}{)} f : ( 0 , 1 ) → ( 0 , 1 ) f ( x ) = { x + 1 2 if x < 1 2 x 2 if x ⩾ 1 2 \htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{x}\htmlData{tutor-start=3,tutor-end=4}{)}\htmlData{tutor-start=4,tutor-end=5}{=} \begin{cases}\htmlData{tutor-start=19,tutor-end=20}{x}\htmlData{tutor-start=20,tutor-end=21}{+}\frac{\htmlData{tutor-start=27,tutor-end=28}{1}}{\htmlData{tutor-start=30,tutor-end=31}{2}} & \text { \htmlData{tutor-start=43,tutor-end=44}{i}\htmlData{tutor-start=44,tutor-end=45}{f} } \htmlData{tutor-start=48,tutor-end=49}{x}\htmlData{tutor-start=49,tutor-end=50}{<}\frac{\htmlData{tutor-start=56,tutor-end=57}{1}}{\htmlData{tutor-start=59,tutor-end=60}{2}} \\ \htmlData{tutor-start=65,tutor-end=66}{x}^{\htmlData{tutor-start=68,tutor-end=69}{2}} & \text { \htmlData{tutor-start=81,tutor-end=82}{i}\htmlData{tutor-start=82,tutor-end=83}{f} } \htmlData{tutor-start=86,tutor-end=87}{x} \htmlData{tutor-start=88,tutor-end=98}{\geqslant }\frac{\htmlData{tutor-start=104,tutor-end=105}{1}}{\htmlData{tutor-start=107,tutor-end=108}{2}}\end{cases} f ( x ) = { x + 2 1 x 2 i f x < 2 1 i f x ⩾ 2 1 a \htmlData{tutor-start=0,tutor-end=1}{a} a b \htmlData{tutor-start=0,tutor-end=1}{b} b 0 < a < b < 1 \htmlData{tutor-start=0,tutor-end=1}{0}\htmlData{tutor-start=1,tutor-end=2}{<}\htmlData{tutor-start=2,tutor-end=3}{a}\htmlData{tutor-start=3,tutor-end=4}{<}\htmlData{tutor-start=4,tutor-end=5}{b}\htmlData{tutor-start=5,tutor-end=6}{<}\htmlData{tutor-start=6,tutor-end=7}{1} 0 < a < b < 1 a n \htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{n}} a n b n \htmlData{tutor-start=0,tutor-end=1}{b}_{\htmlData{tutor-start=3,tutor-end=4}{n}} b n a 0 = a , b 0 = b \htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{0}}\htmlData{tutor-start=5,tutor-end=6}{=}\htmlData{tutor-start=6,tutor-end=7}{a}\htmlData{tutor-start=7,tutor-end=8}{,} \htmlData{tutor-start=9,tutor-end=10}{b}_{\htmlData{tutor-start=12,tutor-end=13}{0}}\htmlData{tutor-start=14,tutor-end=15}{=}\htmlData{tutor-start=15,tutor-end=16}{b} a 0 = a , b 0 = b a n = f ( a n − 1 ) , b n = f ( b n − 1 ) \htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{n}}\htmlData{tutor-start=5,tutor-end=6}{=}\htmlData{tutor-start=6,tutor-end=7}{f}\left(\htmlData{tutor-start=13,tutor-end=14}{a}_{\htmlData{tutor-start=16,tutor-end=17}{n}\htmlData{tutor-start=17,tutor-end=18}{-}\htmlData{tutor-start=18,tutor-end=19}{1}}\right)\htmlData{tutor-start=27,tutor-end=28}{,} \htmlData{tutor-start=29,tutor-end=30}{b}_{\htmlData{tutor-start=32,tutor-end=33}{n}}\htmlData{tutor-start=34,tutor-end=35}{=}\htmlData{tutor-start=35,tutor-end=36}{f}\left(\htmlData{tutor-start=42,tutor-end=43}{b}_{\htmlData{tutor-start=45,tutor-end=46}{n}\htmlData{tutor-start=46,tutor-end=47}{-}\htmlData{tutor-start=47,tutor-end=48}{1}}\right) a n = f ( a n − 1 ) , b n = f ( b n − 1 ) n > 0 \htmlData{tutor-start=0,tutor-end=1}{n}\htmlData{tutor-start=1,tutor-end=2}{>}\htmlData{tutor-start=2,tutor-end=3}{0} n > 0 n \htmlData{tutor-start=0,tutor-end=1}{n} n ( a n − a n − 1 ) ( b n − b n − 1 ) < 0. \left(\htmlData{tutor-start=6,tutor-end=7}{a}_{\htmlData{tutor-start=9,tutor-end=10}{n}}\htmlData{tutor-start=11,tutor-end=12}{-}\htmlData{tutor-start=12,tutor-end=13}{a}_{\htmlData{tutor-start=15,tutor-end=16}{n}\htmlData{tutor-start=16,tutor-end=17}{-}\htmlData{tutor-start=17,tutor-end=18}{1}}\right)\left(\htmlData{tutor-start=32,tutor-end=33}{b}_{\htmlData{tutor-start=35,tutor-end=36}{n}}\htmlData{tutor-start=37,tutor-end=38}{-}\htmlData{tutor-start=38,tutor-end=39}{b}_{\htmlData{tutor-start=41,tutor-end=42}{n}\htmlData{tutor-start=42,tutor-end=43}{-}\htmlData{tutor-start=43,tutor-end=44}{1}}\right)\htmlData{tutor-start=52,tutor-end=53}{<}\htmlData{tutor-start=53,tutor-end=54}{0} \htmlData{tutor-start=55,tutor-end=56}{.} ( a n − a n − 1 ) ( b n − b n − 1 ) < 0 . 题解状态： 标准答案与规范题解待补充
题目标签：2014 IMO Shortlist A2
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2014-A3 2014 · Algebra · IMO/代数
For a sequence x 1 , x 2 , … , x n \htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{x}_{\htmlData{tutor-start=10,tutor-end=11}{2}}\htmlData{tutor-start=12,tutor-end=13}{,} \htmlData{tutor-start=14,tutor-end=20}{\ldots}\htmlData{tutor-start=20,tutor-end=21}{,} \htmlData{tutor-start=22,tutor-end=23}{x}_{\htmlData{tutor-start=25,tutor-end=26}{n}} x 1 , x 2 , … , x n max 1 ⩽ i ⩽ n ∣ x 1 + ⋯ + x i ∣ \htmlData{tutor-start=0,tutor-end=7}{\max _{}\htmlData{tutor-start=7,tutor-end=8}{1} \htmlData{tutor-start=9,tutor-end=19}{\leqslant }\htmlData{tutor-start=19,tutor-end=20}{i} \htmlData{tutor-start=21,tutor-end=31}{\leqslant }\htmlData{tutor-start=31,tutor-end=32}{n}}\left|\htmlData{tutor-start=39,tutor-end=40}{x}_{\htmlData{tutor-start=42,tutor-end=43}{1}}\htmlData{tutor-start=44,tutor-end=45}{+}\cdots\htmlData{tutor-start=51,tutor-end=52}{+}\htmlData{tutor-start=52,tutor-end=53}{x}_{\htmlData{tutor-start=55,tutor-end=56}{i}}\right| max 1 ⩽ i ⩽ n ∣ x 1 + ⋯ + x i ∣ n \htmlData{tutor-start=0,tutor-end=1}{n} n D \htmlData{tutor-start=0,tutor-end=1}{D} D x 1 \htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{1}} x 1 ∣ x 1 ∣ \left|\htmlData{tutor-start=6,tutor-end=7}{x}_{\htmlData{tutor-start=9,tutor-end=10}{1}}\right| ∣ x 1 ∣ x 2 \htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{2}} x 2 ∣ x 1 + x 2 ∣ \left|\htmlData{tutor-start=6,tutor-end=7}{x}_{\htmlData{tutor-start=9,tutor-end=10}{1}}\htmlData{tutor-start=11,tutor-end=12}{+}\htmlData{tutor-start=12,tutor-end=13}{x}_{\htmlData{tutor-start=15,tutor-end=16}{2}}\right| ∣ x 1 + x 2 ∣ i th \htmlData{tutor-start=0,tutor-end=1}{i}^{\text {\htmlData{tutor-start=10,tutor-end=11}{t}\htmlData{tutor-start=11,tutor-end=12}{h} }} i t h x i \htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{i}} x i ∣ x 1 + x 2 + ⋯ + x i ∣ \left|x_{1}+x_{2}+\cdots+x_{i}\right| ∣ x 1 + x 2 + ⋯ + x i ∣ G \htmlData{tutor-start=0,tutor-end=1}{G} G c \htmlData{tutor-start=0,tutor-end=1}{c} c n \htmlData{tutor-start=0,tutor-end=1}{n} n n \htmlData{tutor-start=0,tutor-end=1}{n} n G ⩽ c D \htmlData{tutor-start=0,tutor-end=1}{G} \htmlData{tutor-start=2,tutor-end=12}{\leqslant }\htmlData{tutor-start=12,tutor-end=13}{c} \htmlData{tutor-start=14,tutor-end=15}{D} G ⩽ c D 题解状态： 标准答案与规范题解待补充
题目标签：2014 IMO Shortlist A3
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2014-A4 2014 · Algebra · IMO/代数
Determine all functions f : Z → Z \htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{:} \mathbb{\htmlData{tutor-start=11,tutor-end=12}{Z}} \htmlData{tutor-start=14,tutor-end=26}{\rightarrow }\mathbb{\htmlData{tutor-start=34,tutor-end=35}{Z}} f : Z → Z f ( f ( m ) + n ) + f ( m ) = f ( n ) + f ( 3 m ) + 2014 \htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{f}\htmlData{tutor-start=3,tutor-end=4}{(}\htmlData{tutor-start=4,tutor-end=5}{m}\htmlData{tutor-start=5,tutor-end=6}{)}\htmlData{tutor-start=6,tutor-end=7}{+}\htmlData{tutor-start=7,tutor-end=8}{n}\htmlData{tutor-start=8,tutor-end=9}{)}\htmlData{tutor-start=9,tutor-end=10}{+}\htmlData{tutor-start=10,tutor-end=11}{f}\htmlData{tutor-start=11,tutor-end=12}{(}\htmlData{tutor-start=12,tutor-end=13}{m}\htmlData{tutor-start=13,tutor-end=14}{)}\htmlData{tutor-start=14,tutor-end=15}{=}\htmlData{tutor-start=15,tutor-end=16}{f}\htmlData{tutor-start=16,tutor-end=17}{(}\htmlData{tutor-start=17,tutor-end=18}{n}\htmlData{tutor-start=18,tutor-end=19}{)}\htmlData{tutor-start=19,tutor-end=20}{+}\htmlData{tutor-start=20,tutor-end=21}{f}\htmlData{tutor-start=21,tutor-end=22}{(}\htmlData{tutor-start=22,tutor-end=23}{3} \htmlData{tutor-start=24,tutor-end=25}{m}\htmlData{tutor-start=25,tutor-end=26}{)}\htmlData{tutor-start=26,tutor-end=27}{+}\htmlData{tutor-start=27,tutor-end=28}{2}\htmlData{tutor-start=28,tutor-end=29}{0}\htmlData{tutor-start=29,tutor-end=30}{1}\htmlData{tutor-start=30,tutor-end=31}{4} f ( f ( m ) + n ) + f ( m ) = f ( n ) + f ( 3 m ) + 2 0 1 4 m \htmlData{tutor-start=0,tutor-end=1}{m} m n \htmlData{tutor-start=0,tutor-end=1}{n} n 题解状态： 标准答案与规范题解待补充
题目标签：2014 IMO Shortlist A4
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2014-A5 2014 · Algebra · IMO/代数
Consider all polynomials P ( x ) \htmlData{tutor-start=0,tutor-end=1}{P}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{x}\htmlData{tutor-start=3,tutor-end=4}{)} P ( x ) x \htmlData{tutor-start=0,tutor-end=1}{x} x y \htmlData{tutor-start=0,tutor-end=1}{y} y ∣ y 2 − P ( x ) ∣ ⩽ 2 ∣ x ∣ if and only if ∣ x 2 − P ( y ) ∣ ⩽ 2 ∣ y ∣ \left|\htmlData{tutor-start=6,tutor-end=7}{y}^{\htmlData{tutor-start=9,tutor-end=10}{2}}\htmlData{tutor-start=11,tutor-end=12}{-}\htmlData{tutor-start=12,tutor-end=13}{P}\htmlData{tutor-start=13,tutor-end=14}{(}\htmlData{tutor-start=14,tutor-end=15}{x}\htmlData{tutor-start=15,tutor-end=16}{)}\right| \htmlData{tutor-start=24,tutor-end=34}{\leqslant }\htmlData{tutor-start=34,tutor-end=35}{2}\htmlData{tutor-start=35,tutor-end=36}{|}\htmlData{tutor-start=36,tutor-end=37}{x}\htmlData{tutor-start=37,tutor-end=38}{|} \quad \text { \htmlData{tutor-start=53,tutor-end=54}{i}\htmlData{tutor-start=54,tutor-end=55}{f} \htmlData{tutor-start=56,tutor-end=57}{a}\htmlData{tutor-start=57,tutor-end=58}{n}\htmlData{tutor-start=58,tutor-end=59}{d} \htmlData{tutor-start=60,tutor-end=61}{o}\htmlData{tutor-start=61,tutor-end=62}{n}\htmlData{tutor-start=62,tutor-end=63}{l}\htmlData{tutor-start=63,tutor-end=64}{y} \htmlData{tutor-start=65,tutor-end=66}{i}\htmlData{tutor-start=66,tutor-end=67}{f} } \quad\left|\htmlData{tutor-start=81,tutor-end=82}{x}^{\htmlData{tutor-start=84,tutor-end=85}{2}}\htmlData{tutor-start=86,tutor-end=87}{-}\htmlData{tutor-start=87,tutor-end=88}{P}\htmlData{tutor-start=88,tutor-end=89}{(}\htmlData{tutor-start=89,tutor-end=90}{y}\htmlData{tutor-start=90,tutor-end=91}{)}\right| \htmlData{tutor-start=99,tutor-end=109}{\leqslant }\htmlData{tutor-start=109,tutor-end=110}{2}\htmlData{tutor-start=110,tutor-end=111}{|}\htmlData{tutor-start=111,tutor-end=112}{y}\htmlData{tutor-start=112,tutor-end=113}{|} y 2 − P ( x ) ⩽ 2 ∣ x ∣ i f a n d o n l y i f x 2 − P ( y ) ⩽ 2 ∣ y ∣ P ( 0 ) \htmlData{tutor-start=0,tutor-end=1}{P}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{0}\htmlData{tutor-start=3,tutor-end=4}{)} P ( 0 ) 题解状态： 标准答案与规范题解待补充
题目标签：2014 IMO Shortlist A5
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2014-A6 2014 · Algebra · IMO/代数
Find all functions f : Z → Z \htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{:} \mathbb{\htmlData{tutor-start=11,tutor-end=12}{Z}} \htmlData{tutor-start=14,tutor-end=26}{\rightarrow }\mathbb{\htmlData{tutor-start=34,tutor-end=35}{Z}} f : Z → Z n 2 + 4 f ( n ) = f ( f ( n ) ) 2 \htmlData{tutor-start=0,tutor-end=1}{n}^{\htmlData{tutor-start=3,tutor-end=4}{2}}\htmlData{tutor-start=5,tutor-end=6}{+}\htmlData{tutor-start=6,tutor-end=7}{4} \htmlData{tutor-start=8,tutor-end=9}{f}\htmlData{tutor-start=9,tutor-end=10}{(}\htmlData{tutor-start=10,tutor-end=11}{n}\htmlData{tutor-start=11,tutor-end=12}{)}\htmlData{tutor-start=12,tutor-end=13}{=}\htmlData{tutor-start=13,tutor-end=14}{f}\htmlData{tutor-start=14,tutor-end=15}{(}\htmlData{tutor-start=15,tutor-end=16}{f}\htmlData{tutor-start=16,tutor-end=17}{(}\htmlData{tutor-start=17,tutor-end=18}{n}\htmlData{tutor-start=18,tutor-end=19}{)}\htmlData{tutor-start=19,tutor-end=20}{)}^{\htmlData{tutor-start=22,tutor-end=23}{2}} n 2 + 4 f ( n ) = f ( f ( n ) ) 2 n ∈ Z \htmlData{tutor-start=0,tutor-end=1}{n} \htmlData{tutor-start=2,tutor-end=6}{\in }\mathbb{\htmlData{tutor-start=14,tutor-end=15}{Z}} n ∈ Z 题解状态： 标准答案与规范题解待补充
题目标签：2014 IMO Shortlist A6
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2015-A1 2015 · Algebra · IMO/代数
Suppose that a sequence a 1 , a 2 , … \htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{a}_{\htmlData{tutor-start=10,tutor-end=11}{2}}\htmlData{tutor-start=12,tutor-end=13}{,} \htmlData{tutor-start=14,tutor-end=20}{\ldots} a 1 , a 2 , … a k + 1 ⩾ k a k a k 2 + ( k − 1 ) \htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{k}\htmlData{tutor-start=4,tutor-end=5}{+}\htmlData{tutor-start=5,tutor-end=6}{1}} \htmlData{tutor-start=8,tutor-end=18}{\geqslant }\frac{\htmlData{tutor-start=24,tutor-end=25}{k} \htmlData{tutor-start=26,tutor-end=27}{a}_{\htmlData{tutor-start=29,tutor-end=30}{k}}}{\htmlData{tutor-start=33,tutor-end=34}{a}_{\htmlData{tutor-start=36,tutor-end=37}{k}}^{\htmlData{tutor-start=40,tutor-end=41}{2}}\htmlData{tutor-start=42,tutor-end=43}{+}\htmlData{tutor-start=43,tutor-end=44}{(}\htmlData{tutor-start=44,tutor-end=45}{k}\htmlData{tutor-start=45,tutor-end=46}{-}\htmlData{tutor-start=46,tutor-end=47}{1}\htmlData{tutor-start=47,tutor-end=48}{)}} a k + 1 ⩾ a k 2 + ( k − 1 ) k a k k \htmlData{tutor-start=0,tutor-end=1}{k} k a 1 + a 2 + ⋯ + a n ⩾ n \htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{+}\htmlData{tutor-start=6,tutor-end=7}{a}_{\htmlData{tutor-start=9,tutor-end=10}{2}}\htmlData{tutor-start=11,tutor-end=12}{+}\cdots\htmlData{tutor-start=18,tutor-end=19}{+}\htmlData{tutor-start=19,tutor-end=20}{a}_{\htmlData{tutor-start=22,tutor-end=23}{n}} \htmlData{tutor-start=25,tutor-end=35}{\geqslant }\htmlData{tutor-start=35,tutor-end=36}{n} a 1 + a 2 + ⋯ + a n ⩾ n n ⩾ 2 \htmlData{tutor-start=0,tutor-end=1}{n} \htmlData{tutor-start=2,tutor-end=12}{\geqslant }\htmlData{tutor-start=12,tutor-end=13}{2} n ⩾ 2 题解状态： 标准答案与规范题解待补充
题目标签：2015 IMO Shortlist A1
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2015-A2 2015 · Algebra · IMO/代数
Determine all functions f : Z → Z \htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{:} \mathbb{\htmlData{tutor-start=11,tutor-end=12}{Z}} \htmlData{tutor-start=14,tutor-end=26}{\rightarrow }\mathbb{\htmlData{tutor-start=34,tutor-end=35}{Z}} f : Z → Z f ( x − f ( y ) ) = f ( f ( x ) ) − f ( y ) − 1 \htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{x}\htmlData{tutor-start=3,tutor-end=4}{-}\htmlData{tutor-start=4,tutor-end=5}{f}\htmlData{tutor-start=5,tutor-end=6}{(}\htmlData{tutor-start=6,tutor-end=7}{y}\htmlData{tutor-start=7,tutor-end=8}{)}\htmlData{tutor-start=8,tutor-end=9}{)}\htmlData{tutor-start=9,tutor-end=10}{=}\htmlData{tutor-start=10,tutor-end=11}{f}\htmlData{tutor-start=11,tutor-end=12}{(}\htmlData{tutor-start=12,tutor-end=13}{f}\htmlData{tutor-start=13,tutor-end=14}{(}\htmlData{tutor-start=14,tutor-end=15}{x}\htmlData{tutor-start=15,tutor-end=16}{)}\htmlData{tutor-start=16,tutor-end=17}{)}\htmlData{tutor-start=17,tutor-end=18}{-}\htmlData{tutor-start=18,tutor-end=19}{f}\htmlData{tutor-start=19,tutor-end=20}{(}\htmlData{tutor-start=20,tutor-end=21}{y}\htmlData{tutor-start=21,tutor-end=22}{)}\htmlData{tutor-start=22,tutor-end=23}{-}\htmlData{tutor-start=23,tutor-end=24}{1} f ( x − f ( y ) ) = f ( f ( x ) ) − f ( y ) − 1 x , y ∈ Z \htmlData{tutor-start=0,tutor-end=1}{x}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{y} \htmlData{tutor-start=5,tutor-end=9}{\in }\mathbb{\htmlData{tutor-start=17,tutor-end=18}{Z}} x , y ∈ Z 题解状态： 标准答案与规范题解待补充
题目标签：2015 IMO Shortlist A2
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2015-A3 2015 · Algebra · IMO/代数
Let n \htmlData{tutor-start=0,tutor-end=1}{n} n ∑ 1 ⩽ r < s ⩽ 2 n ( s − r − n ) x r x s \sum_{\htmlData{tutor-start=6,tutor-end=7}{1} \htmlData{tutor-start=8,tutor-end=18}{\leqslant }\htmlData{tutor-start=18,tutor-end=19}{r}\htmlData{tutor-start=19,tutor-end=20}{<}\htmlData{tutor-start=20,tutor-end=21}{s} \htmlData{tutor-start=22,tutor-end=32}{\leqslant }\htmlData{tutor-start=32,tutor-end=33}{2} \htmlData{tutor-start=34,tutor-end=35}{n}}\htmlData{tutor-start=36,tutor-end=37}{(}\htmlData{tutor-start=37,tutor-end=38}{s}\htmlData{tutor-start=38,tutor-end=39}{-}\htmlData{tutor-start=39,tutor-end=40}{r}\htmlData{tutor-start=40,tutor-end=41}{-}\htmlData{tutor-start=41,tutor-end=42}{n}\htmlData{tutor-start=42,tutor-end=43}{)} \htmlData{tutor-start=44,tutor-end=45}{x}_{\htmlData{tutor-start=47,tutor-end=48}{r}} \htmlData{tutor-start=50,tutor-end=51}{x}_{\htmlData{tutor-start=53,tutor-end=54}{s}} ∑ 1 ⩽ r < s ⩽ 2 n ( s − r − n ) x r x s − 1 ⩽ x i ⩽ 1 \htmlData{tutor-start=0,tutor-end=1}{-}\htmlData{tutor-start=1,tutor-end=2}{1} \htmlData{tutor-start=3,tutor-end=13}{\leqslant }\htmlData{tutor-start=13,tutor-end=14}{x}_{\htmlData{tutor-start=16,tutor-end=17}{i}} \htmlData{tutor-start=19,tutor-end=29}{\leqslant }\htmlData{tutor-start=29,tutor-end=30}{1} − 1 ⩽ x i ⩽ 1 i = 1 , 2 , … , 2 n \htmlData{tutor-start=0,tutor-end=1}{i}\htmlData{tutor-start=1,tutor-end=2}{=}\htmlData{tutor-start=2,tutor-end=3}{1}\htmlData{tutor-start=3,tutor-end=4}{,}\htmlData{tutor-start=4,tutor-end=5}{2}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=13}{\ldots}\htmlData{tutor-start=13,tutor-end=14}{,} \htmlData{tutor-start=15,tutor-end=16}{2} \htmlData{tutor-start=17,tutor-end=18}{n} i = 1 , 2 , … , 2 n 题解状态： 标准答案与规范题解待补充
题目标签：2015 IMO Shortlist A3
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2015-A4 2015 · Algebra · IMO/代数
Find all functions f : R → R \htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{:} \mathbb{\htmlData{tutor-start=11,tutor-end=12}{R}} \htmlData{tutor-start=14,tutor-end=26}{\rightarrow }\mathbb{\htmlData{tutor-start=34,tutor-end=35}{R}} f : R → R f ( x + f ( x + y ) ) + f ( x y ) = x + f ( x + y ) + y f ( x ) \htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{x}\htmlData{tutor-start=3,tutor-end=4}{+}\htmlData{tutor-start=4,tutor-end=5}{f}\htmlData{tutor-start=5,tutor-end=6}{(}\htmlData{tutor-start=6,tutor-end=7}{x}\htmlData{tutor-start=7,tutor-end=8}{+}\htmlData{tutor-start=8,tutor-end=9}{y}\htmlData{tutor-start=9,tutor-end=10}{)}\htmlData{tutor-start=10,tutor-end=11}{)}\htmlData{tutor-start=11,tutor-end=12}{+}\htmlData{tutor-start=12,tutor-end=13}{f}\htmlData{tutor-start=13,tutor-end=14}{(}\htmlData{tutor-start=14,tutor-end=15}{x} \htmlData{tutor-start=16,tutor-end=17}{y}\htmlData{tutor-start=17,tutor-end=18}{)}\htmlData{tutor-start=18,tutor-end=19}{=}\htmlData{tutor-start=19,tutor-end=20}{x}\htmlData{tutor-start=20,tutor-end=21}{+}\htmlData{tutor-start=21,tutor-end=22}{f}\htmlData{tutor-start=22,tutor-end=23}{(}\htmlData{tutor-start=23,tutor-end=24}{x}\htmlData{tutor-start=24,tutor-end=25}{+}\htmlData{tutor-start=25,tutor-end=26}{y}\htmlData{tutor-start=26,tutor-end=27}{)}\htmlData{tutor-start=27,tutor-end=28}{+}\htmlData{tutor-start=28,tutor-end=29}{y} \htmlData{tutor-start=30,tutor-end=31}{f}\htmlData{tutor-start=31,tutor-end=32}{(}\htmlData{tutor-start=32,tutor-end=33}{x}\htmlData{tutor-start=33,tutor-end=34}{)} f ( x + f ( x + y ) ) + f ( x y ) = x + f ( x + y ) + y f ( x ) x \htmlData{tutor-start=0,tutor-end=1}{x} x y \htmlData{tutor-start=0,tutor-end=1}{y} y 题解状态： 标准答案与规范题解待补充
题目标签：2015 IMO 正式题 A4
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2015-A5 2015 · Algebra · IMO/代数
Let 2 Z + 1 \htmlData{tutor-start=0,tutor-end=1}{2} \mathbb{\htmlData{tutor-start=10,tutor-end=11}{Z}}\htmlData{tutor-start=12,tutor-end=13}{+}\htmlData{tutor-start=13,tutor-end=14}{1} 2 Z + 1 f : Z → 2 Z + 1 \htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{:} \mathbb{\htmlData{tutor-start=11,tutor-end=12}{Z}} \htmlData{tutor-start=14,tutor-end=26}{\rightarrow }\htmlData{tutor-start=26,tutor-end=27}{2} \mathbb{\htmlData{tutor-start=36,tutor-end=37}{Z}}\htmlData{tutor-start=38,tutor-end=39}{+}\htmlData{tutor-start=39,tutor-end=40}{1} f : Z → 2 Z + 1 f ( x + f ( x ) + y ) + f ( x − f ( x ) − y ) = f ( x + y ) + f ( x − y ) \htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{x}\htmlData{tutor-start=3,tutor-end=4}{+}\htmlData{tutor-start=4,tutor-end=5}{f}\htmlData{tutor-start=5,tutor-end=6}{(}\htmlData{tutor-start=6,tutor-end=7}{x}\htmlData{tutor-start=7,tutor-end=8}{)}\htmlData{tutor-start=8,tutor-end=9}{+}\htmlData{tutor-start=9,tutor-end=10}{y}\htmlData{tutor-start=10,tutor-end=11}{)}\htmlData{tutor-start=11,tutor-end=12}{+}\htmlData{tutor-start=12,tutor-end=13}{f}\htmlData{tutor-start=13,tutor-end=14}{(}\htmlData{tutor-start=14,tutor-end=15}{x}\htmlData{tutor-start=15,tutor-end=16}{-}\htmlData{tutor-start=16,tutor-end=17}{f}\htmlData{tutor-start=17,tutor-end=18}{(}\htmlData{tutor-start=18,tutor-end=19}{x}\htmlData{tutor-start=19,tutor-end=20}{)}\htmlData{tutor-start=20,tutor-end=21}{-}\htmlData{tutor-start=21,tutor-end=22}{y}\htmlData{tutor-start=22,tutor-end=23}{)}\htmlData{tutor-start=23,tutor-end=24}{=}\htmlData{tutor-start=24,tutor-end=25}{f}\htmlData{tutor-start=25,tutor-end=26}{(}\htmlData{tutor-start=26,tutor-end=27}{x}\htmlData{tutor-start=27,tutor-end=28}{+}\htmlData{tutor-start=28,tutor-end=29}{y}\htmlData{tutor-start=29,tutor-end=30}{)}\htmlData{tutor-start=30,tutor-end=31}{+}\htmlData{tutor-start=31,tutor-end=32}{f}\htmlData{tutor-start=32,tutor-end=33}{(}\htmlData{tutor-start=33,tutor-end=34}{x}\htmlData{tutor-start=34,tutor-end=35}{-}\htmlData{tutor-start=35,tutor-end=36}{y}\htmlData{tutor-start=36,tutor-end=37}{)} f ( x + f ( x ) + y ) + f ( x − f ( x ) − y ) = f ( x + y ) + f ( x − y ) x , y ∈ Z \htmlData{tutor-start=0,tutor-end=1}{x}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{y} \htmlData{tutor-start=5,tutor-end=9}{\in }\mathbb{\htmlData{tutor-start=17,tutor-end=18}{Z}} x , y ∈ Z 题解状态： 标准答案与规范题解待补充
题目标签：2015 IMO Shortlist A5
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2015-A6 2015 · Algebra · IMO/代数
Let n \htmlData{tutor-start=0,tutor-end=1}{n} n n ⩾ 2 \htmlData{tutor-start=0,tutor-end=1}{n} \htmlData{tutor-start=2,tutor-end=12}{\geqslant }\htmlData{tutor-start=12,tutor-end=13}{2} n ⩾ 2 P \htmlData{tutor-start=0,tutor-end=1}{P} P Q \htmlData{tutor-start=0,tutor-end=1}{Q} Q i ∈ { 1 , 2 , … , n } \htmlData{tutor-start=0,tutor-end=1}{i} \htmlData{tutor-start=2,tutor-end=5}{\in}\htmlData{tutor-start=5,tutor-end=7}{\{}\htmlData{tutor-start=7,tutor-end=8}{1}\htmlData{tutor-start=8,tutor-end=9}{,}\htmlData{tutor-start=9,tutor-end=10}{2}\htmlData{tutor-start=10,tutor-end=11}{,} \htmlData{tutor-start=12,tutor-end=18}{\ldots}\htmlData{tutor-start=18,tutor-end=19}{,} \htmlData{tutor-start=20,tutor-end=21}{n}\htmlData{tutor-start=21,tutor-end=23}{\}} i ∈ { 1 , 2 , … , n } P ( 2015 i ) , P ( 2015 i − 1 ) , … , P ( 2015 i − 2014 ) and Q ( 2015 i ) , Q ( 2015 i − 1 ) , … , Q ( 2015 i − 2014 ) \begin{aligned}
& \htmlData{tutor-start=18,tutor-end=19}{P}\htmlData{tutor-start=19,tutor-end=20}{(}\htmlData{tutor-start=20,tutor-end=21}{2}\htmlData{tutor-start=21,tutor-end=22}{0}\htmlData{tutor-start=22,tutor-end=23}{1}\htmlData{tutor-start=23,tutor-end=24}{5} \htmlData{tutor-start=25,tutor-end=26}{i}\htmlData{tutor-start=26,tutor-end=27}{)}\htmlData{tutor-start=27,tutor-end=28}{,} \htmlData{tutor-start=29,tutor-end=30}{P}\htmlData{tutor-start=30,tutor-end=31}{(}\htmlData{tutor-start=31,tutor-end=32}{2}\htmlData{tutor-start=32,tutor-end=33}{0}\htmlData{tutor-start=33,tutor-end=34}{1}\htmlData{tutor-start=34,tutor-end=35}{5} \htmlData{tutor-start=36,tutor-end=37}{i}\htmlData{tutor-start=37,tutor-end=38}{-}\htmlData{tutor-start=38,tutor-end=39}{1}\htmlData{tutor-start=39,tutor-end=40}{)}\htmlData{tutor-start=40,tutor-end=41}{,} \htmlData{tutor-start=42,tutor-end=48}{\ldots}\htmlData{tutor-start=48,tutor-end=49}{,} \htmlData{tutor-start=50,tutor-end=51}{P}\htmlData{tutor-start=51,tutor-end=52}{(}\htmlData{tutor-start=52,tutor-end=53}{2}\htmlData{tutor-start=53,tutor-end=54}{0}\htmlData{tutor-start=54,tutor-end=55}{1}\htmlData{tutor-start=55,tutor-end=56}{5} \htmlData{tutor-start=57,tutor-end=58}{i}\htmlData{tutor-start=58,tutor-end=59}{-}\htmlData{tutor-start=59,tutor-end=60}{2}\htmlData{tutor-start=60,tutor-end=61}{0}\htmlData{tutor-start=61,tutor-end=62}{1}\htmlData{tutor-start=62,tutor-end=63}{4}\htmlData{tutor-start=63,tutor-end=64}{)} \quad \text { \htmlData{tutor-start=79,tutor-end=80}{a}\htmlData{tutor-start=80,tutor-end=81}{n}\htmlData{tutor-start=81,tutor-end=82}{d} } \\
& \htmlData{tutor-start=90,tutor-end=91}{Q}\htmlData{tutor-start=91,tutor-end=92}{(}\htmlData{tutor-start=92,tutor-end=93}{2}\htmlData{tutor-start=93,tutor-end=94}{0}\htmlData{tutor-start=94,tutor-end=95}{1}\htmlData{tutor-start=95,tutor-end=96}{5} \htmlData{tutor-start=97,tutor-end=98}{i}\htmlData{tutor-start=98,tutor-end=99}{)}\htmlData{tutor-start=99,tutor-end=100}{,} \htmlData{tutor-start=101,tutor-end=102}{Q}\htmlData{tutor-start=102,tutor-end=103}{(}\htmlData{tutor-start=103,tutor-end=104}{2}\htmlData{tutor-start=104,tutor-end=105}{0}\htmlData{tutor-start=105,tutor-end=106}{1}\htmlData{tutor-start=106,tutor-end=107}{5} \htmlData{tutor-start=108,tutor-end=109}{i}\htmlData{tutor-start=109,tutor-end=110}{-}\htmlData{tutor-start=110,tutor-end=111}{1}\htmlData{tutor-start=111,tutor-end=112}{)}\htmlData{tutor-start=112,tutor-end=113}{,} \htmlData{tutor-start=114,tutor-end=120}{\ldots}\htmlData{tutor-start=120,tutor-end=121}{,} \htmlData{tutor-start=122,tutor-end=123}{Q}\htmlData{tutor-start=123,tutor-end=124}{(}\htmlData{tutor-start=124,tutor-end=125}{2}\htmlData{tutor-start=125,tutor-end=126}{0}\htmlData{tutor-start=126,tutor-end=127}{1}\htmlData{tutor-start=127,tutor-end=128}{5} \htmlData{tutor-start=129,tutor-end=130}{i}\htmlData{tutor-start=130,tutor-end=131}{-}\htmlData{tutor-start=131,tutor-end=132}{2}\htmlData{tutor-start=132,tutor-end=133}{0}\htmlData{tutor-start=133,tutor-end=134}{1}\htmlData{tutor-start=134,tutor-end=135}{4}\htmlData{tutor-start=135,tutor-end=136}{)}
\end{aligned} P ( 2 0 1 5 i ) , P ( 2 0 1 5 i − 1 ) , … , P ( 2 0 1 5 i − 2 0 1 4 ) a n d Q ( 2 0 1 5 i ) , Q ( 2 0 1 5 i − 1 ) , … , Q ( 2 0 1 5 i − 2 0 1 4 ) n + 1 \htmlData{tutor-start=0,tutor-end=1}{n}\htmlData{tutor-start=1,tutor-end=2}{+}\htmlData{tutor-start=2,tutor-end=3}{1} n + 1 n \htmlData{tutor-start=0,tutor-end=1}{n} n 题解状态： 标准答案与规范题解待补充
题目标签：2015 IMO Shortlist A6
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2016-A1 2016 · Algebra · IMO/代数
Let a , b \htmlData{tutor-start=0,tutor-end=1}{a}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{b} a , b c \htmlData{tutor-start=0,tutor-end=1}{c} c min { a b , b c , c a } ⩾ 1 \min \htmlData{tutor-start=5,tutor-end=7}{\{}\htmlData{tutor-start=7,tutor-end=8}{a} \htmlData{tutor-start=9,tutor-end=10}{b}\htmlData{tutor-start=10,tutor-end=11}{,} \htmlData{tutor-start=12,tutor-end=13}{b} \htmlData{tutor-start=14,tutor-end=15}{c}\htmlData{tutor-start=15,tutor-end=16}{,} \htmlData{tutor-start=17,tutor-end=18}{c} \htmlData{tutor-start=19,tutor-end=20}{a}\htmlData{tutor-start=20,tutor-end=22}{\}} \htmlData{tutor-start=23,tutor-end=33}{\geqslant }\htmlData{tutor-start=33,tutor-end=34}{1} min { a b , b c , c a } ⩾ 1 ( a 2 + 1 ) ( b 2 + 1 ) ( c 2 + 1 ) 3 ⩽ ( a + b + c 3 ) 2 + 1 \sqrt[\htmlData{tutor-start=6,tutor-end=7}{3}]{\left(\htmlData{tutor-start=15,tutor-end=16}{a}^{\htmlData{tutor-start=18,tutor-end=19}{2}}\htmlData{tutor-start=20,tutor-end=21}{+}\htmlData{tutor-start=21,tutor-end=22}{1}\right)\left(\htmlData{tutor-start=35,tutor-end=36}{b}^{\htmlData{tutor-start=38,tutor-end=39}{2}}\htmlData{tutor-start=40,tutor-end=41}{+}\htmlData{tutor-start=41,tutor-end=42}{1}\right)\left(\htmlData{tutor-start=55,tutor-end=56}{c}^{\htmlData{tutor-start=58,tutor-end=59}{2}}\htmlData{tutor-start=60,tutor-end=61}{+}\htmlData{tutor-start=61,tutor-end=62}{1}\right)} \htmlData{tutor-start=71,tutor-end=80}{\leqslant}\left(\frac{\htmlData{tutor-start=92,tutor-end=93}{a}\htmlData{tutor-start=93,tutor-end=94}{+}\htmlData{tutor-start=94,tutor-end=95}{b}\htmlData{tutor-start=95,tutor-end=96}{+}\htmlData{tutor-start=96,tutor-end=97}{c}}{\htmlData{tutor-start=99,tutor-end=100}{3}}\right)^{\htmlData{tutor-start=110,tutor-end=111}{2}}\htmlData{tutor-start=112,tutor-end=113}{+}\htmlData{tutor-start=113,tutor-end=114}{1} 3 ( a 2 + 1 ) ( b 2 + 1 ) ( c 2 + 1 ) ⩽ ( 3 a + b + c ) 2 + 1 题解状态： 标准答案与规范题解待补充
题目标签：2016 IMO Shortlist A1
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2016-A2 2016 · Algebra · IMO/代数
Find the smallest real constant C \htmlData{tutor-start=0,tutor-end=1}{C} C a 1 , a 2 , a 3 , a 4 \htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{a}_{\htmlData{tutor-start=10,tutor-end=11}{2}}\htmlData{tutor-start=12,tutor-end=13}{,} \htmlData{tutor-start=14,tutor-end=15}{a}_{\htmlData{tutor-start=17,tutor-end=18}{3}}\htmlData{tutor-start=19,tutor-end=20}{,} \htmlData{tutor-start=21,tutor-end=22}{a}_{\htmlData{tutor-start=24,tutor-end=25}{4}} a 1 , a 2 , a 3 , a 4 a 5 \htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{5}} a 5 i , j , k \htmlData{tutor-start=0,tutor-end=1}{i}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{j}\htmlData{tutor-start=4,tutor-end=5}{,} \htmlData{tutor-start=6,tutor-end=7}{k} i , j , k l \htmlData{tutor-start=0,tutor-end=1}{l} l ∣ a i a j − a k a l ∣ ⩽ C \left|\frac{\htmlData{tutor-start=12,tutor-end=13}{a}_{\htmlData{tutor-start=15,tutor-end=16}{i}}}{\htmlData{tutor-start=19,tutor-end=20}{a}_{\htmlData{tutor-start=22,tutor-end=23}{j}}}\htmlData{tutor-start=25,tutor-end=26}{-}\frac{\htmlData{tutor-start=32,tutor-end=33}{a}_{\htmlData{tutor-start=35,tutor-end=36}{k}}}{\htmlData{tutor-start=39,tutor-end=40}{a}_{\htmlData{tutor-start=42,tutor-end=43}{l}}}\right| \htmlData{tutor-start=53,tutor-end=63}{\leqslant }\htmlData{tutor-start=63,tutor-end=64}{C} a j a i − a l a k ⩽ C 题解状态： 标准答案与规范题解待补充
题目标签：2016 IMO Shortlist A2
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2016-A3 2016 · Algebra · IMO/代数
Find all integers n ⩾ 3 \htmlData{tutor-start=0,tutor-end=1}{n} \htmlData{tutor-start=2,tutor-end=12}{\geqslant }\htmlData{tutor-start=12,tutor-end=13}{3} n ⩾ 3 a 1 , a 2 , … , a n \htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{a}_{\htmlData{tutor-start=10,tutor-end=11}{2}}\htmlData{tutor-start=12,tutor-end=13}{,} \htmlData{tutor-start=14,tutor-end=20}{\ldots}\htmlData{tutor-start=20,tutor-end=21}{,} \htmlData{tutor-start=22,tutor-end=23}{a}_{\htmlData{tutor-start=25,tutor-end=26}{n}} a 1 , a 2 , … , a n b 1 , b 2 , … , b n \htmlData{tutor-start=0,tutor-end=1}{b}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{b}_{\htmlData{tutor-start=10,tutor-end=11}{2}}\htmlData{tutor-start=12,tutor-end=13}{,} \htmlData{tutor-start=14,tutor-end=20}{\ldots}\htmlData{tutor-start=20,tutor-end=21}{,} \htmlData{tutor-start=22,tutor-end=23}{b}_{\htmlData{tutor-start=25,tutor-end=26}{n}} b 1 , b 2 , … , b n ∣ a k ∣ + ∣ b k ∣ = 1 \left|\htmlData{tutor-start=6,tutor-end=7}{a}_{\htmlData{tutor-start=9,tutor-end=10}{k}}\right|\htmlData{tutor-start=18,tutor-end=19}{+}\left|\htmlData{tutor-start=25,tutor-end=26}{b}_{\htmlData{tutor-start=28,tutor-end=29}{k}}\right|\htmlData{tutor-start=37,tutor-end=38}{=}\htmlData{tutor-start=38,tutor-end=39}{1} ∣ a k ∣ + ∣ b k ∣ = 1 1 ⩽ k ⩽ n \htmlData{tutor-start=0,tutor-end=1}{1} \htmlData{tutor-start=2,tutor-end=12}{\leqslant }\htmlData{tutor-start=12,tutor-end=13}{k} \htmlData{tutor-start=14,tutor-end=24}{\leqslant }\htmlData{tutor-start=24,tutor-end=25}{n} 1 ⩽ k ⩽ n x 1 , x 2 , … , x n \htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{x}_{\htmlData{tutor-start=10,tutor-end=11}{2}}\htmlData{tutor-start=12,tutor-end=13}{,} \htmlData{tutor-start=14,tutor-end=20}{\ldots}\htmlData{tutor-start=20,tutor-end=21}{,} \htmlData{tutor-start=22,tutor-end=23}{x}_{\htmlData{tutor-start=25,tutor-end=26}{n}} x 1 , x 2 , … , x n ∣ ∑ k = 1 n x k a k ∣ + ∣ ∑ k = 1 n x k b k ∣ ⩽ 1 \left|\sum_{\htmlData{tutor-start=12,tutor-end=13}{k}\htmlData{tutor-start=13,tutor-end=14}{=}\htmlData{tutor-start=14,tutor-end=15}{1}}^{\htmlData{tutor-start=18,tutor-end=19}{n}} \htmlData{tutor-start=21,tutor-end=22}{x}_{\htmlData{tutor-start=24,tutor-end=25}{k}} \htmlData{tutor-start=27,tutor-end=28}{a}_{\htmlData{tutor-start=30,tutor-end=31}{k}}\right|\htmlData{tutor-start=39,tutor-end=40}{+}\left|\sum_{\htmlData{tutor-start=52,tutor-end=53}{k}\htmlData{tutor-start=53,tutor-end=54}{=}\htmlData{tutor-start=54,tutor-end=55}{1}}^{\htmlData{tutor-start=58,tutor-end=59}{n}} \htmlData{tutor-start=61,tutor-end=62}{x}_{\htmlData{tutor-start=64,tutor-end=65}{k}} \htmlData{tutor-start=67,tutor-end=68}{b}_{\htmlData{tutor-start=70,tutor-end=71}{k}}\right| \htmlData{tutor-start=80,tutor-end=90}{\leqslant }\htmlData{tutor-start=90,tutor-end=91}{1} ∣ ∑ k = 1 n x k a k ∣ + ∣ ∑ k = 1 n x k b k ∣ ⩽ 1 题解状态： 标准答案与规范题解待补充
题目标签：2016 IMO Shortlist A3
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2016-A4 2016 · Algebra · IMO/代数
Denote by R + \mathbb{\htmlData{tutor-start=8,tutor-end=9}{R}}^{\htmlData{tutor-start=12,tutor-end=13}{+}} R + f : R + → R + \htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{:} \mathbb{\htmlData{tutor-start=11,tutor-end=12}{R}}^{\htmlData{tutor-start=15,tutor-end=16}{+}} \htmlData{tutor-start=18,tutor-end=30}{\rightarrow }\mathbb{\htmlData{tutor-start=38,tutor-end=39}{R}}^{\htmlData{tutor-start=42,tutor-end=43}{+}} f : R + → R + x f ( x 2 ) f ( f ( y ) ) + f ( y f ( x ) ) = f ( x y ) ( f ( f ( x 2 ) ) + f ( f ( y 2 ) ) ) \htmlData{tutor-start=0,tutor-end=1}{x} \htmlData{tutor-start=2,tutor-end=3}{f}\left(\htmlData{tutor-start=9,tutor-end=10}{x}^{\htmlData{tutor-start=12,tutor-end=13}{2}}\right) \htmlData{tutor-start=22,tutor-end=23}{f}\htmlData{tutor-start=23,tutor-end=24}{(}\htmlData{tutor-start=24,tutor-end=25}{f}\htmlData{tutor-start=25,tutor-end=26}{(}\htmlData{tutor-start=26,tutor-end=27}{y}\htmlData{tutor-start=27,tutor-end=28}{)}\htmlData{tutor-start=28,tutor-end=29}{)}\htmlData{tutor-start=29,tutor-end=30}{+}\htmlData{tutor-start=30,tutor-end=31}{f}\htmlData{tutor-start=31,tutor-end=32}{(}\htmlData{tutor-start=32,tutor-end=33}{y} \htmlData{tutor-start=34,tutor-end=35}{f}\htmlData{tutor-start=35,tutor-end=36}{(}\htmlData{tutor-start=36,tutor-end=37}{x}\htmlData{tutor-start=37,tutor-end=38}{)}\htmlData{tutor-start=38,tutor-end=39}{)}\htmlData{tutor-start=39,tutor-end=40}{=}\htmlData{tutor-start=40,tutor-end=41}{f}\htmlData{tutor-start=41,tutor-end=42}{(}\htmlData{tutor-start=42,tutor-end=43}{x} \htmlData{tutor-start=44,tutor-end=45}{y}\htmlData{tutor-start=45,tutor-end=46}{)}\left(\htmlData{tutor-start=52,tutor-end=53}{f}\left(\htmlData{tutor-start=59,tutor-end=60}{f}\left(\htmlData{tutor-start=66,tutor-end=67}{x}^{\htmlData{tutor-start=69,tutor-end=70}{2}}\right)\right)\htmlData{tutor-start=85,tutor-end=86}{+}\htmlData{tutor-start=86,tutor-end=87}{f}\left(\htmlData{tutor-start=93,tutor-end=94}{f}\left(\htmlData{tutor-start=100,tutor-end=101}{y}^{\htmlData{tutor-start=103,tutor-end=104}{2}}\right)\right)\right) x f ( x 2 ) f ( f ( y ) ) + f ( y f ( x ) ) = f ( x y ) ( f ( f ( x 2 ) ) + f ( f ( y 2 ) ) ) x \htmlData{tutor-start=0,tutor-end=1}{x} x y \htmlData{tutor-start=0,tutor-end=1}{y} y 题解状态： 标准答案与规范题解待补充
题目标签：2016 IMO Shortlist A4
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2016-A5 2016 · Algebra · IMO/代数
(a) Prove that for every positive integer n \htmlData{tutor-start=0,tutor-end=1}{n} n a b \frac{\htmlData{tutor-start=6,tutor-end=7}{a}}{\htmlData{tutor-start=9,tutor-end=10}{b}} b a a \htmlData{tutor-start=0,tutor-end=1}{a} a b \htmlData{tutor-start=0,tutor-end=1}{b} b 0 < b ⩽ n + 1 \htmlData{tutor-start=0,tutor-end=1}{0}\htmlData{tutor-start=1,tutor-end=2}{<}\htmlData{tutor-start=2,tutor-end=3}{b} \htmlData{tutor-start=4,tutor-end=14}{\leqslant }\sqrt{\htmlData{tutor-start=20,tutor-end=21}{n}}\htmlData{tutor-start=22,tutor-end=23}{+}\htmlData{tutor-start=23,tutor-end=24}{1} 0 < b ⩽ n + 1 n ⩽ a b ⩽ n + 1 \sqrt{\htmlData{tutor-start=6,tutor-end=7}{n}} \htmlData{tutor-start=9,tutor-end=19}{\leqslant }\frac{\htmlData{tutor-start=25,tutor-end=26}{a}}{\htmlData{tutor-start=28,tutor-end=29}{b}} \htmlData{tutor-start=31,tutor-end=41}{\leqslant }\sqrt{\htmlData{tutor-start=47,tutor-end=48}{n}\htmlData{tutor-start=48,tutor-end=49}{+}\htmlData{tutor-start=49,tutor-end=50}{1}} n ⩽ b a ⩽ n + 1 n \htmlData{tutor-start=0,tutor-end=1}{n} n a b \frac{\htmlData{tutor-start=6,tutor-end=7}{a}}{\htmlData{tutor-start=9,tutor-end=10}{b}} b a a \htmlData{tutor-start=0,tutor-end=1}{a} a b \htmlData{tutor-start=0,tutor-end=1}{b} b 0 < b ⩽ n \htmlData{tutor-start=0,tutor-end=1}{0}\htmlData{tutor-start=1,tutor-end=2}{<}\htmlData{tutor-start=2,tutor-end=3}{b} \htmlData{tutor-start=4,tutor-end=14}{\leqslant }\sqrt{\htmlData{tutor-start=20,tutor-end=21}{n}} 0 < b ⩽ n n ⩽ a b ⩽ n + 1 \sqrt{\htmlData{tutor-start=6,tutor-end=7}{n}} \htmlData{tutor-start=9,tutor-end=19}{\leqslant }\frac{\htmlData{tutor-start=25,tutor-end=26}{a}}{\htmlData{tutor-start=28,tutor-end=29}{b}} \htmlData{tutor-start=31,tutor-end=41}{\leqslant }\sqrt{\htmlData{tutor-start=47,tutor-end=48}{n}\htmlData{tutor-start=48,tutor-end=49}{+}\htmlData{tutor-start=49,tutor-end=50}{1}} n ⩽ b a ⩽ n + 1 题解状态： 标准答案与规范题解待补充
题目标签：2016 IMO Shortlist A5
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2016-A6 2016 · Algebra · IMO/代数
The equation
( x − 1 ) ( x − 2 ) ⋯ ( x − 2016 ) = ( x − 1 ) ( x − 2 ) ⋯ ( x − 2016 ) \htmlData{tutor-start=0,tutor-end=1}{(}\htmlData{tutor-start=1,tutor-end=2}{x}\htmlData{tutor-start=2,tutor-end=3}{-}\htmlData{tutor-start=3,tutor-end=4}{1}\htmlData{tutor-start=4,tutor-end=5}{)}\htmlData{tutor-start=5,tutor-end=6}{(}\htmlData{tutor-start=6,tutor-end=7}{x}\htmlData{tutor-start=7,tutor-end=8}{-}\htmlData{tutor-start=8,tutor-end=9}{2}\htmlData{tutor-start=9,tutor-end=10}{)} \cdots\htmlData{tutor-start=17,tutor-end=18}{(}\htmlData{tutor-start=18,tutor-end=19}{x}\htmlData{tutor-start=19,tutor-end=20}{-}\htmlData{tutor-start=20,tutor-end=21}{2}\htmlData{tutor-start=21,tutor-end=22}{0}\htmlData{tutor-start=22,tutor-end=23}{1}\htmlData{tutor-start=23,tutor-end=24}{6}\htmlData{tutor-start=24,tutor-end=25}{)}\htmlData{tutor-start=25,tutor-end=26}{=}\htmlData{tutor-start=26,tutor-end=27}{(}\htmlData{tutor-start=27,tutor-end=28}{x}\htmlData{tutor-start=28,tutor-end=29}{-}\htmlData{tutor-start=29,tutor-end=30}{1}\htmlData{tutor-start=30,tutor-end=31}{)}\htmlData{tutor-start=31,tutor-end=32}{(}\htmlData{tutor-start=32,tutor-end=33}{x}\htmlData{tutor-start=33,tutor-end=34}{-}\htmlData{tutor-start=34,tutor-end=35}{2}\htmlData{tutor-start=35,tutor-end=36}{)} \cdots\htmlData{tutor-start=43,tutor-end=44}{(}\htmlData{tutor-start=44,tutor-end=45}{x}\htmlData{tutor-start=45,tutor-end=46}{-}\htmlData{tutor-start=46,tutor-end=47}{2}\htmlData{tutor-start=47,tutor-end=48}{0}\htmlData{tutor-start=48,tutor-end=49}{1}\htmlData{tutor-start=49,tutor-end=50}{6}\htmlData{tutor-start=50,tutor-end=51}{)} ( x − 1 ) ( x − 2 ) ⋯ ( x − 2 0 1 6 ) = ( x − 1 ) ( x − 2 ) ⋯ ( x − 2 0 1 6 ) 题解状态： 标准答案与规范题解待补充
题目标签：2016 IMO 正式题 A6
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2016-A7 2016 · Algebra · IMO/代数
Denote by R \mathbb{\htmlData{tutor-start=8,tutor-end=9}{R}} R f : R → R \htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{:} \mathbb{\htmlData{tutor-start=11,tutor-end=12}{R}} \htmlData{tutor-start=14,tutor-end=26}{\rightarrow }\mathbb{\htmlData{tutor-start=34,tutor-end=35}{R}} f : R → R f ( 0 ) ≠ 0 \htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{0}\htmlData{tutor-start=3,tutor-end=4}{)} \neq \htmlData{tutor-start=10,tutor-end=11}{0} f ( 0 ) = 0 f ( x + y ) 2 = 2 f ( x ) f ( y ) + max { f ( x 2 ) + f ( y 2 ) , f ( x 2 + y 2 ) } \htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{x}\htmlData{tutor-start=3,tutor-end=4}{+}\htmlData{tutor-start=4,tutor-end=5}{y}\htmlData{tutor-start=5,tutor-end=6}{)}^{\htmlData{tutor-start=8,tutor-end=9}{2}}\htmlData{tutor-start=10,tutor-end=11}{=}\htmlData{tutor-start=11,tutor-end=12}{2} \htmlData{tutor-start=13,tutor-end=14}{f}\htmlData{tutor-start=14,tutor-end=15}{(}\htmlData{tutor-start=15,tutor-end=16}{x}\htmlData{tutor-start=16,tutor-end=17}{)} \htmlData{tutor-start=18,tutor-end=19}{f}\htmlData{tutor-start=19,tutor-end=20}{(}\htmlData{tutor-start=20,tutor-end=21}{y}\htmlData{tutor-start=21,tutor-end=22}{)}\htmlData{tutor-start=22,tutor-end=23}{+}\max \left\{\htmlData{tutor-start=35,tutor-end=36}{f}\left(\htmlData{tutor-start=42,tutor-end=43}{x}^{\htmlData{tutor-start=45,tutor-end=46}{2}}\right)\htmlData{tutor-start=54,tutor-end=55}{+}\htmlData{tutor-start=55,tutor-end=56}{f}\left(\htmlData{tutor-start=62,tutor-end=63}{y}^{\htmlData{tutor-start=65,tutor-end=66}{2}}\right)\htmlData{tutor-start=74,tutor-end=75}{,} \htmlData{tutor-start=76,tutor-end=77}{f}\left(\htmlData{tutor-start=83,tutor-end=84}{x}^{\htmlData{tutor-start=86,tutor-end=87}{2}}\htmlData{tutor-start=88,tutor-end=89}{+}\htmlData{tutor-start=89,tutor-end=90}{y}^{\htmlData{tutor-start=92,tutor-end=93}{2}}\right)\right\} f ( x + y ) 2 = 2 f ( x ) f ( y ) + max { f ( x 2 ) + f ( y 2 ) , f ( x 2 + y 2 ) } x \htmlData{tutor-start=0,tutor-end=1}{x} x y \htmlData{tutor-start=0,tutor-end=1}{y} y 题解状态： 标准答案与规范题解待补充
题目标签：2016 IMO Shortlist A7
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2016-A8 2016 · Algebra · IMO/代数
Determine the largest real number a \htmlData{tutor-start=0,tutor-end=1}{a} a n ⩾ 1 \htmlData{tutor-start=0,tutor-end=1}{n} \htmlData{tutor-start=2,tutor-end=12}{\geqslant }\htmlData{tutor-start=12,tutor-end=13}{1} n ⩾ 1 x 0 , x 1 , … , x n \htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{0}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{x}_{\htmlData{tutor-start=10,tutor-end=11}{1}}\htmlData{tutor-start=12,tutor-end=13}{,} \htmlData{tutor-start=14,tutor-end=20}{\ldots}\htmlData{tutor-start=20,tutor-end=21}{,} \htmlData{tutor-start=22,tutor-end=23}{x}_{\htmlData{tutor-start=25,tutor-end=26}{n}} x 0 , x 1 , … , x n 0 = x 0 < x 1 < x 2 < ⋯ < x n \htmlData{tutor-start=0,tutor-end=1}{0}\htmlData{tutor-start=1,tutor-end=2}{=}\htmlData{tutor-start=2,tutor-end=3}{x}_{\htmlData{tutor-start=5,tutor-end=6}{0}}\htmlData{tutor-start=7,tutor-end=8}{<}\htmlData{tutor-start=8,tutor-end=9}{x}_{\htmlData{tutor-start=11,tutor-end=12}{1}}\htmlData{tutor-start=13,tutor-end=14}{<}\htmlData{tutor-start=14,tutor-end=15}{x}_{\htmlData{tutor-start=17,tutor-end=18}{2}}\htmlData{tutor-start=19,tutor-end=20}{<}\cdots\htmlData{tutor-start=26,tutor-end=27}{<}\htmlData{tutor-start=27,tutor-end=28}{x}_{\htmlData{tutor-start=30,tutor-end=31}{n}} 0 = x 0 < x 1 < x 2 < ⋯ < x n 1 x 1 − x 0 + 1 x 2 − x 1 + ⋯ + 1 x n − x n − 1 ⩾ a ( 2 x 1 + 3 x 2 + ⋯ + n + 1 x n ) . \frac{1}{x_{1}-x_{0}}+\frac{1}{x_{2}-x_{1}}+\cdots+\frac{1}{x_{n}-x_{n-1}} \geqslant a\left(\frac{2}{x_{1}}+\frac{3}{x_{2}}+\cdots+\frac{n+1}{x_{n}}\right) . x 1 − x 0 1 + x 2 − x 1 1 + ⋯ + x n − x n − 1 1 ⩾ a ( x 1 2 + x 2 3 + ⋯ + x n n + 1 ) . 题解状态： 标准答案与规范题解待补充
题目标签：2016 IMO Shortlist A8
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2017-A1 2017 · Algebra · IMO/代数
Let a 1 , a 2 , … , a n , k \htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{a}_{\htmlData{tutor-start=10,tutor-end=11}{2}}\htmlData{tutor-start=12,tutor-end=13}{,} \htmlData{tutor-start=14,tutor-end=20}{\ldots}\htmlData{tutor-start=20,tutor-end=21}{,} \htmlData{tutor-start=22,tutor-end=23}{a}_{\htmlData{tutor-start=25,tutor-end=26}{n}}\htmlData{tutor-start=27,tutor-end=28}{,} \htmlData{tutor-start=29,tutor-end=30}{k} a 1 , a 2 , … , a n , k M \htmlData{tutor-start=0,tutor-end=1}{M} M 1 a 1 + 1 a 2 + ⋯ + 1 a n = k and a 1 a 2 … a n = M \frac{1}{a_{1}}+\frac{1}{a_{2}}+\cdots+\frac{1}{a_{n}}=k \quad \text { and } \quad a_{1} a_{2} \ldots a_{n}=M a 1 1 + a 2 1 + ⋯ + a n 1 = k and a 1 a 2 … a n = M M > 1 \htmlData{tutor-start=0,tutor-end=1}{M}\htmlData{tutor-start=1,tutor-end=2}{>}\htmlData{tutor-start=2,tutor-end=3}{1} M > 1 P ( x ) = M ( x + 1 ) k − ( x + a 1 ) ( x + a 2 ) ⋯ ( x + a n ) \htmlData{tutor-start=0,tutor-end=1}{P}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{x}\htmlData{tutor-start=3,tutor-end=4}{)}\htmlData{tutor-start=4,tutor-end=5}{=}\htmlData{tutor-start=5,tutor-end=6}{M}\htmlData{tutor-start=6,tutor-end=7}{(}\htmlData{tutor-start=7,tutor-end=8}{x}\htmlData{tutor-start=8,tutor-end=9}{+}\htmlData{tutor-start=9,tutor-end=10}{1}\htmlData{tutor-start=10,tutor-end=11}{)}^{\htmlData{tutor-start=13,tutor-end=14}{k}}\htmlData{tutor-start=15,tutor-end=16}{-}\left(\htmlData{tutor-start=22,tutor-end=23}{x}\htmlData{tutor-start=23,tutor-end=24}{+}\htmlData{tutor-start=24,tutor-end=25}{a}_{\htmlData{tutor-start=27,tutor-end=28}{1}}\right)\left(\htmlData{tutor-start=42,tutor-end=43}{x}\htmlData{tutor-start=43,tutor-end=44}{+}\htmlData{tutor-start=44,tutor-end=45}{a}_{\htmlData{tutor-start=47,tutor-end=48}{2}}\right) \cdots\left(\htmlData{tutor-start=69,tutor-end=70}{x}\htmlData{tutor-start=70,tutor-end=71}{+}\htmlData{tutor-start=71,tutor-end=72}{a}_{\htmlData{tutor-start=74,tutor-end=75}{n}}\right) P ( x ) = M ( x + 1 ) k − ( x + a 1 ) ( x + a 2 ) ⋯ ( x + a n ) 题解状态： 标准答案与规范题解待补充
题目标签：2017 IMO Shortlist A1
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2017-A2 2017 · Algebra · IMO/代数
Let q \htmlData{tutor-start=0,tutor-end=1}{q} q a − b \htmlData{tutor-start=0,tutor-end=1}{a}\htmlData{tutor-start=1,tutor-end=2}{-}\htmlData{tutor-start=2,tutor-end=3}{b} a − b a \htmlData{tutor-start=0,tutor-end=1}{a} a b \htmlData{tutor-start=0,tutor-end=1}{b} b q a b \htmlData{tutor-start=0,tutor-end=1}{q} \htmlData{tutor-start=2,tutor-end=3}{a} \htmlData{tutor-start=4,tutor-end=5}{b} q a b a \htmlData{tutor-start=0,tutor-end=1}{a} a b \htmlData{tutor-start=0,tutor-end=1}{b} b a 2 + b 2 − c 2 − d 2 \htmlData{tutor-start=0,tutor-end=1}{a}^{\htmlData{tutor-start=3,tutor-end=4}{2}}\htmlData{tutor-start=5,tutor-end=6}{+}\htmlData{tutor-start=6,tutor-end=7}{b}^{\htmlData{tutor-start=9,tutor-end=10}{2}}\htmlData{tutor-start=11,tutor-end=12}{-}\htmlData{tutor-start=12,tutor-end=13}{c}^{\htmlData{tutor-start=15,tutor-end=16}{2}}\htmlData{tutor-start=17,tutor-end=18}{-}\htmlData{tutor-start=18,tutor-end=19}{d}^{\htmlData{tutor-start=21,tutor-end=22}{2}} a 2 + b 2 − c 2 − d 2 a , b , c , d \htmlData{tutor-start=0,tutor-end=1}{a}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{b}\htmlData{tutor-start=4,tutor-end=5}{,} \htmlData{tutor-start=6,tutor-end=7}{c}\htmlData{tutor-start=7,tutor-end=8}{,} \htmlData{tutor-start=9,tutor-end=10}{d} a , b , c , d q \htmlData{tutor-start=0,tutor-end=1}{q} q 题解状态： 标准答案与规范题解待补充
题目标签：2017 IMO Shortlist A2
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2017-A3 2017 · Algebra · IMO/代数
Let S \htmlData{tutor-start=0,tutor-end=1}{S} S A \mathcal{\htmlData{tutor-start=9,tutor-end=10}{A}} A S \htmlData{tutor-start=0,tutor-end=1}{S} S S \htmlData{tutor-start=0,tutor-end=1}{S} S f \htmlData{tutor-start=0,tutor-end=1}{f} f A \mathcal{\htmlData{tutor-start=9,tutor-end=10}{A}} A T = f ( S ) \htmlData{tutor-start=0,tutor-end=1}{T}\htmlData{tutor-start=1,tutor-end=2}{=}\htmlData{tutor-start=2,tutor-end=3}{f}\htmlData{tutor-start=3,tutor-end=4}{(}\htmlData{tutor-start=4,tutor-end=5}{S}\htmlData{tutor-start=5,tutor-end=6}{)} T = f ( S ) S \htmlData{tutor-start=0,tutor-end=1}{S} S f \htmlData{tutor-start=0,tutor-end=1}{f} f f ∘ g ∘ f ≠ g ∘ f ∘ g \htmlData{tutor-start=0,tutor-end=1}{f} \htmlData{tutor-start=2,tutor-end=8}{\circ }\htmlData{tutor-start=8,tutor-end=9}{g} \htmlData{tutor-start=10,tutor-end=16}{\circ }\htmlData{tutor-start=16,tutor-end=17}{f} \neq \htmlData{tutor-start=23,tutor-end=24}{g} \htmlData{tutor-start=25,tutor-end=31}{\circ }\htmlData{tutor-start=31,tutor-end=32}{f} \htmlData{tutor-start=33,tutor-end=39}{\circ }\htmlData{tutor-start=39,tutor-end=40}{g} f ∘ g ∘ f = g ∘ f ∘ g g \htmlData{tutor-start=0,tutor-end=1}{g} g A \mathcal{\htmlData{tutor-start=9,tutor-end=10}{A}} A g ≠ f \htmlData{tutor-start=0,tutor-end=1}{g} \neq \htmlData{tutor-start=7,tutor-end=8}{f} g = f f ( T ) = T \htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{T}\htmlData{tutor-start=3,tutor-end=4}{)}\htmlData{tutor-start=4,tutor-end=5}{=}\htmlData{tutor-start=5,tutor-end=6}{T} f ( T ) = T 题解状态： 标准答案与规范题解待补充
题目标签：2017 IMO Shortlist A3
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2017-A4 2017 · Algebra · IMO/代数
A sequence of real numbers a 1 , a 2 , … \htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{a}_{\htmlData{tutor-start=10,tutor-end=11}{2}}\htmlData{tutor-start=12,tutor-end=13}{,} \htmlData{tutor-start=14,tutor-end=20}{\ldots} a 1 , a 2 , … a n = − max i + j = n ( a i + a j ) for all n > 2017 \htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{n}}\htmlData{tutor-start=5,tutor-end=6}{=}\htmlData{tutor-start=6,tutor-end=7}{-}\max _{\htmlData{tutor-start=14,tutor-end=15}{i}\htmlData{tutor-start=15,tutor-end=16}{+}\htmlData{tutor-start=16,tutor-end=17}{j}\htmlData{tutor-start=17,tutor-end=18}{=}\htmlData{tutor-start=18,tutor-end=19}{n}}\left(\htmlData{tutor-start=26,tutor-end=27}{a}_{\htmlData{tutor-start=29,tutor-end=30}{i}}\htmlData{tutor-start=31,tutor-end=32}{+}\htmlData{tutor-start=32,tutor-end=33}{a}_{\htmlData{tutor-start=35,tutor-end=36}{j}}\right) \quad \text { \htmlData{tutor-start=59,tutor-end=60}{f}\htmlData{tutor-start=60,tutor-end=61}{o}\htmlData{tutor-start=61,tutor-end=62}{r} \htmlData{tutor-start=63,tutor-end=64}{a}\htmlData{tutor-start=64,tutor-end=65}{l}\htmlData{tutor-start=65,tutor-end=66}{l} } \htmlData{tutor-start=69,tutor-end=70}{n}\htmlData{tutor-start=70,tutor-end=71}{>}\htmlData{tutor-start=71,tutor-end=72}{2}\htmlData{tutor-start=72,tutor-end=73}{0}\htmlData{tutor-start=73,tutor-end=74}{1}\htmlData{tutor-start=74,tutor-end=75}{7} a n = − max i + j = n ( a i + a j ) f o r a l l n > 2 0 1 7 M \htmlData{tutor-start=0,tutor-end=1}{M} M ∣ a n ∣ ⩽ M \left|\htmlData{tutor-start=6,tutor-end=7}{a}_{\htmlData{tutor-start=9,tutor-end=10}{n}}\right| \htmlData{tutor-start=19,tutor-end=29}{\leqslant }\htmlData{tutor-start=29,tutor-end=30}{M} ∣ a n ∣ ⩽ M n \htmlData{tutor-start=0,tutor-end=1}{n} n 题解状态： 标准答案与规范题解待补充
题目标签：2017 IMO Shortlist A4
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2017-A5 2017 · Algebra · IMO/代数
An integer n ⩾ 3 \htmlData{tutor-start=0,tutor-end=1}{n} \htmlData{tutor-start=2,tutor-end=12}{\geqslant }\htmlData{tutor-start=12,tutor-end=13}{3} n ⩾ 3 n \htmlData{tutor-start=0,tutor-end=1}{n} n ( x 1 , x 2 , … , x n ) \left(\htmlData{tutor-start=6,tutor-end=7}{x}_{\htmlData{tutor-start=9,tutor-end=10}{1}}\htmlData{tutor-start=11,tutor-end=12}{,} \htmlData{tutor-start=13,tutor-end=14}{x}_{\htmlData{tutor-start=16,tutor-end=17}{2}}\htmlData{tutor-start=18,tutor-end=19}{,} \htmlData{tutor-start=20,tutor-end=26}{\ldots}\htmlData{tutor-start=26,tutor-end=27}{,} \htmlData{tutor-start=28,tutor-end=29}{x}_{\htmlData{tutor-start=31,tutor-end=32}{n}}\right) ( x 1 , x 2 , … , x n ) y 1 , y 2 , … , y n \htmlData{tutor-start=0,tutor-end=1}{y}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{y}_{\htmlData{tutor-start=10,tutor-end=11}{2}}\htmlData{tutor-start=12,tutor-end=13}{,} \htmlData{tutor-start=14,tutor-end=20}{\ldots}\htmlData{tutor-start=20,tutor-end=21}{,} \htmlData{tutor-start=22,tutor-end=23}{y}_{\htmlData{tutor-start=25,tutor-end=26}{n}} y 1 , y 2 , … , y n ∑ i = 1 n − 1 y i y i + 1 = y 1 y 2 + y 2 y 3 + y 3 y 4 + ⋯ + y n − 1 y n ⩾ − 1 \htmlData{tutor-start=0,tutor-end=7}{\sum_{i}\htmlData{tutor-start=7,tutor-end=8}{=}\htmlData{tutor-start=8,tutor-end=9}{1}}^{\htmlData{tutor-start=12,tutor-end=13}{n}\htmlData{tutor-start=13,tutor-end=14}{-}\htmlData{tutor-start=14,tutor-end=15}{1}} \htmlData{tutor-start=17,tutor-end=18}{y}_{\htmlData{tutor-start=20,tutor-end=21}{i}} \htmlData{tutor-start=23,tutor-end=24}{y}_{\htmlData{tutor-start=26,tutor-end=27}{i}\htmlData{tutor-start=27,tutor-end=28}{+}\htmlData{tutor-start=28,tutor-end=29}{1}}\htmlData{tutor-start=30,tutor-end=31}{=}\htmlData{tutor-start=31,tutor-end=32}{y}_{\htmlData{tutor-start=34,tutor-end=35}{1}} \htmlData{tutor-start=37,tutor-end=38}{y}_{\htmlData{tutor-start=40,tutor-end=41}{2}}\htmlData{tutor-start=42,tutor-end=43}{+}\htmlData{tutor-start=43,tutor-end=44}{y}_{\htmlData{tutor-start=46,tutor-end=47}{2}} \htmlData{tutor-start=49,tutor-end=50}{y}_{\htmlData{tutor-start=52,tutor-end=53}{3}}\htmlData{tutor-start=54,tutor-end=55}{+}\htmlData{tutor-start=55,tutor-end=56}{y}_{\htmlData{tutor-start=58,tutor-end=59}{3}} \htmlData{tutor-start=61,tutor-end=62}{y}_{\htmlData{tutor-start=64,tutor-end=65}{4}}\htmlData{tutor-start=66,tutor-end=67}{+}\cdots\htmlData{tutor-start=73,tutor-end=74}{+}\htmlData{tutor-start=74,tutor-end=75}{y}_{\htmlData{tutor-start=77,tutor-end=78}{n}\htmlData{tutor-start=78,tutor-end=79}{-}\htmlData{tutor-start=79,tutor-end=80}{1}} \htmlData{tutor-start=82,tutor-end=83}{y}_{\htmlData{tutor-start=85,tutor-end=86}{n}} \htmlData{tutor-start=88,tutor-end=97}{\geqslant}\htmlData{tutor-start=97,tutor-end=98}{-}\htmlData{tutor-start=98,tutor-end=99}{1} ∑ i = 1 n − 1 y i y i + 1 = y 1 y 2 + y 2 y 3 + y 3 y 4 + ⋯ + y n − 1 y n ⩾ − 1 K = K ( n ) \htmlData{tutor-start=0,tutor-end=1}{K}\htmlData{tutor-start=1,tutor-end=2}{=}\htmlData{tutor-start=2,tutor-end=3}{K}\htmlData{tutor-start=3,tutor-end=4}{(}\htmlData{tutor-start=4,tutor-end=5}{n}\htmlData{tutor-start=5,tutor-end=6}{)} K = K ( n ) ∑ 1 ⩽ i < j ⩽ n x i x j ⩾ K \sum_{\htmlData{tutor-start=6,tutor-end=7}{1} \htmlData{tutor-start=8,tutor-end=18}{\leqslant }\htmlData{tutor-start=18,tutor-end=19}{i}\htmlData{tutor-start=19,tutor-end=20}{<}\htmlData{tutor-start=20,tutor-end=21}{j} \htmlData{tutor-start=22,tutor-end=32}{\leqslant }\htmlData{tutor-start=32,tutor-end=33}{n}} \htmlData{tutor-start=35,tutor-end=36}{x}_{\htmlData{tutor-start=38,tutor-end=39}{i}} \htmlData{tutor-start=41,tutor-end=42}{x}_{\htmlData{tutor-start=44,tutor-end=45}{j}} \htmlData{tutor-start=47,tutor-end=57}{\geqslant }\htmlData{tutor-start=57,tutor-end=58}{K} ∑ 1 ⩽ i < j ⩽ n x i x j ⩾ K n \htmlData{tutor-start=0,tutor-end=1}{n} n ( x 1 , x 2 , … , x n ) \left(\htmlData{tutor-start=6,tutor-end=7}{x}_{\htmlData{tutor-start=9,tutor-end=10}{1}}\htmlData{tutor-start=11,tutor-end=12}{,} \htmlData{tutor-start=13,tutor-end=14}{x}_{\htmlData{tutor-start=16,tutor-end=17}{2}}\htmlData{tutor-start=18,tutor-end=19}{,} \htmlData{tutor-start=20,tutor-end=26}{\ldots}\htmlData{tutor-start=26,tutor-end=27}{,} \htmlData{tutor-start=28,tutor-end=29}{x}_{\htmlData{tutor-start=31,tutor-end=32}{n}}\right) ( x 1 , x 2 , … , x n ) 题解状态： 标准答案与规范题解待补充
题目标签：2017 IMO Shortlist A5
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2017-A6 2017 · Algebra · IMO/代数
Find all functions f : R → R \htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{:} \mathbb{\htmlData{tutor-start=11,tutor-end=12}{R}} \htmlData{tutor-start=14,tutor-end=26}{\rightarrow }\mathbb{\htmlData{tutor-start=34,tutor-end=35}{R}} f : R → R f ( f ( x ) f ( y ) ) + f ( x + y ) = f ( x y ) \htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{f}\htmlData{tutor-start=3,tutor-end=4}{(}\htmlData{tutor-start=4,tutor-end=5}{x}\htmlData{tutor-start=5,tutor-end=6}{)} \htmlData{tutor-start=7,tutor-end=8}{f}\htmlData{tutor-start=8,tutor-end=9}{(}\htmlData{tutor-start=9,tutor-end=10}{y}\htmlData{tutor-start=10,tutor-end=11}{)}\htmlData{tutor-start=11,tutor-end=12}{)}\htmlData{tutor-start=12,tutor-end=13}{+}\htmlData{tutor-start=13,tutor-end=14}{f}\htmlData{tutor-start=14,tutor-end=15}{(}\htmlData{tutor-start=15,tutor-end=16}{x}\htmlData{tutor-start=16,tutor-end=17}{+}\htmlData{tutor-start=17,tutor-end=18}{y}\htmlData{tutor-start=18,tutor-end=19}{)}\htmlData{tutor-start=19,tutor-end=20}{=}\htmlData{tutor-start=20,tutor-end=21}{f}\htmlData{tutor-start=21,tutor-end=22}{(}\htmlData{tutor-start=22,tutor-end=23}{x} \htmlData{tutor-start=24,tutor-end=25}{y}\htmlData{tutor-start=25,tutor-end=26}{)} f ( f ( x ) f ( y ) ) + f ( x + y ) = f ( x y ) x , y ∈ R \htmlData{tutor-start=0,tutor-end=1}{x}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{y} \htmlData{tutor-start=5,tutor-end=9}{\in }\mathbb{\htmlData{tutor-start=17,tutor-end=18}{R}} x , y ∈ R 题解状态： 标准答案与规范题解待补充
题目标签：2017 IMO 正式题 A6
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2017-A7 2017 · Algebra · IMO/代数
Let a 0 , a 1 , a 2 , … \htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{0}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{a}_{\htmlData{tutor-start=10,tutor-end=11}{1}}\htmlData{tutor-start=12,tutor-end=13}{,} \htmlData{tutor-start=14,tutor-end=15}{a}_{\htmlData{tutor-start=17,tutor-end=18}{2}}\htmlData{tutor-start=19,tutor-end=20}{,} \htmlData{tutor-start=21,tutor-end=27}{\ldots} a 0 , a 1 , a 2 , … b 0 , b 1 , b 2 , … \htmlData{tutor-start=0,tutor-end=1}{b}_{\htmlData{tutor-start=3,tutor-end=4}{0}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{b}_{\htmlData{tutor-start=10,tutor-end=11}{1}}\htmlData{tutor-start=12,tutor-end=13}{,} \htmlData{tutor-start=14,tutor-end=15}{b}_{\htmlData{tutor-start=17,tutor-end=18}{2}}\htmlData{tutor-start=19,tutor-end=20}{,} \htmlData{tutor-start=21,tutor-end=27}{\ldots} b 0 , b 1 , b 2 , … a 0 = 0 , a 1 = 1 \htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{0}}\htmlData{tutor-start=5,tutor-end=6}{=}\htmlData{tutor-start=6,tutor-end=7}{0}\htmlData{tutor-start=7,tutor-end=8}{,} \htmlData{tutor-start=9,tutor-end=10}{a}_{\htmlData{tutor-start=12,tutor-end=13}{1}}\htmlData{tutor-start=14,tutor-end=15}{=}\htmlData{tutor-start=15,tutor-end=16}{1} a 0 = 0 , a 1 = 1 a n + 1 = { a n b n + a n − 1 , if b n − 1 = 1 a n b n − a n − 1 , if b n − 1 > 1 for n = 1 , 2 , … \htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{n}\htmlData{tutor-start=4,tutor-end=5}{+}\htmlData{tutor-start=5,tutor-end=6}{1}}\htmlData{tutor-start=7,tutor-end=8}{=}\left\{\begin{array}{ll}
\htmlData{tutor-start=33,tutor-end=34}{a}_{\htmlData{tutor-start=36,tutor-end=37}{n}} \htmlData{tutor-start=39,tutor-end=40}{b}_{\htmlData{tutor-start=42,tutor-end=43}{n}}\htmlData{tutor-start=44,tutor-end=45}{+}\htmlData{tutor-start=45,tutor-end=46}{a}_{\htmlData{tutor-start=48,tutor-end=49}{n}\htmlData{tutor-start=49,tutor-end=50}{-}\htmlData{tutor-start=50,tutor-end=51}{1}}\htmlData{tutor-start=52,tutor-end=53}{,} & \text { \htmlData{tutor-start=64,tutor-end=65}{i}\htmlData{tutor-start=65,tutor-end=66}{f} } \htmlData{tutor-start=69,tutor-end=70}{b}_{\htmlData{tutor-start=72,tutor-end=73}{n}\htmlData{tutor-start=73,tutor-end=74}{-}\htmlData{tutor-start=74,tutor-end=75}{1}}\htmlData{tutor-start=76,tutor-end=77}{=}\htmlData{tutor-start=77,tutor-end=78}{1} \\
\htmlData{tutor-start=82,tutor-end=83}{a}_{\htmlData{tutor-start=85,tutor-end=86}{n}} \htmlData{tutor-start=88,tutor-end=89}{b}_{\htmlData{tutor-start=91,tutor-end=92}{n}}\htmlData{tutor-start=93,tutor-end=94}{-}\htmlData{tutor-start=94,tutor-end=95}{a}_{\htmlData{tutor-start=97,tutor-end=98}{n}\htmlData{tutor-start=98,tutor-end=99}{-}\htmlData{tutor-start=99,tutor-end=100}{1}}\htmlData{tutor-start=101,tutor-end=102}{,} & \text { \htmlData{tutor-start=113,tutor-end=114}{i}\htmlData{tutor-start=114,tutor-end=115}{f} } \htmlData{tutor-start=118,tutor-end=119}{b}_{\htmlData{tutor-start=121,tutor-end=122}{n}\htmlData{tutor-start=122,tutor-end=123}{-}\htmlData{tutor-start=123,tutor-end=124}{1}}\htmlData{tutor-start=125,tutor-end=126}{>}\htmlData{tutor-start=126,tutor-end=127}{1}
\end{array} \quad \text { \htmlData{tutor-start=154,tutor-end=155}{f}\htmlData{tutor-start=155,tutor-end=156}{o}\htmlData{tutor-start=156,tutor-end=157}{r} } \htmlData{tutor-start=160,tutor-end=161}{n}\htmlData{tutor-start=161,tutor-end=162}{=}\htmlData{tutor-start=162,tutor-end=163}{1}\htmlData{tutor-start=163,tutor-end=164}{,}\htmlData{tutor-start=164,tutor-end=165}{2}\htmlData{tutor-start=165,tutor-end=166}{,} \htmlData{tutor-start=167,tutor-end=173}{\ldots}\right. a n + 1 = { a n b n + a n − 1 , a n b n − a n − 1 , i f b n − 1 = 1 i f b n − 1 > 1 f o r n = 1 , 2 , … a 2017 \htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{2}\htmlData{tutor-start=4,tutor-end=5}{0}\htmlData{tutor-start=5,tutor-end=6}{1}\htmlData{tutor-start=6,tutor-end=7}{7}} a 2 0 1 7 a 2018 \htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{2}\htmlData{tutor-start=4,tutor-end=5}{0}\htmlData{tutor-start=5,tutor-end=6}{1}\htmlData{tutor-start=6,tutor-end=7}{8}} a 2 0 1 8 题解状态： 标准答案与规范题解待补充
题目标签：2017 IMO Shortlist A7
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2017-A8 2017 · Algebra · IMO/代数
Assume that a function f : R → R \htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{:} \mathbb{\htmlData{tutor-start=11,tutor-end=12}{R}} \htmlData{tutor-start=14,tutor-end=26}{\rightarrow }\mathbb{\htmlData{tutor-start=34,tutor-end=35}{R}} f : R → R x , y ∈ R \htmlData{tutor-start=0,tutor-end=1}{x}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{y} \htmlData{tutor-start=5,tutor-end=9}{\in }\mathbb{\htmlData{tutor-start=17,tutor-end=18}{R}} x , y ∈ R ( f ( x ) + y ) ( f ( y ) + x ) > 0 \htmlData{tutor-start=0,tutor-end=1}{(}\htmlData{tutor-start=1,tutor-end=2}{f}\htmlData{tutor-start=2,tutor-end=3}{(}\htmlData{tutor-start=3,tutor-end=4}{x}\htmlData{tutor-start=4,tutor-end=5}{)}\htmlData{tutor-start=5,tutor-end=6}{+}\htmlData{tutor-start=6,tutor-end=7}{y}\htmlData{tutor-start=7,tutor-end=8}{)}\htmlData{tutor-start=8,tutor-end=9}{(}\htmlData{tutor-start=9,tutor-end=10}{f}\htmlData{tutor-start=10,tutor-end=11}{(}\htmlData{tutor-start=11,tutor-end=12}{y}\htmlData{tutor-start=12,tutor-end=13}{)}\htmlData{tutor-start=13,tutor-end=14}{+}\htmlData{tutor-start=14,tutor-end=15}{x}\htmlData{tutor-start=15,tutor-end=16}{)}\htmlData{tutor-start=16,tutor-end=17}{>}\htmlData{tutor-start=17,tutor-end=18}{0} ( f ( x ) + y ) ( f ( y ) + x ) > 0 f ( x ) + y = f ( y ) + x \htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{x}\htmlData{tutor-start=3,tutor-end=4}{)}\htmlData{tutor-start=4,tutor-end=5}{+}\htmlData{tutor-start=5,tutor-end=6}{y}\htmlData{tutor-start=6,tutor-end=7}{=}\htmlData{tutor-start=7,tutor-end=8}{f}\htmlData{tutor-start=8,tutor-end=9}{(}\htmlData{tutor-start=9,tutor-end=10}{y}\htmlData{tutor-start=10,tutor-end=11}{)}\htmlData{tutor-start=11,tutor-end=12}{+}\htmlData{tutor-start=12,tutor-end=13}{x} f ( x ) + y = f ( y ) + x f ( x ) + y ⩽ f ( y ) + x \htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{x}\htmlData{tutor-start=3,tutor-end=4}{)}\htmlData{tutor-start=4,tutor-end=5}{+}\htmlData{tutor-start=5,tutor-end=6}{y} \htmlData{tutor-start=7,tutor-end=17}{\leqslant }\htmlData{tutor-start=17,tutor-end=18}{f}\htmlData{tutor-start=18,tutor-end=19}{(}\htmlData{tutor-start=19,tutor-end=20}{y}\htmlData{tutor-start=20,tutor-end=21}{)}\htmlData{tutor-start=21,tutor-end=22}{+}\htmlData{tutor-start=22,tutor-end=23}{x} f ( x ) + y ⩽ f ( y ) + x x > y \htmlData{tutor-start=0,tutor-end=1}{x}\htmlData{tutor-start=1,tutor-end=2}{>}\htmlData{tutor-start=2,tutor-end=3}{y} x > y 题解状态： 标准答案与规范题解待补充
题目标签：2017 IMO Shortlist A8
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2018-A1 2018 · Algebra · IMO/代数
Let Q > 0 \mathbb{\htmlData{tutor-start=8,tutor-end=9}{Q}}_{\htmlData{tutor-start=12,tutor-end=13}{>}\htmlData{tutor-start=13,tutor-end=14}{0}} Q > 0 f : Q > 0 → Q > 0 \htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{:} \mathbb{\htmlData{tutor-start=11,tutor-end=12}{Q}}_{\htmlData{tutor-start=15,tutor-end=16}{>}\htmlData{tutor-start=16,tutor-end=17}{0}} \htmlData{tutor-start=19,tutor-end=31}{\rightarrow }\mathbb{\htmlData{tutor-start=39,tutor-end=40}{Q}}_{\htmlData{tutor-start=43,tutor-end=44}{>}\htmlData{tutor-start=44,tutor-end=45}{0}} f : Q > 0 → Q > 0 x , y ∈ Q > 0 \htmlData{tutor-start=0,tutor-end=1}{x}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{y} \htmlData{tutor-start=5,tutor-end=9}{\in }\mathbb{\htmlData{tutor-start=17,tutor-end=18}{Q}}_{\htmlData{tutor-start=21,tutor-end=22}{>}\htmlData{tutor-start=22,tutor-end=23}{0}} x , y ∈ Q > 0 f ( x 2 f ( y ) 2 ) = f ( x ) 2 f ( y ) \htmlData{tutor-start=0,tutor-end=1}{f}\left(\htmlData{tutor-start=7,tutor-end=8}{x}^{\htmlData{tutor-start=10,tutor-end=11}{2}} \htmlData{tutor-start=13,tutor-end=14}{f}\htmlData{tutor-start=14,tutor-end=15}{(}\htmlData{tutor-start=15,tutor-end=16}{y}\htmlData{tutor-start=16,tutor-end=17}{)}^{\htmlData{tutor-start=19,tutor-end=20}{2}}\right)\htmlData{tutor-start=28,tutor-end=29}{=}\htmlData{tutor-start=29,tutor-end=30}{f}\htmlData{tutor-start=30,tutor-end=31}{(}\htmlData{tutor-start=31,tutor-end=32}{x}\htmlData{tutor-start=32,tutor-end=33}{)}^{\htmlData{tutor-start=35,tutor-end=36}{2}} \htmlData{tutor-start=38,tutor-end=39}{f}\htmlData{tutor-start=39,tutor-end=40}{(}\htmlData{tutor-start=40,tutor-end=41}{y}\htmlData{tutor-start=41,tutor-end=42}{)} f ( x 2 f ( y ) 2 ) = f ( x ) 2 f ( y ) 题解状态： 标准答案与规范题解待补充
题目标签：2018 IMO Shortlist A1
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2018-A2 2018 · Algebra · IMO/代数
Find all positive integers n ⩾ 3 \htmlData{tutor-start=0,tutor-end=1}{n} \htmlData{tutor-start=2,tutor-end=12}{\geqslant }\htmlData{tutor-start=12,tutor-end=13}{3} n ⩾ 3 a 1 , a 2 , … , a n \htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{a}_{\htmlData{tutor-start=10,tutor-end=11}{2}}\htmlData{tutor-start=12,tutor-end=13}{,} \htmlData{tutor-start=14,tutor-end=20}{\ldots}\htmlData{tutor-start=20,tutor-end=21}{,} \htmlData{tutor-start=22,tutor-end=23}{a}_{\htmlData{tutor-start=25,tutor-end=26}{n}} a 1 , a 2 , … , a n a n + 1 = a 1 , a n + 2 = a 2 \htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{n}\htmlData{tutor-start=4,tutor-end=5}{+}\htmlData{tutor-start=5,tutor-end=6}{1}}\htmlData{tutor-start=7,tutor-end=8}{=}\htmlData{tutor-start=8,tutor-end=9}{a}_{\htmlData{tutor-start=11,tutor-end=12}{1}}\htmlData{tutor-start=13,tutor-end=14}{,} \htmlData{tutor-start=15,tutor-end=16}{a}_{\htmlData{tutor-start=18,tutor-end=19}{n}\htmlData{tutor-start=19,tutor-end=20}{+}\htmlData{tutor-start=20,tutor-end=21}{2}}\htmlData{tutor-start=22,tutor-end=23}{=}\htmlData{tutor-start=23,tutor-end=24}{a}_{\htmlData{tutor-start=26,tutor-end=27}{2}} a n + 1 = a 1 , a n + 2 = a 2 a i a i + 1 + 1 = a i + 2 \htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{i}} \htmlData{tutor-start=6,tutor-end=7}{a}_{\htmlData{tutor-start=9,tutor-end=10}{i}\htmlData{tutor-start=10,tutor-end=11}{+}\htmlData{tutor-start=11,tutor-end=12}{1}}\htmlData{tutor-start=13,tutor-end=14}{+}\htmlData{tutor-start=14,tutor-end=15}{1}\htmlData{tutor-start=15,tutor-end=16}{=}\htmlData{tutor-start=16,tutor-end=17}{a}_{\htmlData{tutor-start=19,tutor-end=20}{i}\htmlData{tutor-start=20,tutor-end=21}{+}\htmlData{tutor-start=21,tutor-end=22}{2}} a i a i + 1 + 1 = a i + 2 i = 1 , 2 , … , n \htmlData{tutor-start=0,tutor-end=1}{i}\htmlData{tutor-start=1,tutor-end=2}{=}\htmlData{tutor-start=2,tutor-end=3}{1}\htmlData{tutor-start=3,tutor-end=4}{,}\htmlData{tutor-start=4,tutor-end=5}{2}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=13}{\ldots}\htmlData{tutor-start=13,tutor-end=14}{,} \htmlData{tutor-start=15,tutor-end=16}{n} i = 1 , 2 , … , n 题解状态： 标准答案与规范题解待补充
题目标签：2018 IMO 正式题 A2
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2018-A3 2018 · Algebra · IMO/代数
Given any set S \htmlData{tutor-start=0,tutor-end=1}{S} S F \htmlData{tutor-start=0,tutor-end=1}{F} F G \htmlData{tutor-start=0,tutor-end=1}{G} G S \htmlData{tutor-start=0,tutor-end=1}{S} S ∑ x ∈ F 1 / x = ∑ x ∈ G 1 / x \sum_{\htmlData{tutor-start=6,tutor-end=7}{x} \htmlData{tutor-start=8,tutor-end=12}{\in }\htmlData{tutor-start=12,tutor-end=13}{F}} \htmlData{tutor-start=15,tutor-end=16}{1} \htmlData{tutor-start=17,tutor-end=18}{/} \htmlData{tutor-start=19,tutor-end=20}{x}\htmlData{tutor-start=20,tutor-end=21}{=}\sum_{\htmlData{tutor-start=27,tutor-end=28}{x} \htmlData{tutor-start=29,tutor-end=33}{\in }\htmlData{tutor-start=33,tutor-end=34}{G}} \htmlData{tutor-start=36,tutor-end=37}{1} \htmlData{tutor-start=38,tutor-end=39}{/} \htmlData{tutor-start=40,tutor-end=41}{x} ∑ x ∈ F 1 / x = ∑ x ∈ G 1 / x r < 1 \htmlData{tutor-start=0,tutor-end=1}{r}\htmlData{tutor-start=1,tutor-end=2}{<}\htmlData{tutor-start=2,tutor-end=3}{1} r < 1 ∑ x ∈ F 1 / x ≠ r \sum_{\htmlData{tutor-start=6,tutor-end=7}{x} \htmlData{tutor-start=8,tutor-end=12}{\in }\htmlData{tutor-start=12,tutor-end=13}{F}} \htmlData{tutor-start=15,tutor-end=16}{1} \htmlData{tutor-start=17,tutor-end=18}{/} \htmlData{tutor-start=19,tutor-end=20}{x} \neq \htmlData{tutor-start=26,tutor-end=27}{r} ∑ x ∈ F 1 / x = r F \htmlData{tutor-start=0,tutor-end=1}{F} F S \htmlData{tutor-start=0,tutor-end=1}{S} S 题解状态： 标准答案与规范题解待补充
题目标签：2018 IMO Shortlist A3
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2018-A4 2018 · Algebra · IMO/代数
Let a 0 , a 1 , a 2 , … \htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{0}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{a}_{\htmlData{tutor-start=10,tutor-end=11}{1}}\htmlData{tutor-start=12,tutor-end=13}{,} \htmlData{tutor-start=14,tutor-end=15}{a}_{\htmlData{tutor-start=17,tutor-end=18}{2}}\htmlData{tutor-start=19,tutor-end=20}{,} \htmlData{tutor-start=21,tutor-end=27}{\ldots} a 0 , a 1 , a 2 , … a 0 = 0 , a 1 = 1 \htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{0}}\htmlData{tutor-start=5,tutor-end=6}{=}\htmlData{tutor-start=6,tutor-end=7}{0}\htmlData{tutor-start=7,tutor-end=8}{,} \htmlData{tutor-start=9,tutor-end=10}{a}_{\htmlData{tutor-start=12,tutor-end=13}{1}}\htmlData{tutor-start=14,tutor-end=15}{=}\htmlData{tutor-start=15,tutor-end=16}{1} a 0 = 0 , a 1 = 1 n ⩾ 2 \htmlData{tutor-start=0,tutor-end=1}{n} \htmlData{tutor-start=2,tutor-end=12}{\geqslant }\htmlData{tutor-start=12,tutor-end=13}{2} n ⩾ 2 1 ⩽ k ⩽ n \htmlData{tutor-start=0,tutor-end=1}{1} \htmlData{tutor-start=2,tutor-end=12}{\leqslant }\htmlData{tutor-start=12,tutor-end=13}{k} \htmlData{tutor-start=14,tutor-end=24}{\leqslant }\htmlData{tutor-start=24,tutor-end=25}{n} 1 ⩽ k ⩽ n a n = a n − 1 + ⋯ + a n − k k \htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{n}}\htmlData{tutor-start=5,tutor-end=6}{=}\frac{\htmlData{tutor-start=12,tutor-end=13}{a}_{\htmlData{tutor-start=15,tutor-end=16}{n}\htmlData{tutor-start=16,tutor-end=17}{-}\htmlData{tutor-start=17,tutor-end=18}{1}}\htmlData{tutor-start=19,tutor-end=20}{+}\cdots\htmlData{tutor-start=26,tutor-end=27}{+}\htmlData{tutor-start=27,tutor-end=28}{a}_{\htmlData{tutor-start=30,tutor-end=31}{n}\htmlData{tutor-start=31,tutor-end=32}{-}\htmlData{tutor-start=32,tutor-end=33}{k}}}{\htmlData{tutor-start=36,tutor-end=37}{k}} a n = k a n − 1 + ⋯ + a n − k a 2018 − a 2017 \htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{2}\htmlData{tutor-start=4,tutor-end=5}{0}\htmlData{tutor-start=5,tutor-end=6}{1}\htmlData{tutor-start=6,tutor-end=7}{8}}\htmlData{tutor-start=8,tutor-end=9}{-}\htmlData{tutor-start=9,tutor-end=10}{a}_{\htmlData{tutor-start=12,tutor-end=13}{2}\htmlData{tutor-start=13,tutor-end=14}{0}\htmlData{tutor-start=14,tutor-end=15}{1}\htmlData{tutor-start=15,tutor-end=16}{7}} a 2 0 1 8 − a 2 0 1 7 题解状态： 标准答案与规范题解待补充
题目标签：2018 IMO Shortlist A4
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2018-A5 2018 · Algebra · IMO/代数
Determine all functions f : ( 0 , ∞ ) → R \htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{:}\htmlData{tutor-start=2,tutor-end=3}{(}\htmlData{tutor-start=3,tutor-end=4}{0}\htmlData{tutor-start=4,tutor-end=5}{,} \htmlData{tutor-start=6,tutor-end=12}{\infty}\htmlData{tutor-start=12,tutor-end=13}{)} \htmlData{tutor-start=14,tutor-end=26}{\rightarrow }\mathbb{\htmlData{tutor-start=34,tutor-end=35}{R}} f : ( 0 , ∞ ) → R ( x + 1 x ) f ( y ) = f ( x y ) + f ( y x ) \left(\htmlData{tutor-start=6,tutor-end=7}{x}\htmlData{tutor-start=7,tutor-end=8}{+}\frac{\htmlData{tutor-start=14,tutor-end=15}{1}}{\htmlData{tutor-start=17,tutor-end=18}{x}}\right) \htmlData{tutor-start=27,tutor-end=28}{f}\htmlData{tutor-start=28,tutor-end=29}{(}\htmlData{tutor-start=29,tutor-end=30}{y}\htmlData{tutor-start=30,tutor-end=31}{)}\htmlData{tutor-start=31,tutor-end=32}{=}\htmlData{tutor-start=32,tutor-end=33}{f}\htmlData{tutor-start=33,tutor-end=34}{(}\htmlData{tutor-start=34,tutor-end=35}{x} \htmlData{tutor-start=36,tutor-end=37}{y}\htmlData{tutor-start=37,tutor-end=38}{)}\htmlData{tutor-start=38,tutor-end=39}{+}\htmlData{tutor-start=39,tutor-end=40}{f}\left(\frac{\htmlData{tutor-start=52,tutor-end=53}{y}}{\htmlData{tutor-start=55,tutor-end=56}{x}}\right) ( x + x 1 ) f ( y ) = f ( x y ) + f ( x y ) x , y > 0 \htmlData{tutor-start=0,tutor-end=1}{x}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{y}\htmlData{tutor-start=4,tutor-end=5}{>}\htmlData{tutor-start=5,tutor-end=6}{0} x , y > 0 题解状态： 标准答案与规范题解待补充
题目标签：2018 IMO Shortlist A5
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2018-A6 2018 · Algebra · IMO/代数
Let m , n ⩾ 2 \htmlData{tutor-start=0,tutor-end=1}{m}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{n} \htmlData{tutor-start=5,tutor-end=15}{\geqslant }\htmlData{tutor-start=15,tutor-end=16}{2} m , n ⩾ 2 f ( x 1 , … , x n ) \htmlData{tutor-start=0,tutor-end=1}{f}\left(\htmlData{tutor-start=7,tutor-end=8}{x}_{\htmlData{tutor-start=10,tutor-end=11}{1}}\htmlData{tutor-start=12,tutor-end=13}{,} \htmlData{tutor-start=14,tutor-end=20}{\ldots}\htmlData{tutor-start=20,tutor-end=21}{,} \htmlData{tutor-start=22,tutor-end=23}{x}_{\htmlData{tutor-start=25,tutor-end=26}{n}}\right) f ( x 1 , … , x n ) f ( x 1 , … , x n ) = ⌊ x 1 + … + x n m ⌋ for every x 1 , … , x n ∈ { 0 , 1 , … , m − 1 } \htmlData{tutor-start=0,tutor-end=1}{f}\left(\htmlData{tutor-start=7,tutor-end=8}{x}_{\htmlData{tutor-start=10,tutor-end=11}{1}}\htmlData{tutor-start=12,tutor-end=13}{,} \htmlData{tutor-start=14,tutor-end=20}{\ldots}\htmlData{tutor-start=20,tutor-end=21}{,} \htmlData{tutor-start=22,tutor-end=23}{x}_{\htmlData{tutor-start=25,tutor-end=26}{n}}\right)\htmlData{tutor-start=34,tutor-end=35}{=}\left\lfloor\frac{\htmlData{tutor-start=53,tutor-end=54}{x}_{\htmlData{tutor-start=56,tutor-end=57}{1}}\htmlData{tutor-start=58,tutor-end=59}{+}\htmlData{tutor-start=59,tutor-end=65}{\ldots}\htmlData{tutor-start=65,tutor-end=66}{+}\htmlData{tutor-start=66,tutor-end=67}{x}_{\htmlData{tutor-start=69,tutor-end=70}{n}}}{\htmlData{tutor-start=73,tutor-end=74}{m}}\right\rfloor \text { \htmlData{tutor-start=97,tutor-end=98}{f}\htmlData{tutor-start=98,tutor-end=99}{o}\htmlData{tutor-start=99,tutor-end=100}{r} \htmlData{tutor-start=101,tutor-end=102}{e}\htmlData{tutor-start=102,tutor-end=103}{v}\htmlData{tutor-start=103,tutor-end=104}{e}\htmlData{tutor-start=104,tutor-end=105}{r}\htmlData{tutor-start=105,tutor-end=106}{y} } \htmlData{tutor-start=109,tutor-end=110}{x}_{\htmlData{tutor-start=112,tutor-end=113}{1}}\htmlData{tutor-start=114,tutor-end=115}{,} \htmlData{tutor-start=116,tutor-end=122}{\ldots}\htmlData{tutor-start=122,tutor-end=123}{,} \htmlData{tutor-start=124,tutor-end=125}{x}_{\htmlData{tutor-start=127,tutor-end=128}{n}} \htmlData{tutor-start=130,tutor-end=133}{\in}\htmlData{tutor-start=133,tutor-end=135}{\{}\htmlData{tutor-start=135,tutor-end=136}{0}\htmlData{tutor-start=136,tutor-end=137}{,}\htmlData{tutor-start=137,tutor-end=138}{1}\htmlData{tutor-start=138,tutor-end=139}{,} \htmlData{tutor-start=140,tutor-end=146}{\ldots}\htmlData{tutor-start=146,tutor-end=147}{,} \htmlData{tutor-start=148,tutor-end=149}{m}\htmlData{tutor-start=149,tutor-end=150}{-}\htmlData{tutor-start=150,tutor-end=151}{1}\htmlData{tutor-start=151,tutor-end=153}{\}} f ( x 1 , … , x n ) = ⌊ m x 1 + … + x n ⌋ f o r e v e r y x 1 , … , x n ∈ { 0 , 1 , … , m − 1 } f \htmlData{tutor-start=0,tutor-end=1}{f} f n \htmlData{tutor-start=0,tutor-end=1}{n} n 题解状态： 标准答案与规范题解待补充
题目标签：2018 IMO Shortlist A6
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2018-A7 2018 · Algebra · IMO/代数
Find the maximal value of
S = a b + 7 3 + b c + 7 3 + c d + 7 3 + d a + 7 3 \htmlData{tutor-start=0,tutor-end=1}{S}\htmlData{tutor-start=1,tutor-end=2}{=}\sqrt[\htmlData{tutor-start=8,tutor-end=9}{3}]{\frac{\htmlData{tutor-start=17,tutor-end=18}{a}}{\htmlData{tutor-start=20,tutor-end=21}{b}\htmlData{tutor-start=21,tutor-end=22}{+}\htmlData{tutor-start=22,tutor-end=23}{7}}}\htmlData{tutor-start=25,tutor-end=26}{+}\sqrt[\htmlData{tutor-start=32,tutor-end=33}{3}]{\frac{\htmlData{tutor-start=41,tutor-end=42}{b}}{\htmlData{tutor-start=44,tutor-end=45}{c}\htmlData{tutor-start=45,tutor-end=46}{+}\htmlData{tutor-start=46,tutor-end=47}{7}}}\htmlData{tutor-start=49,tutor-end=50}{+}\sqrt[\htmlData{tutor-start=56,tutor-end=57}{3}]{\frac{\htmlData{tutor-start=65,tutor-end=66}{c}}{\htmlData{tutor-start=68,tutor-end=69}{d}\htmlData{tutor-start=69,tutor-end=70}{+}\htmlData{tutor-start=70,tutor-end=71}{7}}}\htmlData{tutor-start=73,tutor-end=74}{+}\sqrt[\htmlData{tutor-start=80,tutor-end=81}{3}]{\frac{\htmlData{tutor-start=89,tutor-end=90}{d}}{\htmlData{tutor-start=92,tutor-end=93}{a}\htmlData{tutor-start=93,tutor-end=94}{+}\htmlData{tutor-start=94,tutor-end=95}{7}}} S = 3 b + 7 a + 3 c + 7 b + 3 d + 7 c + 3 a + 7 d a , b , c , d \htmlData{tutor-start=0,tutor-end=1}{a}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{b}\htmlData{tutor-start=4,tutor-end=5}{,} \htmlData{tutor-start=6,tutor-end=7}{c}\htmlData{tutor-start=7,tutor-end=8}{,} \htmlData{tutor-start=9,tutor-end=10}{d} a , b , c , d a + b + c + d = 100 \htmlData{tutor-start=0,tutor-end=1}{a}\htmlData{tutor-start=1,tutor-end=2}{+}\htmlData{tutor-start=2,tutor-end=3}{b}\htmlData{tutor-start=3,tutor-end=4}{+}\htmlData{tutor-start=4,tutor-end=5}{c}\htmlData{tutor-start=5,tutor-end=6}{+}\htmlData{tutor-start=6,tutor-end=7}{d}\htmlData{tutor-start=7,tutor-end=8}{=}\htmlData{tutor-start=8,tutor-end=9}{1}\htmlData{tutor-start=9,tutor-end=10}{0}\htmlData{tutor-start=10,tutor-end=11}{0} a + b + c + d = 1 0 0 题解状态： 标准答案与规范题解待补充
题目标签：2018 IMO Shortlist A7
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2019-A1 2019 · Algebra · IMO/代数
Let Z \mathbb{\htmlData{tutor-start=8,tutor-end=9}{Z}} Z f : Z → Z \htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{:} \mathbb{\htmlData{tutor-start=11,tutor-end=12}{Z}} \htmlData{tutor-start=14,tutor-end=26}{\rightarrow }\mathbb{\htmlData{tutor-start=34,tutor-end=35}{Z}} f : Z → Z a \htmlData{tutor-start=0,tutor-end=1}{a} a b \htmlData{tutor-start=0,tutor-end=1}{b} b f ( 2 a ) + 2 f ( b ) = f ( f ( a + b ) ) \htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{2} \htmlData{tutor-start=4,tutor-end=5}{a}\htmlData{tutor-start=5,tutor-end=6}{)}\htmlData{tutor-start=6,tutor-end=7}{+}\htmlData{tutor-start=7,tutor-end=8}{2} \htmlData{tutor-start=9,tutor-end=10}{f}\htmlData{tutor-start=10,tutor-end=11}{(}\htmlData{tutor-start=11,tutor-end=12}{b}\htmlData{tutor-start=12,tutor-end=13}{)}\htmlData{tutor-start=13,tutor-end=14}{=}\htmlData{tutor-start=14,tutor-end=15}{f}\htmlData{tutor-start=15,tutor-end=16}{(}\htmlData{tutor-start=16,tutor-end=17}{f}\htmlData{tutor-start=17,tutor-end=18}{(}\htmlData{tutor-start=18,tutor-end=19}{a}\htmlData{tutor-start=19,tutor-end=20}{+}\htmlData{tutor-start=20,tutor-end=21}{b}\htmlData{tutor-start=21,tutor-end=22}{)}\htmlData{tutor-start=22,tutor-end=23}{)} f ( 2 a ) + 2 f ( b ) = f ( f ( a + b ) ) 题解状态： 标准答案与规范题解待补充
题目标签：2019 IMO 正式题 A1
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2019-A2 2019 · Algebra · IMO/代数
Let u 1 , u 2 , … , u 2019 \htmlData{tutor-start=0,tutor-end=1}{u}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{u}_{\htmlData{tutor-start=10,tutor-end=11}{2}}\htmlData{tutor-start=12,tutor-end=13}{,} \htmlData{tutor-start=14,tutor-end=20}{\ldots}\htmlData{tutor-start=20,tutor-end=21}{,} \htmlData{tutor-start=22,tutor-end=23}{u}_{\htmlData{tutor-start=25,tutor-end=26}{2}\htmlData{tutor-start=26,tutor-end=27}{0}\htmlData{tutor-start=27,tutor-end=28}{1}\htmlData{tutor-start=28,tutor-end=29}{9}} u 1 , u 2 , … , u 2 0 1 9 u 1 + u 2 + ⋯ + u 2019 = 0 and u 1 2 + u 2 2 + ⋯ + u 2019 2 = 1. \htmlData{tutor-start=0,tutor-end=1}{u}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{+}\htmlData{tutor-start=6,tutor-end=7}{u}_{\htmlData{tutor-start=9,tutor-end=10}{2}}\htmlData{tutor-start=11,tutor-end=12}{+}\cdots\htmlData{tutor-start=18,tutor-end=19}{+}\htmlData{tutor-start=19,tutor-end=20}{u}_{\htmlData{tutor-start=22,tutor-end=23}{2}\htmlData{tutor-start=23,tutor-end=24}{0}\htmlData{tutor-start=24,tutor-end=25}{1}\htmlData{tutor-start=25,tutor-end=26}{9}}\htmlData{tutor-start=27,tutor-end=28}{=}\htmlData{tutor-start=28,tutor-end=29}{0} \quad \text { \htmlData{tutor-start=44,tutor-end=45}{a}\htmlData{tutor-start=45,tutor-end=46}{n}\htmlData{tutor-start=46,tutor-end=47}{d} } \quad \htmlData{tutor-start=56,tutor-end=57}{u}_{\htmlData{tutor-start=59,tutor-end=60}{1}}^{\htmlData{tutor-start=63,tutor-end=64}{2}}\htmlData{tutor-start=65,tutor-end=66}{+}\htmlData{tutor-start=66,tutor-end=67}{u}_{\htmlData{tutor-start=69,tutor-end=70}{2}}^{\htmlData{tutor-start=73,tutor-end=74}{2}}\htmlData{tutor-start=75,tutor-end=76}{+}\cdots\htmlData{tutor-start=82,tutor-end=83}{+}\htmlData{tutor-start=83,tutor-end=84}{u}_{\htmlData{tutor-start=86,tutor-end=87}{2}\htmlData{tutor-start=87,tutor-end=88}{0}\htmlData{tutor-start=88,tutor-end=89}{1}\htmlData{tutor-start=89,tutor-end=90}{9}}^{\htmlData{tutor-start=93,tutor-end=94}{2}}\htmlData{tutor-start=95,tutor-end=96}{=}\htmlData{tutor-start=96,tutor-end=97}{1} \htmlData{tutor-start=98,tutor-end=99}{.} u 1 + u 2 + ⋯ + u 2 0 1 9 = 0 a n d u 1 2 + u 2 2 + ⋯ + u 2 0 1 9 2 = 1 . a = min ( u 1 , u 2 , … , u 2019 ) \htmlData{tutor-start=0,tutor-end=1}{a}\htmlData{tutor-start=1,tutor-end=2}{=}\min \left(\htmlData{tutor-start=13,tutor-end=14}{u}_{\htmlData{tutor-start=16,tutor-end=17}{1}}\htmlData{tutor-start=18,tutor-end=19}{,} \htmlData{tutor-start=20,tutor-end=21}{u}_{\htmlData{tutor-start=23,tutor-end=24}{2}}\htmlData{tutor-start=25,tutor-end=26}{,} \htmlData{tutor-start=27,tutor-end=33}{\ldots}\htmlData{tutor-start=33,tutor-end=34}{,} \htmlData{tutor-start=35,tutor-end=36}{u}_{\htmlData{tutor-start=38,tutor-end=39}{2}\htmlData{tutor-start=39,tutor-end=40}{0}\htmlData{tutor-start=40,tutor-end=41}{1}\htmlData{tutor-start=41,tutor-end=42}{9}}\right) a = min ( u 1 , u 2 , … , u 2 0 1 9 ) b = max ( u 1 , u 2 , … , u 2019 ) \htmlData{tutor-start=0,tutor-end=1}{b}\htmlData{tutor-start=1,tutor-end=2}{=}\max \left(\htmlData{tutor-start=13,tutor-end=14}{u}_{\htmlData{tutor-start=16,tutor-end=17}{1}}\htmlData{tutor-start=18,tutor-end=19}{,} \htmlData{tutor-start=20,tutor-end=21}{u}_{\htmlData{tutor-start=23,tutor-end=24}{2}}\htmlData{tutor-start=25,tutor-end=26}{,} \htmlData{tutor-start=27,tutor-end=33}{\ldots}\htmlData{tutor-start=33,tutor-end=34}{,} \htmlData{tutor-start=35,tutor-end=36}{u}_{\htmlData{tutor-start=38,tutor-end=39}{2}\htmlData{tutor-start=39,tutor-end=40}{0}\htmlData{tutor-start=40,tutor-end=41}{1}\htmlData{tutor-start=41,tutor-end=42}{9}}\right) b = max ( u 1 , u 2 , … , u 2 0 1 9 ) a b ⩽ − 1 2019 \htmlData{tutor-start=0,tutor-end=1}{a} \htmlData{tutor-start=2,tutor-end=3}{b} \htmlData{tutor-start=4,tutor-end=13}{\leqslant}\htmlData{tutor-start=13,tutor-end=14}{-}\frac{\htmlData{tutor-start=20,tutor-end=21}{1}}{\htmlData{tutor-start=23,tutor-end=24}{2}\htmlData{tutor-start=24,tutor-end=25}{0}\htmlData{tutor-start=25,tutor-end=26}{1}\htmlData{tutor-start=26,tutor-end=27}{9}} a b ⩽ − 2 0 1 9 1 题解状态： 标准答案与规范题解待补充
题目标签：2019 IMO Shortlist A2
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2019-A3 2019 · Algebra · IMO/代数
Let n ⩾ 3 \htmlData{tutor-start=0,tutor-end=1}{n} \htmlData{tutor-start=2,tutor-end=12}{\geqslant }\htmlData{tutor-start=12,tutor-end=13}{3} n ⩾ 3 ( a 1 , a 2 , … , a n ) \left(\htmlData{tutor-start=6,tutor-end=7}{a}_{\htmlData{tutor-start=9,tutor-end=10}{1}}\htmlData{tutor-start=11,tutor-end=12}{,} \htmlData{tutor-start=13,tutor-end=14}{a}_{\htmlData{tutor-start=16,tutor-end=17}{2}}\htmlData{tutor-start=18,tutor-end=19}{,} \htmlData{tutor-start=20,tutor-end=26}{\ldots}\htmlData{tutor-start=26,tutor-end=27}{,} \htmlData{tutor-start=28,tutor-end=29}{a}_{\htmlData{tutor-start=31,tutor-end=32}{n}}\right) ( a 1 , a 2 , … , a n ) n \htmlData{tutor-start=0,tutor-end=1}{n} n X \htmlData{tutor-start=0,tutor-end=1}{X} X { 1 , 2 , … , n } \htmlData{tutor-start=0,tutor-end=2}{\{}\htmlData{tutor-start=2,tutor-end=3}{1}\htmlData{tutor-start=3,tutor-end=4}{,}\htmlData{tutor-start=4,tutor-end=5}{2}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=13}{\ldots}\htmlData{tutor-start=13,tutor-end=14}{,} \htmlData{tutor-start=15,tutor-end=16}{n}\htmlData{tutor-start=16,tutor-end=18}{\}} { 1 , 2 , … , n } ∣ 1 − ∑ i ∈ X a i ∣ \left|\htmlData{tutor-start=6,tutor-end=7}{1}\htmlData{tutor-start=7,tutor-end=8}{-}\sum_{\htmlData{tutor-start=14,tutor-end=15}{i} \htmlData{tutor-start=16,tutor-end=20}{\in }\htmlData{tutor-start=20,tutor-end=21}{X}} \htmlData{tutor-start=23,tutor-end=24}{a}_{\htmlData{tutor-start=26,tutor-end=27}{i}}\right| 1 − ∑ i ∈ X a i n \htmlData{tutor-start=0,tutor-end=1}{n} n ( b 1 , b 2 , … , b n ) \left(\htmlData{tutor-start=6,tutor-end=7}{b}_{\htmlData{tutor-start=9,tutor-end=10}{1}}\htmlData{tutor-start=11,tutor-end=12}{,} \htmlData{tutor-start=13,tutor-end=14}{b}_{\htmlData{tutor-start=16,tutor-end=17}{2}}\htmlData{tutor-start=18,tutor-end=19}{,} \htmlData{tutor-start=20,tutor-end=26}{\ldots}\htmlData{tutor-start=26,tutor-end=27}{,} \htmlData{tutor-start=28,tutor-end=29}{b}_{\htmlData{tutor-start=31,tutor-end=32}{n}}\right) ( b 1 , b 2 , … , b n ) ∑ i ∈ X b i = 1 \sum_{\htmlData{tutor-start=6,tutor-end=7}{i} \htmlData{tutor-start=8,tutor-end=12}{\in }\htmlData{tutor-start=12,tutor-end=13}{X}} \htmlData{tutor-start=15,tutor-end=16}{b}_{\htmlData{tutor-start=18,tutor-end=19}{i}}\htmlData{tutor-start=20,tutor-end=21}{=}\htmlData{tutor-start=21,tutor-end=22}{1} ∑ i ∈ X b i = 1 题解状态： 标准答案与规范题解待补充
题目标签：2019 IMO Shortlist A3
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2019-A4 2019 · Algebra · IMO/代数
Let n ⩾ 2 \htmlData{tutor-start=0,tutor-end=1}{n} \htmlData{tutor-start=2,tutor-end=12}{\geqslant }\htmlData{tutor-start=12,tutor-end=13}{2} n ⩾ 2 a 1 , a 2 , … , a n \htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{a}_{\htmlData{tutor-start=10,tutor-end=11}{2}}\htmlData{tutor-start=12,tutor-end=13}{,} \htmlData{tutor-start=14,tutor-end=20}{\ldots}\htmlData{tutor-start=20,tutor-end=21}{,} \htmlData{tutor-start=22,tutor-end=23}{a}_{\htmlData{tutor-start=25,tutor-end=26}{n}} a 1 , a 2 , … , a n a 1 + a 2 + ⋯ + a n = 0 \htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{+}\htmlData{tutor-start=6,tutor-end=7}{a}_{\htmlData{tutor-start=9,tutor-end=10}{2}}\htmlData{tutor-start=11,tutor-end=12}{+}\cdots\htmlData{tutor-start=18,tutor-end=19}{+}\htmlData{tutor-start=19,tutor-end=20}{a}_{\htmlData{tutor-start=22,tutor-end=23}{n}}\htmlData{tutor-start=24,tutor-end=25}{=}\htmlData{tutor-start=25,tutor-end=26}{0} a 1 + a 2 + ⋯ + a n = 0 A \htmlData{tutor-start=0,tutor-end=1}{A} A A = { ( i , j ) ∣ 1 ⩽ i < j ⩽ n , ∣ a i − a j ∣ ⩾ 1 } . \htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{=}\left\{\htmlData{tutor-start=9,tutor-end=10}{(}\htmlData{tutor-start=10,tutor-end=11}{i}\htmlData{tutor-start=11,tutor-end=12}{,} \htmlData{tutor-start=13,tutor-end=14}{j}\htmlData{tutor-start=14,tutor-end=15}{)}\left|\htmlData{tutor-start=21,tutor-end=22}{1} \htmlData{tutor-start=23,tutor-end=33}{\leqslant }\htmlData{tutor-start=33,tutor-end=34}{i}\htmlData{tutor-start=34,tutor-end=35}{<}\htmlData{tutor-start=35,tutor-end=36}{j} \htmlData{tutor-start=37,tutor-end=47}{\leqslant }\htmlData{tutor-start=47,tutor-end=48}{n}\htmlData{tutor-start=48,tutor-end=49}{,}\left|\htmlData{tutor-start=55,tutor-end=56}{a}_{\htmlData{tutor-start=58,tutor-end=59}{i}}\htmlData{tutor-start=60,tutor-end=61}{-}\htmlData{tutor-start=61,tutor-end=62}{a}_{\htmlData{tutor-start=64,tutor-end=65}{j}}\right| \htmlData{tutor-start=74,tutor-end=84}{\geqslant }\htmlData{tutor-start=84,tutor-end=85}{1}\right\} \htmlData{tutor-start=94,tutor-end=95}{.}\right. A = { ( i , j ) ∣ 1 ⩽ i < j ⩽ n , ∣ a i − a j ∣ ⩾ 1 } . A \htmlData{tutor-start=0,tutor-end=1}{A} A ∑ ( i , j ) ∈ A a i a j < 0 \sum_{\htmlData{tutor-start=6,tutor-end=7}{(}\htmlData{tutor-start=7,tutor-end=8}{i}\htmlData{tutor-start=8,tutor-end=9}{,} \htmlData{tutor-start=10,tutor-end=11}{j}\htmlData{tutor-start=11,tutor-end=12}{)} \htmlData{tutor-start=13,tutor-end=17}{\in }\htmlData{tutor-start=17,tutor-end=18}{A}} \htmlData{tutor-start=20,tutor-end=21}{a}_{\htmlData{tutor-start=23,tutor-end=24}{i}} \htmlData{tutor-start=26,tutor-end=27}{a}_{\htmlData{tutor-start=29,tutor-end=30}{j}}\htmlData{tutor-start=31,tutor-end=32}{<}\htmlData{tutor-start=32,tutor-end=33}{0} ∑ ( i , j ) ∈ A a i a j < 0 题解状态： 标准答案与规范题解待补充
题目标签：2019 IMO Shortlist A4
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2019-A5 2019 · Algebra · IMO/代数
Let x 1 , x 2 , … , x n \htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{x}_{\htmlData{tutor-start=10,tutor-end=11}{2}}\htmlData{tutor-start=12,tutor-end=13}{,} \htmlData{tutor-start=14,tutor-end=20}{\ldots}\htmlData{tutor-start=20,tutor-end=21}{,} \htmlData{tutor-start=22,tutor-end=23}{x}_{\htmlData{tutor-start=25,tutor-end=26}{n}} x 1 , x 2 , … , x n ∑ 1 ⩽ i ⩽ n ∏ j ≠ i 1 − x i x j x i − x j = { 0 , if n is even 1 , if n is odd \sum_{1 \leqslant i \leqslant n} \prod_{j \neq i} \frac{1-x_{i} x_{j}}{x_{i}-x_{j}}= \begin{cases}0, & \text { if } n \text { is even } \\ 1, & \text { if } n \text { is odd }\end{cases} ∑ 1 ⩽ i ⩽ n ∏ j = i x i − x j 1 − x i x j = { 0 , 1 , if n is even if n is odd 题解状态： 标准答案与规范题解待补充
题目标签：2019 IMO Shortlist A5
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2019-A6 2019 · Algebra · IMO/代数
A polynomial P ( x , y , z ) \htmlData{tutor-start=0,tutor-end=1}{P}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{x}\htmlData{tutor-start=3,tutor-end=4}{,} \htmlData{tutor-start=5,tutor-end=6}{y}\htmlData{tutor-start=6,tutor-end=7}{,} \htmlData{tutor-start=8,tutor-end=9}{z}\htmlData{tutor-start=9,tutor-end=10}{)} P ( x , y , z ) P ( x , y , z ) = P ( x , y , x y − z ) = P ( x , z x − y , z ) = P ( y z − x , y , z ) . \htmlData{tutor-start=0,tutor-end=1}{P}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{x}\htmlData{tutor-start=3,tutor-end=4}{,} \htmlData{tutor-start=5,tutor-end=6}{y}\htmlData{tutor-start=6,tutor-end=7}{,} \htmlData{tutor-start=8,tutor-end=9}{z}\htmlData{tutor-start=9,tutor-end=10}{)}\htmlData{tutor-start=10,tutor-end=11}{=}\htmlData{tutor-start=11,tutor-end=12}{P}\htmlData{tutor-start=12,tutor-end=13}{(}\htmlData{tutor-start=13,tutor-end=14}{x}\htmlData{tutor-start=14,tutor-end=15}{,} \htmlData{tutor-start=16,tutor-end=17}{y}\htmlData{tutor-start=17,tutor-end=18}{,} \htmlData{tutor-start=19,tutor-end=20}{x} \htmlData{tutor-start=21,tutor-end=22}{y}\htmlData{tutor-start=22,tutor-end=23}{-}\htmlData{tutor-start=23,tutor-end=24}{z}\htmlData{tutor-start=24,tutor-end=25}{)}\htmlData{tutor-start=25,tutor-end=26}{=}\htmlData{tutor-start=26,tutor-end=27}{P}\htmlData{tutor-start=27,tutor-end=28}{(}\htmlData{tutor-start=28,tutor-end=29}{x}\htmlData{tutor-start=29,tutor-end=30}{,} \htmlData{tutor-start=31,tutor-end=32}{z} \htmlData{tutor-start=33,tutor-end=34}{x}\htmlData{tutor-start=34,tutor-end=35}{-}\htmlData{tutor-start=35,tutor-end=36}{y}\htmlData{tutor-start=36,tutor-end=37}{,} \htmlData{tutor-start=38,tutor-end=39}{z}\htmlData{tutor-start=39,tutor-end=40}{)}\htmlData{tutor-start=40,tutor-end=41}{=}\htmlData{tutor-start=41,tutor-end=42}{P}\htmlData{tutor-start=42,tutor-end=43}{(}\htmlData{tutor-start=43,tutor-end=44}{y} \htmlData{tutor-start=45,tutor-end=46}{z}\htmlData{tutor-start=46,tutor-end=47}{-}\htmlData{tutor-start=47,tutor-end=48}{x}\htmlData{tutor-start=48,tutor-end=49}{,} \htmlData{tutor-start=50,tutor-end=51}{y}\htmlData{tutor-start=51,tutor-end=52}{,} \htmlData{tutor-start=53,tutor-end=54}{z}\htmlData{tutor-start=54,tutor-end=55}{)} \htmlData{tutor-start=56,tutor-end=57}{.} P ( x , y , z ) = P ( x , y , x y − z ) = P ( x , z x − y , z ) = P ( y z − x , y , z ) . F ( t ) \htmlData{tutor-start=0,tutor-end=1}{F}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{t}\htmlData{tutor-start=3,tutor-end=4}{)} F ( t ) P ( x , y , z ) = F ( x 2 + y 2 + z 2 − x y z ) . \htmlData{tutor-start=0,tutor-end=1}{P}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{x}\htmlData{tutor-start=3,tutor-end=4}{,} \htmlData{tutor-start=5,tutor-end=6}{y}\htmlData{tutor-start=6,tutor-end=7}{,} \htmlData{tutor-start=8,tutor-end=9}{z}\htmlData{tutor-start=9,tutor-end=10}{)}\htmlData{tutor-start=10,tutor-end=11}{=}\htmlData{tutor-start=11,tutor-end=12}{F}\left(\htmlData{tutor-start=18,tutor-end=19}{x}^{\htmlData{tutor-start=21,tutor-end=22}{2}}\htmlData{tutor-start=23,tutor-end=24}{+}\htmlData{tutor-start=24,tutor-end=25}{y}^{\htmlData{tutor-start=27,tutor-end=28}{2}}\htmlData{tutor-start=29,tutor-end=30}{+}\htmlData{tutor-start=30,tutor-end=31}{z}^{\htmlData{tutor-start=33,tutor-end=34}{2}}\htmlData{tutor-start=35,tutor-end=36}{-}\htmlData{tutor-start=36,tutor-end=37}{x} \htmlData{tutor-start=38,tutor-end=39}{y} \htmlData{tutor-start=40,tutor-end=41}{z}\right) \htmlData{tutor-start=49,tutor-end=50}{.} P ( x , y , z ) = F ( x 2 + y 2 + z 2 − x y z ) . 题解状态： 标准答案与规范题解待补充
题目标签：2019 IMO Shortlist A6
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2019-A7 2019 · Algebra · IMO/代数
Let Z \mathbb{\htmlData{tutor-start=8,tutor-end=9}{Z}} Z f : Z → Z \htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{:} \mathbb{\htmlData{tutor-start=11,tutor-end=12}{Z}} \htmlData{tutor-start=14,tutor-end=26}{\rightarrow }\mathbb{\htmlData{tutor-start=34,tutor-end=35}{Z}} f : Z → Z f ( f ( x + y ) + y ) = f ( f ( x ) + y ) \htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{f}\htmlData{tutor-start=3,tutor-end=4}{(}\htmlData{tutor-start=4,tutor-end=5}{x}\htmlData{tutor-start=5,tutor-end=6}{+}\htmlData{tutor-start=6,tutor-end=7}{y}\htmlData{tutor-start=7,tutor-end=8}{)}\htmlData{tutor-start=8,tutor-end=9}{+}\htmlData{tutor-start=9,tutor-end=10}{y}\htmlData{tutor-start=10,tutor-end=11}{)}\htmlData{tutor-start=11,tutor-end=12}{=}\htmlData{tutor-start=12,tutor-end=13}{f}\htmlData{tutor-start=13,tutor-end=14}{(}\htmlData{tutor-start=14,tutor-end=15}{f}\htmlData{tutor-start=15,tutor-end=16}{(}\htmlData{tutor-start=16,tutor-end=17}{x}\htmlData{tutor-start=17,tutor-end=18}{)}\htmlData{tutor-start=18,tutor-end=19}{+}\htmlData{tutor-start=19,tutor-end=20}{y}\htmlData{tutor-start=20,tutor-end=21}{)} f ( f ( x + y ) + y ) = f ( f ( x ) + y ) x \htmlData{tutor-start=0,tutor-end=1}{x} x y \htmlData{tutor-start=0,tutor-end=1}{y} y v \htmlData{tutor-start=0,tutor-end=1}{v} v f \htmlData{tutor-start=0,tutor-end=1}{f} f X v = { x ∈ Z : f ( x ) = v } \htmlData{tutor-start=0,tutor-end=1}{X}_{\htmlData{tutor-start=3,tutor-end=4}{v}}\htmlData{tutor-start=5,tutor-end=6}{=}\htmlData{tutor-start=6,tutor-end=8}{\{}\htmlData{tutor-start=8,tutor-end=9}{x} \htmlData{tutor-start=10,tutor-end=14}{\in }\mathbb{\htmlData{tutor-start=22,tutor-end=23}{Z}}\htmlData{tutor-start=24,tutor-end=25}{:} \htmlData{tutor-start=26,tutor-end=27}{f}\htmlData{tutor-start=27,tutor-end=28}{(}\htmlData{tutor-start=28,tutor-end=29}{x}\htmlData{tutor-start=29,tutor-end=30}{)}\htmlData{tutor-start=30,tutor-end=31}{=}\htmlData{tutor-start=31,tutor-end=32}{v}\htmlData{tutor-start=32,tutor-end=34}{\}} X v = { x ∈ Z : f ( x ) = v } f \htmlData{tutor-start=0,tutor-end=1}{f} f f \htmlData{tutor-start=0,tutor-end=1}{f} f f \htmlData{tutor-start=0,tutor-end=1}{f} f f \htmlData{tutor-start=0,tutor-end=1}{f} f 题解状态： 标准答案与规范题解待补充
题目标签：2019 IMO Shortlist A7
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2020-A1 2020 · Algebra · IMO/代数
Version 1. Let n \htmlData{tutor-start=0,tutor-end=1}{n} n N = 2 n \htmlData{tutor-start=0,tutor-end=1}{N}\htmlData{tutor-start=1,tutor-end=2}{=}\htmlData{tutor-start=2,tutor-end=3}{2}^{\htmlData{tutor-start=5,tutor-end=6}{n}} N = 2 n a n \htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{n}} a n x \htmlData{tutor-start=0,tutor-end=1}{x} x x 2 N + 1 2 N ⩽ a n ( x − 1 ) 2 + x \sqrt[\htmlData{tutor-start=6,tutor-end=7}{N}]{\frac{\htmlData{tutor-start=15,tutor-end=16}{x}^{\htmlData{tutor-start=18,tutor-end=19}{2} \htmlData{tutor-start=20,tutor-end=21}{N}}\htmlData{tutor-start=22,tutor-end=23}{+}\htmlData{tutor-start=23,tutor-end=24}{1}}{\htmlData{tutor-start=26,tutor-end=27}{2}}} \htmlData{tutor-start=30,tutor-end=40}{\leqslant }\htmlData{tutor-start=40,tutor-end=41}{a}_{\htmlData{tutor-start=43,tutor-end=44}{n}}\htmlData{tutor-start=45,tutor-end=46}{(}\htmlData{tutor-start=46,tutor-end=47}{x}\htmlData{tutor-start=47,tutor-end=48}{-}\htmlData{tutor-start=48,tutor-end=49}{1}\htmlData{tutor-start=49,tutor-end=50}{)}^{\htmlData{tutor-start=52,tutor-end=53}{2}}\htmlData{tutor-start=54,tutor-end=55}{+}\htmlData{tutor-start=55,tutor-end=56}{x} N 2 x 2 N + 1 ⩽ a n ( x − 1 ) 2 + x N \htmlData{tutor-start=0,tutor-end=1}{N} N b N \htmlData{tutor-start=0,tutor-end=1}{b}_{\htmlData{tutor-start=3,tutor-end=4}{N}} b N x \htmlData{tutor-start=0,tutor-end=1}{x} x x 2 N + 1 2 N ⩽ b N ( x − 1 ) 2 + x \sqrt[\htmlData{tutor-start=6,tutor-end=7}{N}]{\frac{\htmlData{tutor-start=15,tutor-end=16}{x}^{\htmlData{tutor-start=18,tutor-end=19}{2} \htmlData{tutor-start=20,tutor-end=21}{N}}\htmlData{tutor-start=22,tutor-end=23}{+}\htmlData{tutor-start=23,tutor-end=24}{1}}{\htmlData{tutor-start=26,tutor-end=27}{2}}} \htmlData{tutor-start=30,tutor-end=40}{\leqslant }\htmlData{tutor-start=40,tutor-end=41}{b}_{\htmlData{tutor-start=43,tutor-end=44}{N}}\htmlData{tutor-start=45,tutor-end=46}{(}\htmlData{tutor-start=46,tutor-end=47}{x}\htmlData{tutor-start=47,tutor-end=48}{-}\htmlData{tutor-start=48,tutor-end=49}{1}\htmlData{tutor-start=49,tutor-end=50}{)}^{\htmlData{tutor-start=52,tutor-end=53}{2}}\htmlData{tutor-start=54,tutor-end=55}{+}\htmlData{tutor-start=55,tutor-end=56}{x} N 2 x 2 N + 1 ⩽ b N ( x − 1 ) 2 + x 题解状态： 标准答案与规范题解待补充
题目标签：2020 IMO Shortlist A1
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2020-A2 2020 · Algebra · IMO/代数
Let A \mathcal{\htmlData{tutor-start=9,tutor-end=10}{A}} A x , y , z \htmlData{tutor-start=0,tutor-end=1}{x}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{y}\htmlData{tutor-start=4,tutor-end=5}{,} \htmlData{tutor-start=6,tutor-end=7}{z} x , y , z B \mathcal{\htmlData{tutor-start=9,tutor-end=10}{B}} B A \mathcal{\htmlData{tutor-start=9,tutor-end=10}{A}} A ( x + y + z ) P ( x , y , z ) + ( x y + y z + z x ) Q ( x , y , z ) + x y z R ( x , y , z ) \htmlData{tutor-start=0,tutor-end=1}{(}\htmlData{tutor-start=1,tutor-end=2}{x}\htmlData{tutor-start=2,tutor-end=3}{+}\htmlData{tutor-start=3,tutor-end=4}{y}\htmlData{tutor-start=4,tutor-end=5}{+}\htmlData{tutor-start=5,tutor-end=6}{z}\htmlData{tutor-start=6,tutor-end=7}{)} \htmlData{tutor-start=8,tutor-end=9}{P}\htmlData{tutor-start=9,tutor-end=10}{(}\htmlData{tutor-start=10,tutor-end=11}{x}\htmlData{tutor-start=11,tutor-end=12}{,} \htmlData{tutor-start=13,tutor-end=14}{y}\htmlData{tutor-start=14,tutor-end=15}{,} \htmlData{tutor-start=16,tutor-end=17}{z}\htmlData{tutor-start=17,tutor-end=18}{)}\htmlData{tutor-start=18,tutor-end=19}{+}\htmlData{tutor-start=19,tutor-end=20}{(}\htmlData{tutor-start=20,tutor-end=21}{x} \htmlData{tutor-start=22,tutor-end=23}{y}\htmlData{tutor-start=23,tutor-end=24}{+}\htmlData{tutor-start=24,tutor-end=25}{y} \htmlData{tutor-start=26,tutor-end=27}{z}\htmlData{tutor-start=27,tutor-end=28}{+}\htmlData{tutor-start=28,tutor-end=29}{z} \htmlData{tutor-start=30,tutor-end=31}{x}\htmlData{tutor-start=31,tutor-end=32}{)} \htmlData{tutor-start=33,tutor-end=34}{Q}\htmlData{tutor-start=34,tutor-end=35}{(}\htmlData{tutor-start=35,tutor-end=36}{x}\htmlData{tutor-start=36,tutor-end=37}{,} \htmlData{tutor-start=38,tutor-end=39}{y}\htmlData{tutor-start=39,tutor-end=40}{,} \htmlData{tutor-start=41,tutor-end=42}{z}\htmlData{tutor-start=42,tutor-end=43}{)}\htmlData{tutor-start=43,tutor-end=44}{+}\htmlData{tutor-start=44,tutor-end=45}{x} \htmlData{tutor-start=46,tutor-end=47}{y} \htmlData{tutor-start=48,tutor-end=49}{z} \htmlData{tutor-start=50,tutor-end=51}{R}\htmlData{tutor-start=51,tutor-end=52}{(}\htmlData{tutor-start=52,tutor-end=53}{x}\htmlData{tutor-start=53,tutor-end=54}{,} \htmlData{tutor-start=55,tutor-end=56}{y}\htmlData{tutor-start=56,tutor-end=57}{,} \htmlData{tutor-start=58,tutor-end=59}{z}\htmlData{tutor-start=59,tutor-end=60}{)} ( x + y + z ) P ( x , y , z ) + ( x y + y z + z x ) Q ( x , y , z ) + x y z R ( x , y , z ) P , Q , R ∈ A \htmlData{tutor-start=0,tutor-end=1}{P}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{Q}\htmlData{tutor-start=4,tutor-end=5}{,} \htmlData{tutor-start=6,tutor-end=7}{R} \htmlData{tutor-start=8,tutor-end=12}{\in }\mathcal{\htmlData{tutor-start=21,tutor-end=22}{A}} P , Q , R ∈ A n \htmlData{tutor-start=0,tutor-end=1}{n} n x i y j z k ∈ B \htmlData{tutor-start=0,tutor-end=1}{x}^{\htmlData{tutor-start=3,tutor-end=4}{i}} \htmlData{tutor-start=6,tutor-end=7}{y}^{\htmlData{tutor-start=9,tutor-end=10}{j}} \htmlData{tutor-start=12,tutor-end=13}{z}^{\htmlData{tutor-start=15,tutor-end=16}{k}} \htmlData{tutor-start=18,tutor-end=22}{\in }\mathcal{\htmlData{tutor-start=31,tutor-end=32}{B}} x i y j z k ∈ B i , j , k \htmlData{tutor-start=0,tutor-end=1}{i}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{j}\htmlData{tutor-start=4,tutor-end=5}{,} \htmlData{tutor-start=6,tutor-end=7}{k} i , j , k i + j + k ⩾ n \htmlData{tutor-start=0,tutor-end=1}{i}\htmlData{tutor-start=1,tutor-end=2}{+}\htmlData{tutor-start=2,tutor-end=3}{j}\htmlData{tutor-start=3,tutor-end=4}{+}\htmlData{tutor-start=4,tutor-end=5}{k} \htmlData{tutor-start=6,tutor-end=16}{\geqslant }\htmlData{tutor-start=16,tutor-end=17}{n} i + j + k ⩾ n 题解状态： 标准答案与规范题解待补充
题目标签：2020 IMO Shortlist A2
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2020-A3 2020 · Algebra · IMO/代数
Suppose that a , b , c , d \htmlData{tutor-start=0,tutor-end=1}{a}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{b}\htmlData{tutor-start=4,tutor-end=5}{,} \htmlData{tutor-start=6,tutor-end=7}{c}\htmlData{tutor-start=7,tutor-end=8}{,} \htmlData{tutor-start=9,tutor-end=10}{d} a , b , c , d ( a + c ) ( b + d ) = a c + b d \htmlData{tutor-start=0,tutor-end=1}{(}\htmlData{tutor-start=1,tutor-end=2}{a}\htmlData{tutor-start=2,tutor-end=3}{+}\htmlData{tutor-start=3,tutor-end=4}{c}\htmlData{tutor-start=4,tutor-end=5}{)}\htmlData{tutor-start=5,tutor-end=6}{(}\htmlData{tutor-start=6,tutor-end=7}{b}\htmlData{tutor-start=7,tutor-end=8}{+}\htmlData{tutor-start=8,tutor-end=9}{d}\htmlData{tutor-start=9,tutor-end=10}{)}\htmlData{tutor-start=10,tutor-end=11}{=}\htmlData{tutor-start=11,tutor-end=12}{a} \htmlData{tutor-start=13,tutor-end=14}{c}\htmlData{tutor-start=14,tutor-end=15}{+}\htmlData{tutor-start=15,tutor-end=16}{b} \htmlData{tutor-start=17,tutor-end=18}{d} ( a + c ) ( b + d ) = a c + b d a b + b c + c d + d a \frac{\htmlData{tutor-start=6,tutor-end=7}{a}}{\htmlData{tutor-start=9,tutor-end=10}{b}}\htmlData{tutor-start=11,tutor-end=12}{+}\frac{\htmlData{tutor-start=18,tutor-end=19}{b}}{\htmlData{tutor-start=21,tutor-end=22}{c}}\htmlData{tutor-start=23,tutor-end=24}{+}\frac{\htmlData{tutor-start=30,tutor-end=31}{c}}{\htmlData{tutor-start=33,tutor-end=34}{d}}\htmlData{tutor-start=35,tutor-end=36}{+}\frac{\htmlData{tutor-start=42,tutor-end=43}{d}}{\htmlData{tutor-start=45,tutor-end=46}{a}} b a + c b + d c + a d 题解状态： 标准答案与规范题解待补充
题目标签：2020 IMO Shortlist A3
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2020-A4 2020 · Algebra · IMO/代数
Let a , b , c , d \htmlData{tutor-start=0,tutor-end=1}{a}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{b}\htmlData{tutor-start=4,tutor-end=5}{,} \htmlData{tutor-start=6,tutor-end=7}{c}\htmlData{tutor-start=7,tutor-end=8}{,} \htmlData{tutor-start=9,tutor-end=10}{d} a , b , c , d a ⩾ b ⩾ c ⩾ d > 0 \htmlData{tutor-start=0,tutor-end=1}{a} \htmlData{tutor-start=2,tutor-end=12}{\geqslant }\htmlData{tutor-start=12,tutor-end=13}{b} \htmlData{tutor-start=14,tutor-end=24}{\geqslant }\htmlData{tutor-start=24,tutor-end=25}{c} \htmlData{tutor-start=26,tutor-end=36}{\geqslant }\htmlData{tutor-start=36,tutor-end=37}{d}\htmlData{tutor-start=37,tutor-end=38}{>}\htmlData{tutor-start=38,tutor-end=39}{0} a ⩾ b ⩾ c ⩾ d > 0 a + b + c + d = 1 \htmlData{tutor-start=0,tutor-end=1}{a}\htmlData{tutor-start=1,tutor-end=2}{+}\htmlData{tutor-start=2,tutor-end=3}{b}\htmlData{tutor-start=3,tutor-end=4}{+}\htmlData{tutor-start=4,tutor-end=5}{c}\htmlData{tutor-start=5,tutor-end=6}{+}\htmlData{tutor-start=6,tutor-end=7}{d}\htmlData{tutor-start=7,tutor-end=8}{=}\htmlData{tutor-start=8,tutor-end=9}{1} a + b + c + d = 1 ( a + 2 b + 3 c + 4 d ) a a b b c c d d < 1 \htmlData{tutor-start=0,tutor-end=1}{(}\htmlData{tutor-start=1,tutor-end=2}{a}\htmlData{tutor-start=2,tutor-end=3}{+}\htmlData{tutor-start=3,tutor-end=4}{2} \htmlData{tutor-start=5,tutor-end=6}{b}\htmlData{tutor-start=6,tutor-end=7}{+}\htmlData{tutor-start=7,tutor-end=8}{3} \htmlData{tutor-start=9,tutor-end=10}{c}\htmlData{tutor-start=10,tutor-end=11}{+}\htmlData{tutor-start=11,tutor-end=12}{4} \htmlData{tutor-start=13,tutor-end=14}{d}\htmlData{tutor-start=14,tutor-end=15}{)} \htmlData{tutor-start=16,tutor-end=17}{a}^{\htmlData{tutor-start=19,tutor-end=20}{a}} \htmlData{tutor-start=22,tutor-end=23}{b}^{\htmlData{tutor-start=25,tutor-end=26}{b}} \htmlData{tutor-start=28,tutor-end=29}{c}^{\htmlData{tutor-start=31,tutor-end=32}{c}} \htmlData{tutor-start=34,tutor-end=35}{d}^{\htmlData{tutor-start=37,tutor-end=38}{d}}\htmlData{tutor-start=39,tutor-end=40}{<}\htmlData{tutor-start=40,tutor-end=41}{1} ( a + 2 b + 3 c + 4 d ) a a b b c c d d < 1 题解状态： 标准答案与规范题解待补充
题目标签：2020 IMO 正式题 A4
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2020-A5 2020 · Algebra · IMO/代数
A magician intends to perform the following trick. She announces a positive integer n \htmlData{tutor-start=0,tutor-end=1}{n} n 2 n \htmlData{tutor-start=0,tutor-end=1}{2} \htmlData{tutor-start=2,tutor-end=3}{n} 2 n x 1 < … < x 2 n \htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{<}\htmlData{tutor-start=6,tutor-end=12}{\ldots}\htmlData{tutor-start=12,tutor-end=13}{<}\htmlData{tutor-start=13,tutor-end=14}{x}_{\htmlData{tutor-start=16,tutor-end=17}{2} \htmlData{tutor-start=18,tutor-end=19}{n}} x 1 < … < x 2 n P ( x ) \htmlData{tutor-start=0,tutor-end=1}{P}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{x}\htmlData{tutor-start=3,tutor-end=4}{)} P ( x ) n \htmlData{tutor-start=0,tutor-end=1}{n} n 2 n \htmlData{tutor-start=0,tutor-end=1}{2} \htmlData{tutor-start=2,tutor-end=3}{n} 2 n P ( x 1 ) , … , P ( x 2 n ) \htmlData{tutor-start=0,tutor-end=1}{P}\left(\htmlData{tutor-start=7,tutor-end=8}{x}_{\htmlData{tutor-start=10,tutor-end=11}{1}}\right)\htmlData{tutor-start=19,tutor-end=20}{,} \htmlData{tutor-start=21,tutor-end=27}{\ldots}\htmlData{tutor-start=27,tutor-end=28}{,} \htmlData{tutor-start=29,tutor-end=30}{P}\left(\htmlData{tutor-start=36,tutor-end=37}{x}_{\htmlData{tutor-start=39,tutor-end=40}{2} \htmlData{tutor-start=41,tutor-end=42}{n}}\right) P ( x 1 ) , … , P ( x 2 n ) 2 n \htmlData{tutor-start=0,tutor-end=1}{2} \htmlData{tutor-start=2,tutor-end=3}{n} 2 n 题解状态： 标准答案与规范题解待补充
题目标签：2020 IMO Shortlist A5
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2020-A6 2020 · Algebra · IMO/代数
Determine all functions f : Z → Z \htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{:} \mathbb{\htmlData{tutor-start=11,tutor-end=12}{Z}} \htmlData{tutor-start=14,tutor-end=26}{\rightarrow }\mathbb{\htmlData{tutor-start=34,tutor-end=35}{Z}} f : Z → Z f a 2 + b 2 ( a + b ) = a f ( a ) + b f ( b ) for every a , b ∈ Z \htmlData{tutor-start=0,tutor-end=1}{f}^{\htmlData{tutor-start=3,tutor-end=4}{a}^{\htmlData{tutor-start=6,tutor-end=7}{2}}\htmlData{tutor-start=8,tutor-end=9}{+}\htmlData{tutor-start=9,tutor-end=10}{b}^{\htmlData{tutor-start=12,tutor-end=13}{2}}}\htmlData{tutor-start=15,tutor-end=16}{(}\htmlData{tutor-start=16,tutor-end=17}{a}\htmlData{tutor-start=17,tutor-end=18}{+}\htmlData{tutor-start=18,tutor-end=19}{b}\htmlData{tutor-start=19,tutor-end=20}{)}\htmlData{tutor-start=20,tutor-end=21}{=}\htmlData{tutor-start=21,tutor-end=22}{a} \htmlData{tutor-start=23,tutor-end=24}{f}\htmlData{tutor-start=24,tutor-end=25}{(}\htmlData{tutor-start=25,tutor-end=26}{a}\htmlData{tutor-start=26,tutor-end=27}{)}\htmlData{tutor-start=27,tutor-end=28}{+}\htmlData{tutor-start=28,tutor-end=29}{b} \htmlData{tutor-start=30,tutor-end=31}{f}\htmlData{tutor-start=31,tutor-end=32}{(}\htmlData{tutor-start=32,tutor-end=33}{b}\htmlData{tutor-start=33,tutor-end=34}{)} \quad \text { \htmlData{tutor-start=49,tutor-end=50}{f}\htmlData{tutor-start=50,tutor-end=51}{o}\htmlData{tutor-start=51,tutor-end=52}{r} \htmlData{tutor-start=53,tutor-end=54}{e}\htmlData{tutor-start=54,tutor-end=55}{v}\htmlData{tutor-start=55,tutor-end=56}{e}\htmlData{tutor-start=56,tutor-end=57}{r}\htmlData{tutor-start=57,tutor-end=58}{y} } \htmlData{tutor-start=61,tutor-end=62}{a}\htmlData{tutor-start=62,tutor-end=63}{,} \htmlData{tutor-start=64,tutor-end=65}{b} \htmlData{tutor-start=66,tutor-end=70}{\in }\mathbb{\htmlData{tutor-start=78,tutor-end=79}{Z}} f a 2 + b 2 ( a + b ) = a f ( a ) + b f ( b ) f o r e v e r y a , b ∈ Z f n \htmlData{tutor-start=0,tutor-end=1}{f}^{\htmlData{tutor-start=3,tutor-end=4}{n}} f n n th \htmlData{tutor-start=0,tutor-end=1}{n}^{\text {\htmlData{tutor-start=10,tutor-end=11}{t}\htmlData{tutor-start=11,tutor-end=12}{h} }} n t h f \htmlData{tutor-start=0,tutor-end=1}{f} f f 0 ( x ) = x \htmlData{tutor-start=0,tutor-end=1}{f}^{\htmlData{tutor-start=3,tutor-end=4}{0}}\htmlData{tutor-start=5,tutor-end=6}{(}\htmlData{tutor-start=6,tutor-end=7}{x}\htmlData{tutor-start=7,tutor-end=8}{)}\htmlData{tutor-start=8,tutor-end=9}{=}\htmlData{tutor-start=9,tutor-end=10}{x} f 0 ( x ) = x f n + 1 ( x ) = f ( f n ( x ) ) \htmlData{tutor-start=0,tutor-end=1}{f}^{\htmlData{tutor-start=3,tutor-end=4}{n}\htmlData{tutor-start=4,tutor-end=5}{+}\htmlData{tutor-start=5,tutor-end=6}{1}}\htmlData{tutor-start=7,tutor-end=8}{(}\htmlData{tutor-start=8,tutor-end=9}{x}\htmlData{tutor-start=9,tutor-end=10}{)}\htmlData{tutor-start=10,tutor-end=11}{=}\htmlData{tutor-start=11,tutor-end=12}{f}\left(\htmlData{tutor-start=18,tutor-end=19}{f}^{\htmlData{tutor-start=21,tutor-end=22}{n}}\htmlData{tutor-start=23,tutor-end=24}{(}\htmlData{tutor-start=24,tutor-end=25}{x}\htmlData{tutor-start=25,tutor-end=26}{)}\right) f n + 1 ( x ) = f ( f n ( x ) ) n ⩾ 0 \htmlData{tutor-start=0,tutor-end=1}{n} \htmlData{tutor-start=2,tutor-end=12}{\geqslant }\htmlData{tutor-start=12,tutor-end=13}{0} n ⩾ 0 题解状态： 标准答案与规范题解待补充
题目标签：2020 IMO Shortlist A6
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2020-A7 2020 · Algebra · IMO/代数
Let n \htmlData{tutor-start=0,tutor-end=1}{n} n k \htmlData{tutor-start=0,tutor-end=1}{k} k a 1 , … , a n ∈ [ 1 , 2 k ] \htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=13}{\ldots}\htmlData{tutor-start=13,tutor-end=14}{,} \htmlData{tutor-start=15,tutor-end=16}{a}_{\htmlData{tutor-start=18,tutor-end=19}{n}} \htmlData{tutor-start=21,tutor-end=24}{\in}\left[\htmlData{tutor-start=30,tutor-end=31}{1}\htmlData{tutor-start=31,tutor-end=32}{,}\htmlData{tutor-start=32,tutor-end=33}{2}^{\htmlData{tutor-start=35,tutor-end=36}{k}}\right] a 1 , … , a n ∈ [ 1 , 2 k ] ∑ i = 1 n a i a 1 2 + … + a i 2 ⩽ 4 k n \sum_{\htmlData{tutor-start=6,tutor-end=7}{i}\htmlData{tutor-start=7,tutor-end=8}{=}\htmlData{tutor-start=8,tutor-end=9}{1}}^{\htmlData{tutor-start=12,tutor-end=13}{n}} \frac{\htmlData{tutor-start=21,tutor-end=22}{a}_{\htmlData{tutor-start=24,tutor-end=25}{i}}}{\sqrt{\htmlData{tutor-start=34,tutor-end=35}{a}_{\htmlData{tutor-start=37,tutor-end=38}{1}}^{\htmlData{tutor-start=41,tutor-end=42}{2}}\htmlData{tutor-start=43,tutor-end=44}{+}\htmlData{tutor-start=44,tutor-end=50}{\ldots}\htmlData{tutor-start=50,tutor-end=51}{+}\htmlData{tutor-start=51,tutor-end=52}{a}_{\htmlData{tutor-start=54,tutor-end=55}{i}}^{\htmlData{tutor-start=58,tutor-end=59}{2}}}} \htmlData{tutor-start=63,tutor-end=73}{\leqslant }\htmlData{tutor-start=73,tutor-end=74}{4} \sqrt{\htmlData{tutor-start=81,tutor-end=82}{k} \htmlData{tutor-start=83,tutor-end=84}{n}} ∑ i = 1 n a 1 2 + … + a i 2 a i ⩽ 4 k n 题解状态： 标准答案与规范题解待补充
题目标签：2020 IMO Shortlist A7
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2020-A8 2020 · Algebra · IMO/代数
Let R + \mathbb{\htmlData{tutor-start=8,tutor-end=9}{R}}^{\htmlData{tutor-start=12,tutor-end=13}{+}} R + f : R + → R + \htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{:} \mathbb{\htmlData{tutor-start=11,tutor-end=12}{R}}^{\htmlData{tutor-start=15,tutor-end=16}{+}} \htmlData{tutor-start=18,tutor-end=30}{\rightarrow }\mathbb{\htmlData{tutor-start=38,tutor-end=39}{R}}^{\htmlData{tutor-start=42,tutor-end=43}{+}} f : R + → R + x \htmlData{tutor-start=0,tutor-end=1}{x} x y \htmlData{tutor-start=0,tutor-end=1}{y} y f ( x + f ( x y ) ) + y = f ( x ) f ( y ) + 1 \htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{x}\htmlData{tutor-start=3,tutor-end=4}{+}\htmlData{tutor-start=4,tutor-end=5}{f}\htmlData{tutor-start=5,tutor-end=6}{(}\htmlData{tutor-start=6,tutor-end=7}{x} \htmlData{tutor-start=8,tutor-end=9}{y}\htmlData{tutor-start=9,tutor-end=10}{)}\htmlData{tutor-start=10,tutor-end=11}{)}\htmlData{tutor-start=11,tutor-end=12}{+}\htmlData{tutor-start=12,tutor-end=13}{y}\htmlData{tutor-start=13,tutor-end=14}{=}\htmlData{tutor-start=14,tutor-end=15}{f}\htmlData{tutor-start=15,tutor-end=16}{(}\htmlData{tutor-start=16,tutor-end=17}{x}\htmlData{tutor-start=17,tutor-end=18}{)} \htmlData{tutor-start=19,tutor-end=20}{f}\htmlData{tutor-start=20,tutor-end=21}{(}\htmlData{tutor-start=21,tutor-end=22}{y}\htmlData{tutor-start=22,tutor-end=23}{)}\htmlData{tutor-start=23,tutor-end=24}{+}\htmlData{tutor-start=24,tutor-end=25}{1} f ( x + f ( x y ) ) + y = f ( x ) f ( y ) + 1 题解状态： 标准答案与规范题解待补充
题目标签：2020 IMO Shortlist A8
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2021-A1 2021 · Algebra · IMO/代数
Let n \htmlData{tutor-start=0,tutor-end=1}{n} n A \htmlData{tutor-start=0,tutor-end=1}{A} A { 0 , 1 , 2 , 3 , … , 5 n } \left\{\htmlData{tutor-start=7,tutor-end=8}{0}\htmlData{tutor-start=8,tutor-end=9}{,}\htmlData{tutor-start=9,tutor-end=10}{1}\htmlData{tutor-start=10,tutor-end=11}{,}\htmlData{tutor-start=11,tutor-end=12}{2}\htmlData{tutor-start=12,tutor-end=13}{,}\htmlData{tutor-start=13,tutor-end=14}{3}\htmlData{tutor-start=14,tutor-end=15}{,} \htmlData{tutor-start=16,tutor-end=22}{\ldots}\htmlData{tutor-start=22,tutor-end=23}{,} \htmlData{tutor-start=24,tutor-end=25}{5}^{\htmlData{tutor-start=27,tutor-end=28}{n}}\right\} { 0 , 1 , 2 , 3 , … , 5 n } 4 n + 2 \htmlData{tutor-start=0,tutor-end=1}{4} \htmlData{tutor-start=2,tutor-end=3}{n}\htmlData{tutor-start=3,tutor-end=4}{+}\htmlData{tutor-start=4,tutor-end=5}{2} 4 n + 2 a , b , c ∈ A \htmlData{tutor-start=0,tutor-end=1}{a}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{b}\htmlData{tutor-start=4,tutor-end=5}{,} \htmlData{tutor-start=6,tutor-end=7}{c} \htmlData{tutor-start=8,tutor-end=12}{\in }\htmlData{tutor-start=12,tutor-end=13}{A} a , b , c ∈ A a < b < c \htmlData{tutor-start=0,tutor-end=1}{a}\htmlData{tutor-start=1,tutor-end=2}{<}\htmlData{tutor-start=2,tutor-end=3}{b}\htmlData{tutor-start=3,tutor-end=4}{<}\htmlData{tutor-start=4,tutor-end=5}{c} a < b < c c + 2 a > 3 b \htmlData{tutor-start=0,tutor-end=1}{c}\htmlData{tutor-start=1,tutor-end=2}{+}\htmlData{tutor-start=2,tutor-end=3}{2} \htmlData{tutor-start=4,tutor-end=5}{a}\htmlData{tutor-start=5,tutor-end=6}{>}\htmlData{tutor-start=6,tutor-end=7}{3} \htmlData{tutor-start=8,tutor-end=9}{b} c + 2 a > 3 b 题解状态： 标准答案与规范题解待补充
题目标签：2021 IMO Shortlist A1
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2021-A2 2021 · Algebra · IMO/代数
For every integer n ⩾ 1 \htmlData{tutor-start=0,tutor-end=1}{n} \htmlData{tutor-start=2,tutor-end=12}{\geqslant }\htmlData{tutor-start=12,tutor-end=13}{1} n ⩾ 1 n × n \htmlData{tutor-start=0,tutor-end=1}{n} \htmlData{tutor-start=2,tutor-end=9}{\times }\htmlData{tutor-start=9,tutor-end=10}{n} n × n ⌊ i j n + 1 ⌋ \left\lfloor\frac{\htmlData{tutor-start=18,tutor-end=19}{i} \htmlData{tutor-start=20,tutor-end=21}{j}}{\htmlData{tutor-start=23,tutor-end=24}{n}\htmlData{tutor-start=24,tutor-end=25}{+}\htmlData{tutor-start=25,tutor-end=26}{1}}\right\rfloor ⌊ n + 1 i j ⌋ i \htmlData{tutor-start=0,tutor-end=1}{i} i j \htmlData{tutor-start=0,tutor-end=1}{j} j i = 1 , … , n \htmlData{tutor-start=0,tutor-end=1}{i}\htmlData{tutor-start=1,tutor-end=2}{=}\htmlData{tutor-start=2,tutor-end=3}{1}\htmlData{tutor-start=3,tutor-end=4}{,} \htmlData{tutor-start=5,tutor-end=11}{\ldots}\htmlData{tutor-start=11,tutor-end=12}{,} \htmlData{tutor-start=13,tutor-end=14}{n} i = 1 , … , n j = 1 , … , n \htmlData{tutor-start=0,tutor-end=1}{j}\htmlData{tutor-start=1,tutor-end=2}{=}\htmlData{tutor-start=2,tutor-end=3}{1}\htmlData{tutor-start=3,tutor-end=4}{,} \htmlData{tutor-start=5,tutor-end=11}{\ldots}\htmlData{tutor-start=11,tutor-end=12}{,} \htmlData{tutor-start=13,tutor-end=14}{n} j = 1 , … , n n ⩾ 1 \htmlData{tutor-start=0,tutor-end=1}{n} \htmlData{tutor-start=2,tutor-end=12}{\geqslant }\htmlData{tutor-start=12,tutor-end=13}{1} n ⩾ 1 n 2 \htmlData{tutor-start=0,tutor-end=1}{n}^{\htmlData{tutor-start=3,tutor-end=4}{2}} n 2 1 4 n 2 ( n − 1 ) \frac{\htmlData{tutor-start=6,tutor-end=7}{1}}{\htmlData{tutor-start=9,tutor-end=10}{4}} \htmlData{tutor-start=12,tutor-end=13}{n}^{\htmlData{tutor-start=15,tutor-end=16}{2}}\htmlData{tutor-start=17,tutor-end=18}{(}\htmlData{tutor-start=18,tutor-end=19}{n}\htmlData{tutor-start=19,tutor-end=20}{-}\htmlData{tutor-start=20,tutor-end=21}{1}\htmlData{tutor-start=21,tutor-end=22}{)} 4 1 n 2 ( n − 1 ) 题解状态： 标准答案与规范题解待补充
题目标签：2021 IMO Shortlist A2
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2021-A3 2021 · Algebra · IMO/代数
Given a positive integer n \htmlData{tutor-start=0,tutor-end=1}{n} n ⌊ a 1 1 ⌋ + ⌊ a 2 2 ⌋ + ⋯ + ⌊ a n n ⌋ \left\lfloor\frac{a_{1}}{1}\right\rfloor+\left\lfloor\frac{a_{2}}{2}\right\rfloor+\cdots+\left\lfloor\frac{a_{n}}{n}\right\rfloor ⌊ 1 a 1 ⌋ + ⌊ 2 a 2 ⌋ + ⋯ + ⌊ n a n ⌋ ( a 1 , a 2 , … , a n ) \left(\htmlData{tutor-start=6,tutor-end=7}{a}_{\htmlData{tutor-start=9,tutor-end=10}{1}}\htmlData{tutor-start=11,tutor-end=12}{,} \htmlData{tutor-start=13,tutor-end=14}{a}_{\htmlData{tutor-start=16,tutor-end=17}{2}}\htmlData{tutor-start=18,tutor-end=19}{,} \htmlData{tutor-start=20,tutor-end=26}{\ldots}\htmlData{tutor-start=26,tutor-end=27}{,} \htmlData{tutor-start=28,tutor-end=29}{a}_{\htmlData{tutor-start=31,tutor-end=32}{n}}\right) ( a 1 , a 2 , … , a n ) ( 1 , 2 , … , n ) \htmlData{tutor-start=0,tutor-end=1}{(}\htmlData{tutor-start=1,tutor-end=2}{1}\htmlData{tutor-start=2,tutor-end=3}{,}\htmlData{tutor-start=3,tutor-end=4}{2}\htmlData{tutor-start=4,tutor-end=5}{,} \htmlData{tutor-start=6,tutor-end=12}{\ldots}\htmlData{tutor-start=12,tutor-end=13}{,} \htmlData{tutor-start=14,tutor-end=15}{n}\htmlData{tutor-start=15,tutor-end=16}{)} ( 1 , 2 , … , n ) 题解状态： 标准答案与规范题解待补充
题目标签：2021 IMO Shortlist A3
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2021-A4 2021 · Algebra · IMO/代数
Show that for all real numbers x 1 , … , x n \htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=13}{\ldots}\htmlData{tutor-start=13,tutor-end=14}{,} \htmlData{tutor-start=15,tutor-end=16}{x}_{\htmlData{tutor-start=18,tutor-end=19}{n}} x 1 , … , x n ∑ i = 1 n ∑ j = 1 n ∣ x i − x j ∣ ⩽ ∑ i = 1 n ∑ j = 1 n ∣ x i + x j ∣ \sum_{\htmlData{tutor-start=6,tutor-end=7}{i}\htmlData{tutor-start=7,tutor-end=8}{=}\htmlData{tutor-start=8,tutor-end=9}{1}}^{\htmlData{tutor-start=12,tutor-end=13}{n}} \sum_{\htmlData{tutor-start=21,tutor-end=22}{j}\htmlData{tutor-start=22,tutor-end=23}{=}\htmlData{tutor-start=23,tutor-end=24}{1}}^{\htmlData{tutor-start=27,tutor-end=28}{n}} \sqrt{\left|\htmlData{tutor-start=42,tutor-end=43}{x}_{\htmlData{tutor-start=45,tutor-end=46}{i}}\htmlData{tutor-start=47,tutor-end=48}{-}\htmlData{tutor-start=48,tutor-end=49}{x}_{\htmlData{tutor-start=51,tutor-end=52}{j}}\right|} \htmlData{tutor-start=62,tutor-end=72}{\leqslant }\sum_{\htmlData{tutor-start=78,tutor-end=79}{i}\htmlData{tutor-start=79,tutor-end=80}{=}\htmlData{tutor-start=80,tutor-end=81}{1}}^{\htmlData{tutor-start=84,tutor-end=85}{n}} \sum_{\htmlData{tutor-start=93,tutor-end=94}{j}\htmlData{tutor-start=94,tutor-end=95}{=}\htmlData{tutor-start=95,tutor-end=96}{1}}^{\htmlData{tutor-start=99,tutor-end=100}{n}} \sqrt{\left|\htmlData{tutor-start=114,tutor-end=115}{x}_{\htmlData{tutor-start=117,tutor-end=118}{i}}\htmlData{tutor-start=119,tutor-end=120}{+}\htmlData{tutor-start=120,tutor-end=121}{x}_{\htmlData{tutor-start=123,tutor-end=124}{j}}\right|} ∑ i = 1 n ∑ j = 1 n ∣ x i − x j ∣ ⩽ ∑ i = 1 n ∑ j = 1 n ∣ x i + x j ∣ 题解状态： 标准答案与规范题解待补充
题目标签：2021 IMO 正式题 A4
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2021-A5 2021 · Algebra · IMO/代数
Let n ⩾ 2 \htmlData{tutor-start=0,tutor-end=1}{n} \htmlData{tutor-start=2,tutor-end=12}{\geqslant }\htmlData{tutor-start=12,tutor-end=13}{2} n ⩾ 2 a 1 , a 2 , … , a n \htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{a}_{\htmlData{tutor-start=10,tutor-end=11}{2}}\htmlData{tutor-start=12,tutor-end=13}{,} \htmlData{tutor-start=14,tutor-end=20}{\ldots}\htmlData{tutor-start=20,tutor-end=21}{,} \htmlData{tutor-start=22,tutor-end=23}{a}_{\htmlData{tutor-start=25,tutor-end=26}{n}} a 1 , a 2 , … , a n a 1 + a 2 + ⋯ + a n = 1 \htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{+}\htmlData{tutor-start=6,tutor-end=7}{a}_{\htmlData{tutor-start=9,tutor-end=10}{2}}\htmlData{tutor-start=11,tutor-end=12}{+}\cdots\htmlData{tutor-start=18,tutor-end=19}{+}\htmlData{tutor-start=19,tutor-end=20}{a}_{\htmlData{tutor-start=22,tutor-end=23}{n}}\htmlData{tutor-start=24,tutor-end=25}{=}\htmlData{tutor-start=25,tutor-end=26}{1} a 1 + a 2 + ⋯ + a n = 1 ∑ k = 1 n a k 1 − a k ( a 1 + a 2 + ⋯ + a k − 1 ) 2 < 1 3 \htmlData{tutor-start=0,tutor-end=7}{\sum_{k}\htmlData{tutor-start=7,tutor-end=8}{=}\htmlData{tutor-start=8,tutor-end=9}{1}}^{\htmlData{tutor-start=12,tutor-end=13}{n}} \frac{\htmlData{tutor-start=21,tutor-end=22}{a}_{\htmlData{tutor-start=24,tutor-end=25}{k}}}{\htmlData{tutor-start=28,tutor-end=29}{1}\htmlData{tutor-start=29,tutor-end=30}{-}\htmlData{tutor-start=30,tutor-end=31}{a}_{\htmlData{tutor-start=33,tutor-end=34}{k}}}\left(\htmlData{tutor-start=42,tutor-end=43}{a}_{\htmlData{tutor-start=45,tutor-end=46}{1}}\htmlData{tutor-start=47,tutor-end=48}{+}\htmlData{tutor-start=48,tutor-end=49}{a}_{\htmlData{tutor-start=51,tutor-end=52}{2}}\htmlData{tutor-start=53,tutor-end=54}{+}\cdots\htmlData{tutor-start=60,tutor-end=61}{+}\htmlData{tutor-start=61,tutor-end=62}{a}_{\htmlData{tutor-start=64,tutor-end=65}{k}\htmlData{tutor-start=65,tutor-end=66}{-}\htmlData{tutor-start=66,tutor-end=67}{1}}\right)^{\htmlData{tutor-start=77,tutor-end=78}{2}}\htmlData{tutor-start=79,tutor-end=80}{<}\frac{\htmlData{tutor-start=86,tutor-end=87}{1}}{\htmlData{tutor-start=89,tutor-end=90}{3}} ∑ k = 1 n 1 − a k a k ( a 1 + a 2 + ⋯ + a k − 1 ) 2 < 3 1 题解状态： 标准答案与规范题解待补充
题目标签：2021 IMO Shortlist A5
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2021-A6 2021 · Algebra · IMO/代数
Let A \htmlData{tutor-start=0,tutor-end=1}{A} A m ⩾ 2 \htmlData{tutor-start=0,tutor-end=1}{m} \htmlData{tutor-start=2,tutor-end=12}{\geqslant }\htmlData{tutor-start=12,tutor-end=13}{2} m ⩾ 2 B 1 , B 2 , B 3 , … , B m \htmlData{tutor-start=0,tutor-end=1}{B}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{B}_{\htmlData{tutor-start=10,tutor-end=11}{2}}\htmlData{tutor-start=12,tutor-end=13}{,} \htmlData{tutor-start=14,tutor-end=15}{B}_{\htmlData{tutor-start=17,tutor-end=18}{3}}\htmlData{tutor-start=19,tutor-end=20}{,} \htmlData{tutor-start=21,tutor-end=27}{\ldots}\htmlData{tutor-start=27,tutor-end=28}{,} \htmlData{tutor-start=29,tutor-end=30}{B}_{\htmlData{tutor-start=32,tutor-end=33}{m}} B 1 , B 2 , B 3 , … , B m A \htmlData{tutor-start=0,tutor-end=1}{A} A m 1 , m 2 , m 3 , … , m m \htmlData{tutor-start=0,tutor-end=1}{m}^{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{m}^{\htmlData{tutor-start=10,tutor-end=11}{2}}\htmlData{tutor-start=12,tutor-end=13}{,} \htmlData{tutor-start=14,tutor-end=15}{m}^{\htmlData{tutor-start=17,tutor-end=18}{3}}\htmlData{tutor-start=19,tutor-end=20}{,} \htmlData{tutor-start=21,tutor-end=27}{\ldots}\htmlData{tutor-start=27,tutor-end=28}{,} \htmlData{tutor-start=29,tutor-end=30}{m}^{\htmlData{tutor-start=32,tutor-end=33}{m}} m 1 , m 2 , m 3 , … , m m A \htmlData{tutor-start=0,tutor-end=1}{A} A m / 2 \htmlData{tutor-start=0,tutor-end=1}{m} \htmlData{tutor-start=2,tutor-end=3}{/} \htmlData{tutor-start=4,tutor-end=5}{2} m / 2 题解状态： 标准答案与规范题解待补充
题目标签：2021 IMO 正式题 A6
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2021-A7 2021 · Algebra · IMO/代数
Let n ⩾ 1 \htmlData{tutor-start=0,tutor-end=1}{n} \htmlData{tutor-start=2,tutor-end=12}{\geqslant }\htmlData{tutor-start=12,tutor-end=13}{1} n ⩾ 1 x 0 , x 1 , … , x n + 1 \htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{0}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{x}_{\htmlData{tutor-start=10,tutor-end=11}{1}}\htmlData{tutor-start=12,tutor-end=13}{,} \htmlData{tutor-start=14,tutor-end=20}{\ldots}\htmlData{tutor-start=20,tutor-end=21}{,} \htmlData{tutor-start=22,tutor-end=23}{x}_{\htmlData{tutor-start=25,tutor-end=26}{n}\htmlData{tutor-start=26,tutor-end=27}{+}\htmlData{tutor-start=27,tutor-end=28}{1}} x 0 , x 1 , … , x n + 1 n + 2 \htmlData{tutor-start=0,tutor-end=1}{n}\htmlData{tutor-start=1,tutor-end=2}{+}\htmlData{tutor-start=2,tutor-end=3}{2} n + 2 x i x i + 1 − x i − 1 2 ⩾ 1 \htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{i}} \htmlData{tutor-start=6,tutor-end=7}{x}_{\htmlData{tutor-start=9,tutor-end=10}{i}\htmlData{tutor-start=10,tutor-end=11}{+}\htmlData{tutor-start=11,tutor-end=12}{1}}\htmlData{tutor-start=13,tutor-end=14}{-}\htmlData{tutor-start=14,tutor-end=15}{x}_{\htmlData{tutor-start=17,tutor-end=18}{i}\htmlData{tutor-start=18,tutor-end=19}{-}\htmlData{tutor-start=19,tutor-end=20}{1}}^{\htmlData{tutor-start=23,tutor-end=24}{2}} \htmlData{tutor-start=26,tutor-end=36}{\geqslant }\htmlData{tutor-start=36,tutor-end=37}{1} x i x i + 1 − x i − 1 2 ⩾ 1 i = 1 , 2 , … , n \htmlData{tutor-start=0,tutor-end=1}{i}\htmlData{tutor-start=1,tutor-end=2}{=}\htmlData{tutor-start=2,tutor-end=3}{1}\htmlData{tutor-start=3,tutor-end=4}{,}\htmlData{tutor-start=4,tutor-end=5}{2}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=13}{\ldots}\htmlData{tutor-start=13,tutor-end=14}{,} \htmlData{tutor-start=15,tutor-end=16}{n} i = 1 , 2 , … , n x 0 + x 1 + ⋯ + x n + x n + 1 > ( 2 n 3 ) 3 / 2 \htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{0}}\htmlData{tutor-start=5,tutor-end=6}{+}\htmlData{tutor-start=6,tutor-end=7}{x}_{\htmlData{tutor-start=9,tutor-end=10}{1}}\htmlData{tutor-start=11,tutor-end=12}{+}\cdots\htmlData{tutor-start=18,tutor-end=19}{+}\htmlData{tutor-start=19,tutor-end=20}{x}_{\htmlData{tutor-start=22,tutor-end=23}{n}}\htmlData{tutor-start=24,tutor-end=25}{+}\htmlData{tutor-start=25,tutor-end=26}{x}_{\htmlData{tutor-start=28,tutor-end=29}{n}\htmlData{tutor-start=29,tutor-end=30}{+}\htmlData{tutor-start=30,tutor-end=31}{1}}\htmlData{tutor-start=32,tutor-end=33}{>}\left(\frac{\htmlData{tutor-start=45,tutor-end=46}{2} \htmlData{tutor-start=47,tutor-end=48}{n}}{\htmlData{tutor-start=50,tutor-end=51}{3}}\right)^{\htmlData{tutor-start=61,tutor-end=62}{3} \htmlData{tutor-start=63,tutor-end=64}{/} \htmlData{tutor-start=65,tutor-end=66}{2}} x 0 + x 1 + ⋯ + x n + x n + 1 > ( 3 2 n ) 3 / 2 题解状态： 标准答案与规范题解待补充
题目标签：2021 IMO Shortlist A7
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2021-A8 2021 · Algebra · IMO/代数
Determine all functions f : R → R \htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{:} \mathbb{\htmlData{tutor-start=11,tutor-end=12}{R}} \htmlData{tutor-start=14,tutor-end=26}{\rightarrow }\mathbb{\htmlData{tutor-start=34,tutor-end=35}{R}} f : R → R ( f ( a ) − f ( b ) ) ( f ( b ) − f ( c ) ) ( f ( c ) − f ( a ) ) = f ( a b 2 + b c 2 + c a 2 ) − f ( a 2 b + b 2 c + c 2 a ) \htmlData{tutor-start=0,tutor-end=1}{(}\htmlData{tutor-start=1,tutor-end=2}{f}\htmlData{tutor-start=2,tutor-end=3}{(}\htmlData{tutor-start=3,tutor-end=4}{a}\htmlData{tutor-start=4,tutor-end=5}{)}\htmlData{tutor-start=5,tutor-end=6}{-}\htmlData{tutor-start=6,tutor-end=7}{f}\htmlData{tutor-start=7,tutor-end=8}{(}\htmlData{tutor-start=8,tutor-end=9}{b}\htmlData{tutor-start=9,tutor-end=10}{)}\htmlData{tutor-start=10,tutor-end=11}{)}\htmlData{tutor-start=11,tutor-end=12}{(}\htmlData{tutor-start=12,tutor-end=13}{f}\htmlData{tutor-start=13,tutor-end=14}{(}\htmlData{tutor-start=14,tutor-end=15}{b}\htmlData{tutor-start=15,tutor-end=16}{)}\htmlData{tutor-start=16,tutor-end=17}{-}\htmlData{tutor-start=17,tutor-end=18}{f}\htmlData{tutor-start=18,tutor-end=19}{(}\htmlData{tutor-start=19,tutor-end=20}{c}\htmlData{tutor-start=20,tutor-end=21}{)}\htmlData{tutor-start=21,tutor-end=22}{)}\htmlData{tutor-start=22,tutor-end=23}{(}\htmlData{tutor-start=23,tutor-end=24}{f}\htmlData{tutor-start=24,tutor-end=25}{(}\htmlData{tutor-start=25,tutor-end=26}{c}\htmlData{tutor-start=26,tutor-end=27}{)}\htmlData{tutor-start=27,tutor-end=28}{-}\htmlData{tutor-start=28,tutor-end=29}{f}\htmlData{tutor-start=29,tutor-end=30}{(}\htmlData{tutor-start=30,tutor-end=31}{a}\htmlData{tutor-start=31,tutor-end=32}{)}\htmlData{tutor-start=32,tutor-end=33}{)}\htmlData{tutor-start=33,tutor-end=34}{=}\htmlData{tutor-start=34,tutor-end=35}{f}\left(\htmlData{tutor-start=41,tutor-end=42}{a} \htmlData{tutor-start=43,tutor-end=44}{b}^{\htmlData{tutor-start=46,tutor-end=47}{2}}\htmlData{tutor-start=48,tutor-end=49}{+}\htmlData{tutor-start=49,tutor-end=50}{b} \htmlData{tutor-start=51,tutor-end=52}{c}^{\htmlData{tutor-start=54,tutor-end=55}{2}}\htmlData{tutor-start=56,tutor-end=57}{+}\htmlData{tutor-start=57,tutor-end=58}{c} \htmlData{tutor-start=59,tutor-end=60}{a}^{\htmlData{tutor-start=62,tutor-end=63}{2}}\right)\htmlData{tutor-start=71,tutor-end=72}{-}\htmlData{tutor-start=72,tutor-end=73}{f}\left(\htmlData{tutor-start=79,tutor-end=80}{a}^{\htmlData{tutor-start=82,tutor-end=83}{2}} \htmlData{tutor-start=85,tutor-end=86}{b}\htmlData{tutor-start=86,tutor-end=87}{+}\htmlData{tutor-start=87,tutor-end=88}{b}^{\htmlData{tutor-start=90,tutor-end=91}{2}} \htmlData{tutor-start=93,tutor-end=94}{c}\htmlData{tutor-start=94,tutor-end=95}{+}\htmlData{tutor-start=95,tutor-end=96}{c}^{\htmlData{tutor-start=98,tutor-end=99}{2}} \htmlData{tutor-start=101,tutor-end=102}{a}\right) ( f ( a ) − f ( b ) ) ( f ( b ) − f ( c ) ) ( f ( c ) − f ( a ) ) = f ( a b 2 + b c 2 + c a 2 ) − f ( a 2 b + b 2 c + c 2 a ) a , b , c \htmlData{tutor-start=0,tutor-end=1}{a}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{b}\htmlData{tutor-start=4,tutor-end=5}{,} \htmlData{tutor-start=6,tutor-end=7}{c} a , b , c 题解状态： 标准答案与规范题解待补充
题目标签：2021 IMO Shortlist A8
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2022-A1 2022 · Algebra · IMO/代数
Let ( a n ) n ⩾ 1 \left(\htmlData{tutor-start=6,tutor-end=7}{a}_{\htmlData{tutor-start=9,tutor-end=10}{n}}\right)_{\htmlData{tutor-start=20,tutor-end=21}{n} \htmlData{tutor-start=22,tutor-end=32}{\geqslant }\htmlData{tutor-start=32,tutor-end=33}{1}} ( a n ) n ⩾ 1 ( a n + 1 ) 2 + a n a n + 2 ⩽ a n + a n + 2 \left(\htmlData{tutor-start=6,tutor-end=7}{a}_{\htmlData{tutor-start=9,tutor-end=10}{n}\htmlData{tutor-start=10,tutor-end=11}{+}\htmlData{tutor-start=11,tutor-end=12}{1}}\right)^{\htmlData{tutor-start=22,tutor-end=23}{2}}\htmlData{tutor-start=24,tutor-end=25}{+}\htmlData{tutor-start=25,tutor-end=26}{a}_{\htmlData{tutor-start=28,tutor-end=29}{n}} \htmlData{tutor-start=31,tutor-end=32}{a}_{\htmlData{tutor-start=34,tutor-end=35}{n}\htmlData{tutor-start=35,tutor-end=36}{+}\htmlData{tutor-start=36,tutor-end=37}{2}} \htmlData{tutor-start=39,tutor-end=49}{\leqslant }\htmlData{tutor-start=49,tutor-end=50}{a}_{\htmlData{tutor-start=52,tutor-end=53}{n}}\htmlData{tutor-start=54,tutor-end=55}{+}\htmlData{tutor-start=55,tutor-end=56}{a}_{\htmlData{tutor-start=58,tutor-end=59}{n}\htmlData{tutor-start=59,tutor-end=60}{+}\htmlData{tutor-start=60,tutor-end=61}{2}} ( a n + 1 ) 2 + a n a n + 2 ⩽ a n + a n + 2 n \htmlData{tutor-start=0,tutor-end=1}{n} n a 2022 ⩽ 1 \htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{2}\htmlData{tutor-start=4,tutor-end=5}{0}\htmlData{tutor-start=5,tutor-end=6}{2}\htmlData{tutor-start=6,tutor-end=7}{2}} \htmlData{tutor-start=9,tutor-end=19}{\leqslant }\htmlData{tutor-start=19,tutor-end=20}{1} a 2 0 2 2 ⩽ 1 题解状态： 标准答案与规范题解待补充
题目标签：2022 IMO Shortlist A1
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2022-A2 2022 · Algebra · IMO/代数
Let k ⩾ 2 \htmlData{tutor-start=0,tutor-end=1}{k} \htmlData{tutor-start=2,tutor-end=12}{\geqslant }\htmlData{tutor-start=12,tutor-end=13}{2} k ⩾ 2 n ⩾ k + 1 \htmlData{tutor-start=0,tutor-end=1}{n} \htmlData{tutor-start=2,tutor-end=12}{\geqslant }\htmlData{tutor-start=12,tutor-end=13}{k}\htmlData{tutor-start=13,tutor-end=14}{+}\htmlData{tutor-start=14,tutor-end=15}{1} n ⩾ k + 1 n \htmlData{tutor-start=0,tutor-end=1}{n} n k \htmlData{tutor-start=0,tutor-end=1}{k} k 题解状态： 标准答案与规范题解待补充
题目标签：2022 IMO Shortlist A2
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2022-A3 2022 · Algebra · IMO/代数
Let R > 0 \mathbb{\htmlData{tutor-start=8,tutor-end=9}{R}}_{\htmlData{tutor-start=12,tutor-end=13}{>}\htmlData{tutor-start=13,tutor-end=14}{0}} R > 0 f : R > 0 → R > 0 \htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{:} \mathbb{\htmlData{tutor-start=11,tutor-end=12}{R}}_{\htmlData{tutor-start=15,tutor-end=16}{>}\htmlData{tutor-start=16,tutor-end=17}{0}} \htmlData{tutor-start=19,tutor-end=31}{\rightarrow }\mathbb{\htmlData{tutor-start=39,tutor-end=40}{R}}_{\htmlData{tutor-start=43,tutor-end=44}{>}\htmlData{tutor-start=44,tutor-end=45}{0}} f : R > 0 → R > 0 x ∈ R > 0 \htmlData{tutor-start=0,tutor-end=1}{x} \htmlData{tutor-start=2,tutor-end=6}{\in }\mathbb{\htmlData{tutor-start=14,tutor-end=15}{R}}_{\htmlData{tutor-start=18,tutor-end=19}{>}\htmlData{tutor-start=19,tutor-end=20}{0}} x ∈ R > 0 y ∈ R > 0 \htmlData{tutor-start=0,tutor-end=1}{y} \htmlData{tutor-start=2,tutor-end=6}{\in }\mathbb{\htmlData{tutor-start=14,tutor-end=15}{R}}_{\htmlData{tutor-start=18,tutor-end=19}{>}\htmlData{tutor-start=19,tutor-end=20}{0}} y ∈ R > 0 x f ( y ) + y f ( x ) ⩽ 2 \htmlData{tutor-start=0,tutor-end=1}{x} \htmlData{tutor-start=2,tutor-end=3}{f}\htmlData{tutor-start=3,tutor-end=4}{(}\htmlData{tutor-start=4,tutor-end=5}{y}\htmlData{tutor-start=5,tutor-end=6}{)}\htmlData{tutor-start=6,tutor-end=7}{+}\htmlData{tutor-start=7,tutor-end=8}{y} \htmlData{tutor-start=9,tutor-end=10}{f}\htmlData{tutor-start=10,tutor-end=11}{(}\htmlData{tutor-start=11,tutor-end=12}{x}\htmlData{tutor-start=12,tutor-end=13}{)} \htmlData{tutor-start=14,tutor-end=24}{\leqslant }\htmlData{tutor-start=24,tutor-end=25}{2} x f ( y ) + y f ( x ) ⩽ 2 题解状态： 标准答案与规范题解待补充
题目标签：2022 IMO 正式题 A3
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2022-A4 2022 · Algebra · IMO/代数
Let n ⩾ 3 \htmlData{tutor-start=0,tutor-end=1}{n} \htmlData{tutor-start=2,tutor-end=12}{\geqslant }\htmlData{tutor-start=12,tutor-end=13}{3} n ⩾ 3 x 1 , x 2 , … , x n \htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{x}_{\htmlData{tutor-start=10,tutor-end=11}{2}}\htmlData{tutor-start=12,tutor-end=13}{,} \htmlData{tutor-start=14,tutor-end=20}{\ldots}\htmlData{tutor-start=20,tutor-end=21}{,} \htmlData{tutor-start=22,tutor-end=23}{x}_{\htmlData{tutor-start=25,tutor-end=26}{n}} x 1 , x 2 , … , x n [ 0 , 1 ] \htmlData{tutor-start=0,tutor-end=1}{[}\htmlData{tutor-start=1,tutor-end=2}{0}\htmlData{tutor-start=2,tutor-end=3}{,}\htmlData{tutor-start=3,tutor-end=4}{1}\htmlData{tutor-start=4,tutor-end=5}{]} [ 0 , 1 ] s = x 1 + x 2 + … + x n \htmlData{tutor-start=0,tutor-end=1}{s}\htmlData{tutor-start=1,tutor-end=2}{=}\htmlData{tutor-start=2,tutor-end=3}{x}_{\htmlData{tutor-start=5,tutor-end=6}{1}}\htmlData{tutor-start=7,tutor-end=8}{+}\htmlData{tutor-start=8,tutor-end=9}{x}_{\htmlData{tutor-start=11,tutor-end=12}{2}}\htmlData{tutor-start=13,tutor-end=14}{+}\htmlData{tutor-start=14,tutor-end=20}{\ldots}\htmlData{tutor-start=20,tutor-end=21}{+}\htmlData{tutor-start=21,tutor-end=22}{x}_{\htmlData{tutor-start=24,tutor-end=25}{n}} s = x 1 + x 2 + … + x n s ⩾ 3 \htmlData{tutor-start=0,tutor-end=1}{s} \htmlData{tutor-start=2,tutor-end=12}{\geqslant }\htmlData{tutor-start=12,tutor-end=13}{3} s ⩾ 3 i \htmlData{tutor-start=0,tutor-end=1}{i} i j \htmlData{tutor-start=0,tutor-end=1}{j} j 1 ⩽ i < j ⩽ n \htmlData{tutor-start=0,tutor-end=1}{1} \htmlData{tutor-start=2,tutor-end=12}{\leqslant }\htmlData{tutor-start=12,tutor-end=13}{i}\htmlData{tutor-start=13,tutor-end=14}{<}\htmlData{tutor-start=14,tutor-end=15}{j} \htmlData{tutor-start=16,tutor-end=26}{\leqslant }\htmlData{tutor-start=26,tutor-end=27}{n} 1 ⩽ i < j ⩽ n 2 j − i x i x j > 2 s − 3 \htmlData{tutor-start=0,tutor-end=1}{2}^{\htmlData{tutor-start=3,tutor-end=4}{j}\htmlData{tutor-start=4,tutor-end=5}{-}\htmlData{tutor-start=5,tutor-end=6}{i}} \htmlData{tutor-start=8,tutor-end=9}{x}_{\htmlData{tutor-start=11,tutor-end=12}{i}} \htmlData{tutor-start=14,tutor-end=15}{x}_{\htmlData{tutor-start=17,tutor-end=18}{j}}\htmlData{tutor-start=19,tutor-end=20}{>}\htmlData{tutor-start=20,tutor-end=21}{2}^{\htmlData{tutor-start=23,tutor-end=24}{s}\htmlData{tutor-start=24,tutor-end=25}{-}\htmlData{tutor-start=25,tutor-end=26}{3}} 2 j − i x i x j > 2 s − 3 题解状态： 标准答案与规范题解待补充
题目标签：2022 IMO Shortlist A4
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2022-A5 2022 · Algebra · IMO/代数
Find all positive integers n ⩾ 2 \htmlData{tutor-start=0,tutor-end=1}{n} \htmlData{tutor-start=2,tutor-end=12}{\geqslant }\htmlData{tutor-start=12,tutor-end=13}{2} n ⩾ 2 n \htmlData{tutor-start=0,tutor-end=1}{n} n a 1 < ⋯ < a n \htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{<}\cdots\htmlData{tutor-start=12,tutor-end=13}{<}\htmlData{tutor-start=13,tutor-end=14}{a}_{\htmlData{tutor-start=16,tutor-end=17}{n}} a 1 < ⋯ < a n r > 0 \htmlData{tutor-start=0,tutor-end=1}{r}\htmlData{tutor-start=1,tutor-end=2}{>}\htmlData{tutor-start=2,tutor-end=3}{0} r > 0 1 2 n ( n − 1 ) \frac{\htmlData{tutor-start=6,tutor-end=7}{1}}{\htmlData{tutor-start=9,tutor-end=10}{2}} \htmlData{tutor-start=12,tutor-end=13}{n}\htmlData{tutor-start=13,tutor-end=14}{(}\htmlData{tutor-start=14,tutor-end=15}{n}\htmlData{tutor-start=15,tutor-end=16}{-}\htmlData{tutor-start=16,tutor-end=17}{1}\htmlData{tutor-start=17,tutor-end=18}{)} 2 1 n ( n − 1 ) a j − a i \htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{j}}\htmlData{tutor-start=5,tutor-end=6}{-}\htmlData{tutor-start=6,tutor-end=7}{a}_{\htmlData{tutor-start=9,tutor-end=10}{i}} a j − a i 1 ⩽ i < j ⩽ n \htmlData{tutor-start=0,tutor-end=1}{1} \htmlData{tutor-start=2,tutor-end=12}{\leqslant }\htmlData{tutor-start=12,tutor-end=13}{i}\htmlData{tutor-start=13,tutor-end=14}{<}\htmlData{tutor-start=14,tutor-end=15}{j} \htmlData{tutor-start=16,tutor-end=26}{\leqslant }\htmlData{tutor-start=26,tutor-end=27}{n} 1 ⩽ i < j ⩽ n r 1 , r 2 , … , r 1 2 n ( n − 1 ) \htmlData{tutor-start=0,tutor-end=1}{r}^{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{r}^{\htmlData{tutor-start=10,tutor-end=11}{2}}\htmlData{tutor-start=12,tutor-end=13}{,} \htmlData{tutor-start=14,tutor-end=20}{\ldots}\htmlData{tutor-start=20,tutor-end=21}{,} \htmlData{tutor-start=22,tutor-end=23}{r}^{\frac{\htmlData{tutor-start=31,tutor-end=32}{1}}{\htmlData{tutor-start=34,tutor-end=35}{2}} \htmlData{tutor-start=37,tutor-end=38}{n}\htmlData{tutor-start=38,tutor-end=39}{(}\htmlData{tutor-start=39,tutor-end=40}{n}\htmlData{tutor-start=40,tutor-end=41}{-}\htmlData{tutor-start=41,tutor-end=42}{1}\htmlData{tutor-start=42,tutor-end=43}{)}} r 1 , r 2 , … , r 2 1 n ( n − 1 ) 题解状态： 标准答案与规范题解待补充
题目标签：2022 IMO Shortlist A5
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2022-A6 2022 · Algebra · IMO/代数
Let R \mathbb{\htmlData{tutor-start=8,tutor-end=9}{R}} R F \mathcal{\htmlData{tutor-start=9,tutor-end=10}{F}} F f : R → R \htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{:} \mathbb{\htmlData{tutor-start=11,tutor-end=12}{R}} \htmlData{tutor-start=14,tutor-end=26}{\rightarrow }\mathbb{\htmlData{tutor-start=34,tutor-end=35}{R}} f : R → R f ( x + f ( y ) ) = f ( x ) + f ( y ) \htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{x}\htmlData{tutor-start=3,tutor-end=4}{+}\htmlData{tutor-start=4,tutor-end=5}{f}\htmlData{tutor-start=5,tutor-end=6}{(}\htmlData{tutor-start=6,tutor-end=7}{y}\htmlData{tutor-start=7,tutor-end=8}{)}\htmlData{tutor-start=8,tutor-end=9}{)}\htmlData{tutor-start=9,tutor-end=10}{=}\htmlData{tutor-start=10,tutor-end=11}{f}\htmlData{tutor-start=11,tutor-end=12}{(}\htmlData{tutor-start=12,tutor-end=13}{x}\htmlData{tutor-start=13,tutor-end=14}{)}\htmlData{tutor-start=14,tutor-end=15}{+}\htmlData{tutor-start=15,tutor-end=16}{f}\htmlData{tutor-start=16,tutor-end=17}{(}\htmlData{tutor-start=17,tutor-end=18}{y}\htmlData{tutor-start=18,tutor-end=19}{)} f ( x + f ( y ) ) = f ( x ) + f ( y ) x , y ∈ R \htmlData{tutor-start=0,tutor-end=1}{x}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{y} \htmlData{tutor-start=5,tutor-end=9}{\in }\mathbb{\htmlData{tutor-start=17,tutor-end=18}{R}} x , y ∈ R q \htmlData{tutor-start=0,tutor-end=1}{q} q f ∈ F \htmlData{tutor-start=0,tutor-end=1}{f} \htmlData{tutor-start=2,tutor-end=6}{\in }\mathcal{\htmlData{tutor-start=15,tutor-end=16}{F}} f ∈ F z ∈ R \htmlData{tutor-start=0,tutor-end=1}{z} \htmlData{tutor-start=2,tutor-end=6}{\in }\mathbb{\htmlData{tutor-start=14,tutor-end=15}{R}} z ∈ R f ( z ) = q z \htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{z}\htmlData{tutor-start=3,tutor-end=4}{)}\htmlData{tutor-start=4,tutor-end=5}{=}\htmlData{tutor-start=5,tutor-end=6}{q} \htmlData{tutor-start=7,tutor-end=8}{z} f ( z ) = q z 题解状态： 标准答案与规范题解待补充
题目标签：2022 IMO Shortlist A6
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2022-A7 2022 · Algebra · IMO/代数
For a positive integer n \htmlData{tutor-start=0,tutor-end=1}{n} n s ( n ) \htmlData{tutor-start=0,tutor-end=1}{s}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{n}\htmlData{tutor-start=3,tutor-end=4}{)} s ( n ) n \htmlData{tutor-start=0,tutor-end=1}{n} n P ( x ) = \htmlData{tutor-start=0,tutor-end=1}{P}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{x}\htmlData{tutor-start=3,tutor-end=4}{)}\htmlData{tutor-start=4,tutor-end=5}{=} P ( x ) = x n + a n − 1 x n − 1 + ⋯ + a 1 x + a 0 \htmlData{tutor-start=0,tutor-end=1}{x}^{\htmlData{tutor-start=3,tutor-end=4}{n}}\htmlData{tutor-start=5,tutor-end=6}{+}\htmlData{tutor-start=6,tutor-end=7}{a}_{\htmlData{tutor-start=9,tutor-end=10}{n}\htmlData{tutor-start=10,tutor-end=11}{-}\htmlData{tutor-start=11,tutor-end=12}{1}} \htmlData{tutor-start=14,tutor-end=15}{x}^{\htmlData{tutor-start=17,tutor-end=18}{n}\htmlData{tutor-start=18,tutor-end=19}{-}\htmlData{tutor-start=19,tutor-end=20}{1}}\htmlData{tutor-start=21,tutor-end=22}{+}\cdots\htmlData{tutor-start=28,tutor-end=29}{+}\htmlData{tutor-start=29,tutor-end=30}{a}_{\htmlData{tutor-start=32,tutor-end=33}{1}} \htmlData{tutor-start=35,tutor-end=36}{x}\htmlData{tutor-start=36,tutor-end=37}{+}\htmlData{tutor-start=37,tutor-end=38}{a}_{\htmlData{tutor-start=40,tutor-end=41}{0}} x n + a n − 1 x n − 1 + ⋯ + a 1 x + a 0 n ⩾ 2 \htmlData{tutor-start=0,tutor-end=1}{n} \htmlData{tutor-start=2,tutor-end=12}{\geqslant }\htmlData{tutor-start=12,tutor-end=13}{2} n ⩾ 2 a i \htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{i}} a i 0 ⩽ i ⩽ n − 1 \htmlData{tutor-start=0,tutor-end=1}{0} \htmlData{tutor-start=2,tutor-end=12}{\leqslant }\htmlData{tutor-start=12,tutor-end=13}{i} \htmlData{tutor-start=14,tutor-end=24}{\leqslant }\htmlData{tutor-start=24,tutor-end=25}{n}\htmlData{tutor-start=25,tutor-end=26}{-}\htmlData{tutor-start=26,tutor-end=27}{1} 0 ⩽ i ⩽ n − 1 k , s ( k ) \htmlData{tutor-start=0,tutor-end=1}{k}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{s}\htmlData{tutor-start=4,tutor-end=5}{(}\htmlData{tutor-start=5,tutor-end=6}{k}\htmlData{tutor-start=6,tutor-end=7}{)} k , s ( k ) s ( P ( k ) ) \htmlData{tutor-start=0,tutor-end=1}{s}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{P}\htmlData{tutor-start=3,tutor-end=4}{(}\htmlData{tutor-start=4,tutor-end=5}{k}\htmlData{tutor-start=5,tutor-end=6}{)}\htmlData{tutor-start=6,tutor-end=7}{)} s ( P ( k ) ) 题解状态： 标准答案与规范题解待补充
题目标签：2022 IMO Shortlist A7
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2022-A8 2022 · Algebra · IMO/代数
For a positive integer n \htmlData{tutor-start=0,tutor-end=1}{n} n n \htmlData{tutor-start=0,tutor-end=1}{n} n ( a 0 , … , a n ) \left(\htmlData{tutor-start=6,tutor-end=7}{a}_{\htmlData{tutor-start=9,tutor-end=10}{0}}\htmlData{tutor-start=11,tutor-end=12}{,} \htmlData{tutor-start=13,tutor-end=19}{\ldots}\htmlData{tutor-start=19,tutor-end=20}{,} \htmlData{tutor-start=21,tutor-end=22}{a}_{\htmlData{tutor-start=24,tutor-end=25}{n}}\right) ( a 0 , … , a n ) i \htmlData{tutor-start=0,tutor-end=1}{i} i j \htmlData{tutor-start=0,tutor-end=1}{j} j i + j ⩽ n \htmlData{tutor-start=0,tutor-end=1}{i}\htmlData{tutor-start=1,tutor-end=2}{+}\htmlData{tutor-start=2,tutor-end=3}{j} \htmlData{tutor-start=4,tutor-end=14}{\leqslant }\htmlData{tutor-start=14,tutor-end=15}{n} i + j ⩽ n a i + a j ⩽ n \htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{i}}\htmlData{tutor-start=5,tutor-end=6}{+}\htmlData{tutor-start=6,tutor-end=7}{a}_{\htmlData{tutor-start=9,tutor-end=10}{j}} \htmlData{tutor-start=12,tutor-end=22}{\leqslant }\htmlData{tutor-start=22,tutor-end=23}{n} a i + a j ⩽ n a a i + a j = a i + j \htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{a}_{\htmlData{tutor-start=6,tutor-end=7}{i}}\htmlData{tutor-start=8,tutor-end=9}{+}\htmlData{tutor-start=9,tutor-end=10}{a}_{\htmlData{tutor-start=12,tutor-end=13}{j}}}\htmlData{tutor-start=15,tutor-end=16}{=}\htmlData{tutor-start=16,tutor-end=17}{a}_{\htmlData{tutor-start=19,tutor-end=20}{i}\htmlData{tutor-start=20,tutor-end=21}{+}\htmlData{tutor-start=21,tutor-end=22}{j}} a a i + a j = a i + j f ( n ) \htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{n}\htmlData{tutor-start=3,tutor-end=4}{)} f ( n ) n \htmlData{tutor-start=0,tutor-end=1}{n} n c 1 , c 2 \htmlData{tutor-start=0,tutor-end=1}{c}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{c}_{\htmlData{tutor-start=10,tutor-end=11}{2}} c 1 , c 2 λ \htmlData{tutor-start=0,tutor-end=7}{\lambda} λ c 1 λ n < f ( n ) < c 2 λ n \htmlData{tutor-start=0,tutor-end=1}{c}_{\htmlData{tutor-start=3,tutor-end=4}{1}} \htmlData{tutor-start=6,tutor-end=13}{\lambda}^{\htmlData{tutor-start=15,tutor-end=16}{n}}\htmlData{tutor-start=17,tutor-end=18}{<}\htmlData{tutor-start=18,tutor-end=19}{f}\htmlData{tutor-start=19,tutor-end=20}{(}\htmlData{tutor-start=20,tutor-end=21}{n}\htmlData{tutor-start=21,tutor-end=22}{)}\htmlData{tutor-start=22,tutor-end=23}{<}\htmlData{tutor-start=23,tutor-end=24}{c}_{\htmlData{tutor-start=26,tutor-end=27}{2}} \htmlData{tutor-start=29,tutor-end=36}{\lambda}^{\htmlData{tutor-start=38,tutor-end=39}{n}} c 1 λ n < f ( n ) < c 2 λ n n \htmlData{tutor-start=0,tutor-end=1}{n} n 题解状态： 标准答案与规范题解待补充
题目标签：2022 IMO Shortlist A8
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2023-A1 2023 · Algebra · IMO/代数
Professor Oak is feeding his 100 Pokémon. Each Pokémon has a bowl whose capacity is a positive real number of kilograms. These capacities are known to Professor Oak. The total capacity of all the bowls is 100 kilograms. Professor Oak distributes 100 kilograms of food in such a way that each Pokémon receives a non-negative integer number of kilograms of food (which may be larger than the capacity of their bowl). The dissatisfaction level of a Pokémon who received N \htmlData{tutor-start=0,tutor-end=1}{N} N C \htmlData{tutor-start=0,tutor-end=1}{C} C ∣ N − C ∣ \htmlData{tutor-start=0,tutor-end=1}{|}\htmlData{tutor-start=1,tutor-end=2}{N}\htmlData{tutor-start=2,tutor-end=3}{-}\htmlData{tutor-start=3,tutor-end=4}{C}\htmlData{tutor-start=4,tutor-end=5}{|} ∣ N − C ∣ D \htmlData{tutor-start=0,tutor-end=1}{D} D D \htmlData{tutor-start=0,tutor-end=1}{D} D 题解状态： 标准答案与规范题解待补充
题目标签：2023 IMO Shortlist A1
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2023-A2 2023 · Algebra · IMO/代数
Let R \mathbb{\htmlData{tutor-start=8,tutor-end=9}{R}} R f : R → R \htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{:} \mathbb{\htmlData{tutor-start=11,tutor-end=12}{R}} \htmlData{tutor-start=14,tutor-end=26}{\rightarrow }\mathbb{\htmlData{tutor-start=34,tutor-end=35}{R}} f : R → R f ( x + y ) f ( x − y ) ⩾ f ( x ) 2 − f ( y ) 2 \htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{x}\htmlData{tutor-start=3,tutor-end=4}{+}\htmlData{tutor-start=4,tutor-end=5}{y}\htmlData{tutor-start=5,tutor-end=6}{)} \htmlData{tutor-start=7,tutor-end=8}{f}\htmlData{tutor-start=8,tutor-end=9}{(}\htmlData{tutor-start=9,tutor-end=10}{x}\htmlData{tutor-start=10,tutor-end=11}{-}\htmlData{tutor-start=11,tutor-end=12}{y}\htmlData{tutor-start=12,tutor-end=13}{)} \htmlData{tutor-start=14,tutor-end=24}{\geqslant }\htmlData{tutor-start=24,tutor-end=25}{f}\htmlData{tutor-start=25,tutor-end=26}{(}\htmlData{tutor-start=26,tutor-end=27}{x}\htmlData{tutor-start=27,tutor-end=28}{)}^{\htmlData{tutor-start=30,tutor-end=31}{2}}\htmlData{tutor-start=32,tutor-end=33}{-}\htmlData{tutor-start=33,tutor-end=34}{f}\htmlData{tutor-start=34,tutor-end=35}{(}\htmlData{tutor-start=35,tutor-end=36}{y}\htmlData{tutor-start=36,tutor-end=37}{)}^{\htmlData{tutor-start=39,tutor-end=40}{2}} f ( x + y ) f ( x − y ) ⩾ f ( x ) 2 − f ( y ) 2 x , y ∈ R \htmlData{tutor-start=0,tutor-end=1}{x}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{y} \htmlData{tutor-start=5,tutor-end=9}{\in }\mathbb{\htmlData{tutor-start=17,tutor-end=18}{R}} x , y ∈ R x 0 , y 0 ∈ R \htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{0}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{y}_{\htmlData{tutor-start=10,tutor-end=11}{0}} \htmlData{tutor-start=13,tutor-end=17}{\in }\mathbb{\htmlData{tutor-start=25,tutor-end=26}{R}} x 0 , y 0 ∈ R f ( x ) ⩾ 0 \htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{x}\htmlData{tutor-start=3,tutor-end=4}{)} \htmlData{tutor-start=5,tutor-end=15}{\geqslant }\htmlData{tutor-start=15,tutor-end=16}{0} f ( x ) ⩾ 0 x ∈ R \htmlData{tutor-start=0,tutor-end=1}{x} \htmlData{tutor-start=2,tutor-end=6}{\in }\mathbb{\htmlData{tutor-start=14,tutor-end=15}{R}} x ∈ R f ( x ) ⩽ 0 \htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{x}\htmlData{tutor-start=3,tutor-end=4}{)} \htmlData{tutor-start=5,tutor-end=15}{\leqslant }\htmlData{tutor-start=15,tutor-end=16}{0} f ( x ) ⩽ 0 x ∈ R \htmlData{tutor-start=0,tutor-end=1}{x} \htmlData{tutor-start=2,tutor-end=6}{\in }\mathbb{\htmlData{tutor-start=14,tutor-end=15}{R}} x ∈ R 题解状态： 标准答案与规范题解待补充
题目标签：2023 IMO Shortlist A2
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2023-A3 2023 · Algebra · IMO/代数
Let x 1 , x 2 , … , x 2023 \htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{x}_{\htmlData{tutor-start=10,tutor-end=11}{2}}\htmlData{tutor-start=12,tutor-end=13}{,} \htmlData{tutor-start=14,tutor-end=20}{\ldots}\htmlData{tutor-start=20,tutor-end=21}{,} \htmlData{tutor-start=22,tutor-end=23}{x}_{\htmlData{tutor-start=25,tutor-end=26}{2}\htmlData{tutor-start=26,tutor-end=27}{0}\htmlData{tutor-start=27,tutor-end=28}{2}\htmlData{tutor-start=28,tutor-end=29}{3}} x 1 , x 2 , … , x 2 0 2 3 a n = ( x 1 + x 2 + ⋯ + x n ) ( 1 x 1 + 1 x 2 + ⋯ + 1 x n ) \htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{n}}\htmlData{tutor-start=5,tutor-end=6}{=}\sqrt{\left(\htmlData{tutor-start=18,tutor-end=19}{x}_{\htmlData{tutor-start=21,tutor-end=22}{1}}\htmlData{tutor-start=23,tutor-end=24}{+}\htmlData{tutor-start=24,tutor-end=25}{x}_{\htmlData{tutor-start=27,tutor-end=28}{2}}\htmlData{tutor-start=29,tutor-end=30}{+}\cdots\htmlData{tutor-start=36,tutor-end=37}{+}\htmlData{tutor-start=37,tutor-end=38}{x}_{\htmlData{tutor-start=40,tutor-end=41}{n}}\right)\left(\frac{\htmlData{tutor-start=61,tutor-end=62}{1}}{\htmlData{tutor-start=64,tutor-end=65}{x}_{\htmlData{tutor-start=67,tutor-end=68}{1}}}\htmlData{tutor-start=70,tutor-end=71}{+}\frac{\htmlData{tutor-start=77,tutor-end=78}{1}}{\htmlData{tutor-start=80,tutor-end=81}{x}_{\htmlData{tutor-start=83,tutor-end=84}{2}}}\htmlData{tutor-start=86,tutor-end=87}{+}\cdots\htmlData{tutor-start=93,tutor-end=94}{+}\frac{\htmlData{tutor-start=100,tutor-end=101}{1}}{\htmlData{tutor-start=103,tutor-end=104}{x}_{\htmlData{tutor-start=106,tutor-end=107}{n}}}\right)} a n = ( x 1 + x 2 + ⋯ + x n ) ( x 1 1 + x 2 1 + ⋯ + x n 1 ) n = 1 , 2 , … , 2023 \htmlData{tutor-start=0,tutor-end=1}{n}\htmlData{tutor-start=1,tutor-end=2}{=}\htmlData{tutor-start=2,tutor-end=3}{1}\htmlData{tutor-start=3,tutor-end=4}{,}\htmlData{tutor-start=4,tutor-end=5}{2}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=13}{\ldots}\htmlData{tutor-start=13,tutor-end=14}{,} \htmlData{tutor-start=15,tutor-end=16}{2}\htmlData{tutor-start=16,tutor-end=17}{0}\htmlData{tutor-start=17,tutor-end=18}{2}\htmlData{tutor-start=18,tutor-end=19}{3} n = 1 , 2 , … , 2 0 2 3 a 2023 ⩾ 3034 \htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{2}\htmlData{tutor-start=4,tutor-end=5}{0}\htmlData{tutor-start=5,tutor-end=6}{2}\htmlData{tutor-start=6,tutor-end=7}{3}} \htmlData{tutor-start=9,tutor-end=19}{\geqslant }\htmlData{tutor-start=19,tutor-end=20}{3}\htmlData{tutor-start=20,tutor-end=21}{0}\htmlData{tutor-start=21,tutor-end=22}{3}\htmlData{tutor-start=22,tutor-end=23}{4} a 2 0 2 3 ⩾ 3 0 3 4 题解状态： 标准答案与规范题解待补充
题目标签：2023 IMO 正式题 A3
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2023-A4 2023 · Algebra · IMO/代数
Let R > 0 \mathbb{\htmlData{tutor-start=8,tutor-end=9}{R}}_{\htmlData{tutor-start=12,tutor-end=13}{>}\htmlData{tutor-start=13,tutor-end=14}{0}} R > 0 f : R > 0 → R > 0 \htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{:} \mathbb{\htmlData{tutor-start=11,tutor-end=12}{R}}_{\htmlData{tutor-start=15,tutor-end=16}{>}\htmlData{tutor-start=16,tutor-end=17}{0}} \htmlData{tutor-start=19,tutor-end=31}{\rightarrow }\mathbb{\htmlData{tutor-start=39,tutor-end=40}{R}}_{\htmlData{tutor-start=43,tutor-end=44}{>}\htmlData{tutor-start=44,tutor-end=45}{0}} f : R > 0 → R > 0 x ( f ( x ) + f ( y ) ) ⩾ ( f ( f ( x ) ) + y ) f ( y ) \htmlData{tutor-start=0,tutor-end=1}{x}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{f}\htmlData{tutor-start=3,tutor-end=4}{(}\htmlData{tutor-start=4,tutor-end=5}{x}\htmlData{tutor-start=5,tutor-end=6}{)}\htmlData{tutor-start=6,tutor-end=7}{+}\htmlData{tutor-start=7,tutor-end=8}{f}\htmlData{tutor-start=8,tutor-end=9}{(}\htmlData{tutor-start=9,tutor-end=10}{y}\htmlData{tutor-start=10,tutor-end=11}{)}\htmlData{tutor-start=11,tutor-end=12}{)} \htmlData{tutor-start=13,tutor-end=22}{\geqslant}\htmlData{tutor-start=22,tutor-end=23}{(}\htmlData{tutor-start=23,tutor-end=24}{f}\htmlData{tutor-start=24,tutor-end=25}{(}\htmlData{tutor-start=25,tutor-end=26}{f}\htmlData{tutor-start=26,tutor-end=27}{(}\htmlData{tutor-start=27,tutor-end=28}{x}\htmlData{tutor-start=28,tutor-end=29}{)}\htmlData{tutor-start=29,tutor-end=30}{)}\htmlData{tutor-start=30,tutor-end=31}{+}\htmlData{tutor-start=31,tutor-end=32}{y}\htmlData{tutor-start=32,tutor-end=33}{)} \htmlData{tutor-start=34,tutor-end=35}{f}\htmlData{tutor-start=35,tutor-end=36}{(}\htmlData{tutor-start=36,tutor-end=37}{y}\htmlData{tutor-start=37,tutor-end=38}{)} x ( f ( x ) + f ( y ) ) ⩾ ( f ( f ( x ) ) + y ) f ( y ) x , y ∈ R > 0 \htmlData{tutor-start=0,tutor-end=1}{x}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{y} \htmlData{tutor-start=5,tutor-end=9}{\in }\mathbb{\htmlData{tutor-start=17,tutor-end=18}{R}}_{\htmlData{tutor-start=21,tutor-end=22}{>}\htmlData{tutor-start=22,tutor-end=23}{0}} x , y ∈ R > 0 题解状态： 标准答案与规范题解待补充
题目标签：2023 IMO Shortlist A4
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2023-A5 2023 · Algebra · IMO/代数
Let a 1 , a 2 , … , a 2023 \htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{a}_{\htmlData{tutor-start=10,tutor-end=11}{2}}\htmlData{tutor-start=12,tutor-end=13}{,} \htmlData{tutor-start=14,tutor-end=20}{\ldots}\htmlData{tutor-start=20,tutor-end=21}{,} \htmlData{tutor-start=22,tutor-end=23}{a}_{\htmlData{tutor-start=25,tutor-end=26}{2}\htmlData{tutor-start=26,tutor-end=27}{0}\htmlData{tutor-start=27,tutor-end=28}{2}\htmlData{tutor-start=28,tutor-end=29}{3}} a 1 , a 2 , … , a 2 0 2 3 a 1 , a 2 , … , a 2023 \htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{a}_{\htmlData{tutor-start=10,tutor-end=11}{2}}\htmlData{tutor-start=12,tutor-end=13}{,} \htmlData{tutor-start=14,tutor-end=20}{\ldots}\htmlData{tutor-start=20,tutor-end=21}{,} \htmlData{tutor-start=22,tutor-end=23}{a}_{\htmlData{tutor-start=25,tutor-end=26}{2}\htmlData{tutor-start=26,tutor-end=27}{0}\htmlData{tutor-start=27,tutor-end=28}{2}\htmlData{tutor-start=28,tutor-end=29}{3}} a 1 , a 2 , … , a 2 0 2 3 1 , 2 , … , 2023 \htmlData{tutor-start=0,tutor-end=1}{1}\htmlData{tutor-start=1,tutor-end=2}{,}\htmlData{tutor-start=2,tutor-end=3}{2}\htmlData{tutor-start=3,tutor-end=4}{,} \htmlData{tutor-start=5,tutor-end=11}{\ldots}\htmlData{tutor-start=11,tutor-end=12}{,} \htmlData{tutor-start=13,tutor-end=14}{2}\htmlData{tutor-start=14,tutor-end=15}{0}\htmlData{tutor-start=15,tutor-end=16}{2}\htmlData{tutor-start=16,tutor-end=17}{3} 1 , 2 , … , 2 0 2 3 ∣ a 1 − a 2 ∣ , ∣ a 2 − a 3 ∣ , … , ∣ a 2022 − a 2023 ∣ \left|\htmlData{tutor-start=6,tutor-end=7}{a}_{\htmlData{tutor-start=9,tutor-end=10}{1}}\htmlData{tutor-start=11,tutor-end=12}{-}\htmlData{tutor-start=12,tutor-end=13}{a}_{\htmlData{tutor-start=15,tutor-end=16}{2}}\right|\htmlData{tutor-start=24,tutor-end=25}{,}\left|\htmlData{tutor-start=31,tutor-end=32}{a}_{\htmlData{tutor-start=34,tutor-end=35}{2}}\htmlData{tutor-start=36,tutor-end=37}{-}\htmlData{tutor-start=37,tutor-end=38}{a}_{\htmlData{tutor-start=40,tutor-end=41}{3}}\right|\htmlData{tutor-start=49,tutor-end=50}{,} \htmlData{tutor-start=51,tutor-end=57}{\ldots}\htmlData{tutor-start=57,tutor-end=58}{,}\left|\htmlData{tutor-start=64,tutor-end=65}{a}_{\htmlData{tutor-start=67,tutor-end=68}{2}\htmlData{tutor-start=68,tutor-end=69}{0}\htmlData{tutor-start=69,tutor-end=70}{2}\htmlData{tutor-start=70,tutor-end=71}{2}}\htmlData{tutor-start=72,tutor-end=73}{-}\htmlData{tutor-start=73,tutor-end=74}{a}_{\htmlData{tutor-start=76,tutor-end=77}{2}\htmlData{tutor-start=77,tutor-end=78}{0}\htmlData{tutor-start=78,tutor-end=79}{2}\htmlData{tutor-start=79,tutor-end=80}{3}}\right| ∣ a 1 − a 2 ∣ , ∣ a 2 − a 3 ∣ , … , ∣ a 2 0 2 2 − a 2 0 2 3 ∣ 1 , 2 , … , 2022 \htmlData{tutor-start=0,tutor-end=1}{1}\htmlData{tutor-start=1,tutor-end=2}{,}\htmlData{tutor-start=2,tutor-end=3}{2}\htmlData{tutor-start=3,tutor-end=4}{,} \htmlData{tutor-start=5,tutor-end=11}{\ldots}\htmlData{tutor-start=11,tutor-end=12}{,} \htmlData{tutor-start=13,tutor-end=14}{2}\htmlData{tutor-start=14,tutor-end=15}{0}\htmlData{tutor-start=15,tutor-end=16}{2}\htmlData{tutor-start=16,tutor-end=17}{2} 1 , 2 , … , 2 0 2 2 max ( a 1 , a 2023 ) ⩾ 507 \max \left(\htmlData{tutor-start=11,tutor-end=12}{a}_{\htmlData{tutor-start=14,tutor-end=15}{1}}\htmlData{tutor-start=16,tutor-end=17}{,} \htmlData{tutor-start=18,tutor-end=19}{a}_{\htmlData{tutor-start=21,tutor-end=22}{2}\htmlData{tutor-start=22,tutor-end=23}{0}\htmlData{tutor-start=23,tutor-end=24}{2}\htmlData{tutor-start=24,tutor-end=25}{3}}\right) \htmlData{tutor-start=34,tutor-end=44}{\geqslant }\htmlData{tutor-start=44,tutor-end=45}{5}\htmlData{tutor-start=45,tutor-end=46}{0}\htmlData{tutor-start=46,tutor-end=47}{7} max ( a 1 , a 2 0 2 3 ) ⩾ 5 0 7 题解状态： 标准答案与规范题解待补充
题目标签：2023 IMO Shortlist A5
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2023-A6 2023 · Algebra · IMO/代数
Let k ⩾ 2 \htmlData{tutor-start=0,tutor-end=1}{k} \htmlData{tutor-start=2,tutor-end=12}{\geqslant }\htmlData{tutor-start=12,tutor-end=13}{2} k ⩾ 2 a 1 , a 2 , … \htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{a}_{\htmlData{tutor-start=10,tutor-end=11}{2}}\htmlData{tutor-start=12,tutor-end=13}{,} \htmlData{tutor-start=14,tutor-end=20}{\ldots} a 1 , a 2 , … P \htmlData{tutor-start=0,tutor-end=1}{P} P k \htmlData{tutor-start=0,tutor-end=1}{k} k P ( a n ) = a n + 1 a n + 2 ⋯ a n + k \htmlData{tutor-start=0,tutor-end=1}{P}\left(\htmlData{tutor-start=7,tutor-end=8}{a}_{\htmlData{tutor-start=10,tutor-end=11}{n}}\right)\htmlData{tutor-start=19,tutor-end=20}{=}\htmlData{tutor-start=20,tutor-end=21}{a}_{\htmlData{tutor-start=23,tutor-end=24}{n}\htmlData{tutor-start=24,tutor-end=25}{+}\htmlData{tutor-start=25,tutor-end=26}{1}} \htmlData{tutor-start=28,tutor-end=29}{a}_{\htmlData{tutor-start=31,tutor-end=32}{n}\htmlData{tutor-start=32,tutor-end=33}{+}\htmlData{tutor-start=33,tutor-end=34}{2}} \cdots \htmlData{tutor-start=43,tutor-end=44}{a}_{\htmlData{tutor-start=46,tutor-end=47}{n}\htmlData{tutor-start=47,tutor-end=48}{+}\htmlData{tutor-start=48,tutor-end=49}{k}} P ( a n ) = a n + 1 a n + 2 ⋯ a n + k n ⩾ 1 \htmlData{tutor-start=0,tutor-end=1}{n} \htmlData{tutor-start=2,tutor-end=12}{\geqslant }\htmlData{tutor-start=12,tutor-end=13}{1} n ⩾ 1 题解状态： 标准答案与规范题解待补充
题目标签：2023 IMO 正式题 A6
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2023-A7 2023 · Algebra · IMO/代数
Let N \htmlData{tutor-start=0,tutor-end=1}{N} N a 1 , a 2 , … , a N \htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{a}_{\htmlData{tutor-start=10,tutor-end=11}{2}}\htmlData{tutor-start=12,tutor-end=13}{,} \htmlData{tutor-start=14,tutor-end=20}{\ldots}\htmlData{tutor-start=20,tutor-end=21}{,} \htmlData{tutor-start=22,tutor-end=23}{a}_{\htmlData{tutor-start=25,tutor-end=26}{N}} a 1 , a 2 , … , a N b 1 , b 2 , … , b N \htmlData{tutor-start=0,tutor-end=1}{b}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{b}_{\htmlData{tutor-start=10,tutor-end=11}{2}}\htmlData{tutor-start=12,tutor-end=13}{,} \htmlData{tutor-start=14,tutor-end=20}{\ldots}\htmlData{tutor-start=20,tutor-end=21}{,} \htmlData{tutor-start=22,tutor-end=23}{b}_{\htmlData{tutor-start=25,tutor-end=26}{N}} b 1 , b 2 , … , b N c 1 , c 2 , … , c N \htmlData{tutor-start=0,tutor-end=1}{c}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{c}_{\htmlData{tutor-start=10,tutor-end=11}{2}}\htmlData{tutor-start=12,tutor-end=13}{,} \htmlData{tutor-start=14,tutor-end=20}{\ldots}\htmlData{tutor-start=20,tutor-end=21}{,} \htmlData{tutor-start=22,tutor-end=23}{c}_{\htmlData{tutor-start=25,tutor-end=26}{N}} c 1 , c 2 , … , c N 1 , 2 , … , N \htmlData{tutor-start=0,tutor-end=1}{1}\htmlData{tutor-start=1,tutor-end=2}{,}\htmlData{tutor-start=2,tutor-end=3}{2}\htmlData{tutor-start=3,tutor-end=4}{,} \htmlData{tutor-start=5,tutor-end=11}{\ldots}\htmlData{tutor-start=11,tutor-end=12}{,} \htmlData{tutor-start=13,tutor-end=14}{N} 1 , 2 , … , N ∣ a k + b k + c k − 2 N ∣ < 2023 \left|\sqrt{\htmlData{tutor-start=12,tutor-end=13}{a}_{\htmlData{tutor-start=15,tutor-end=16}{k}}}\htmlData{tutor-start=18,tutor-end=19}{+}\sqrt{\htmlData{tutor-start=25,tutor-end=26}{b}_{\htmlData{tutor-start=28,tutor-end=29}{k}}}\htmlData{tutor-start=31,tutor-end=32}{+}\sqrt{\htmlData{tutor-start=38,tutor-end=39}{c}_{\htmlData{tutor-start=41,tutor-end=42}{k}}}\htmlData{tutor-start=44,tutor-end=45}{-}\htmlData{tutor-start=45,tutor-end=46}{2} \sqrt{\htmlData{tutor-start=53,tutor-end=54}{N}}\right|\htmlData{tutor-start=62,tutor-end=63}{<}\htmlData{tutor-start=63,tutor-end=64}{2}\htmlData{tutor-start=64,tutor-end=65}{0}\htmlData{tutor-start=65,tutor-end=66}{2}\htmlData{tutor-start=66,tutor-end=67}{3} a k + b k + c k − 2 N < 2 0 2 3 k = 1 , 2 , … , N \htmlData{tutor-start=0,tutor-end=1}{k}\htmlData{tutor-start=1,tutor-end=2}{=}\htmlData{tutor-start=2,tutor-end=3}{1}\htmlData{tutor-start=3,tutor-end=4}{,}\htmlData{tutor-start=4,tutor-end=5}{2}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=13}{\ldots}\htmlData{tutor-start=13,tutor-end=14}{,} \htmlData{tutor-start=15,tutor-end=16}{N} k = 1 , 2 , … , N 题解状态： 标准答案与规范题解待补充
题目标签：2023 IMO Shortlist A7
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2024-A1 2024 · Algebra · IMO/代数
Determine all real numbers α \htmlData{tutor-start=0,tutor-end=6}{\alpha} α [ α ] + [ 2 α ] + … + [ n α ] \htmlData{tutor-start=0,tutor-end=1}{[}\htmlData{tutor-start=1,tutor-end=7}{\alpha}\htmlData{tutor-start=7,tutor-end=8}{]} \htmlData{tutor-start=9,tutor-end=10}{+} \htmlData{tutor-start=11,tutor-end=12}{[}\htmlData{tutor-start=12,tutor-end=13}{2}\htmlData{tutor-start=13,tutor-end=19}{\alpha}\htmlData{tutor-start=19,tutor-end=20}{]} \htmlData{tutor-start=21,tutor-end=22}{+} \dots \htmlData{tutor-start=29,tutor-end=30}{+} \htmlData{tutor-start=31,tutor-end=32}{[}\htmlData{tutor-start=32,tutor-end=33}{n}\htmlData{tutor-start=33,tutor-end=39}{\alpha}\htmlData{tutor-start=39,tutor-end=40}{]} [ α ] + [ 2 α ] + … + [ n α ] n \htmlData{tutor-start=0,tutor-end=1}{n} n n \htmlData{tutor-start=0,tutor-end=1}{n} n [ z ] \htmlData{tutor-start=0,tutor-end=1}{[}\htmlData{tutor-start=1,tutor-end=2}{z}\htmlData{tutor-start=2,tutor-end=3}{]} [ z ] z \htmlData{tutor-start=0,tutor-end=1}{z} z 题解状态： 标准答案与规范题解待补充
题目标签：2024 IMO 正式题 A1
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2024-A2 2024 · Algebra · IMO/代数
Let n \htmlData{tutor-start=0,tutor-end=1}{n} n S = 2 0 x 0 2 + 2 1 x 1 2 + … + 2 n x n 2 , \htmlData{tutor-start=0,tutor-end=1}{S} \htmlData{tutor-start=2,tutor-end=3}{=} \htmlData{tutor-start=4,tutor-end=5}{2}^{\htmlData{tutor-start=7,tutor-end=8}{0}} \htmlData{tutor-start=10,tutor-end=11}{x}_{\htmlData{tutor-start=13,tutor-end=14}{0}}^{\htmlData{tutor-start=17,tutor-end=18}{2}} \htmlData{tutor-start=20,tutor-end=21}{+} \htmlData{tutor-start=22,tutor-end=23}{2}^{\htmlData{tutor-start=25,tutor-end=26}{1}} \htmlData{tutor-start=28,tutor-end=29}{x}_{\htmlData{tutor-start=31,tutor-end=32}{1}}^{\htmlData{tutor-start=35,tutor-end=36}{2}} \htmlData{tutor-start=38,tutor-end=39}{+} \dots \htmlData{tutor-start=46,tutor-end=47}{+} \htmlData{tutor-start=48,tutor-end=49}{2}^{\htmlData{tutor-start=51,tutor-end=52}{n}} \htmlData{tutor-start=54,tutor-end=55}{x}_{\htmlData{tutor-start=57,tutor-end=58}{n}}^{\htmlData{tutor-start=61,tutor-end=62}{2}}\htmlData{tutor-start=63,tutor-end=64}{,} S = 2 0 x 0 2 + 2 1 x 1 2 + … + 2 n x n 2 , x 0 , x 1 , … , x n \htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{0}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{x}_{\htmlData{tutor-start=10,tutor-end=11}{1}}\htmlData{tutor-start=12,tutor-end=13}{,} \dots\htmlData{tutor-start=19,tutor-end=20}{,} \htmlData{tutor-start=21,tutor-end=22}{x}_{\htmlData{tutor-start=24,tutor-end=25}{n}} x 0 , x 1 , … , x n x 0 + x 1 + … + x n = n \htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{0}} \htmlData{tutor-start=6,tutor-end=7}{+} \htmlData{tutor-start=8,tutor-end=9}{x}_{\htmlData{tutor-start=11,tutor-end=12}{1}} \htmlData{tutor-start=14,tutor-end=15}{+} \dots \htmlData{tutor-start=22,tutor-end=23}{+} \htmlData{tutor-start=24,tutor-end=25}{x}_{\htmlData{tutor-start=27,tutor-end=28}{n}} \htmlData{tutor-start=30,tutor-end=31}{=} \htmlData{tutor-start=32,tutor-end=33}{n} x 0 + x 1 + … + x n = n 题解状态： 标准答案与规范题解待补充
题目标签：2024 IMO Shortlist A2
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2024-A3 2024 · Algebra · IMO/代数
Decide whether for every sequence ( a n ) \htmlData{tutor-start=0,tutor-end=1}{(}\htmlData{tutor-start=1,tutor-end=2}{a}_{\htmlData{tutor-start=4,tutor-end=5}{n}}\htmlData{tutor-start=6,tutor-end=7}{)} ( a n ) 3 a 1 + 3 a 2 + ⋯ + 3 a n ( 2 a 1 + 2 a 2 + ⋯ + 2 a n ) 2 < 1 2024 \frac{3^{a_{1}} + 3^{a_{2}} + \dots + 3^{a_{n}}}{(2^{a_{1}} + 2^{a_{2}} + \dots + 2^{a_{n}})^{2}} < \frac{1}{2024} ( 2 a 1 + 2 a 2 + ⋯ + 2 a n ) 2 3 a 1 + 3 a 2 + ⋯ + 3 a n < 2024 1 n \htmlData{tutor-start=0,tutor-end=1}{n} n 题解状态： 标准答案与规范题解待补充
题目标签：2024 IMO Shortlist A3
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2024-A4 2024 · Algebra · IMO/代数
Let Z > 0 \mathbb{\htmlData{tutor-start=8,tutor-end=9}{Z}}_{\htmlData{tutor-start=12,tutor-end=13}{>}\htmlData{tutor-start=13,tutor-end=14}{0}} Z > 0 S \mathcal{\htmlData{tutor-start=9,tutor-end=10}{S}} S { 2 0 , 2 1 , 2 2 , … } \htmlData{tutor-start=0,tutor-end=2}{\{}\htmlData{tutor-start=2,tutor-end=3}{2}^{\htmlData{tutor-start=5,tutor-end=6}{0}}\htmlData{tutor-start=7,tutor-end=8}{,} \htmlData{tutor-start=9,tutor-end=10}{2}^{\htmlData{tutor-start=12,tutor-end=13}{1}}\htmlData{tutor-start=14,tutor-end=15}{,} \htmlData{tutor-start=16,tutor-end=17}{2}^{\htmlData{tutor-start=19,tutor-end=20}{2}}\htmlData{tutor-start=21,tutor-end=22}{,} \dots\htmlData{tutor-start=28,tutor-end=30}{\}} { 2 0 , 2 1 , 2 2 , … } f : Z > 0 → Z > 0 \htmlData{tutor-start=0,tutor-end=1}{f} \colon \mathbb{\htmlData{tutor-start=17,tutor-end=18}{Z}}_{\htmlData{tutor-start=21,tutor-end=22}{>}\htmlData{tutor-start=22,tutor-end=23}{0}} \htmlData{tutor-start=25,tutor-end=29}{\to }\mathbb{\htmlData{tutor-start=37,tutor-end=38}{Z}}_{\htmlData{tutor-start=41,tutor-end=42}{>}\htmlData{tutor-start=42,tutor-end=43}{0}} f : Z > 0 → Z > 0 S = { f ( a + b ) − f ( a ) − f ( b ) ∣ a , b ∈ Z > 0 } . \mathcal{\htmlData{tutor-start=9,tutor-end=10}{S}} \htmlData{tutor-start=12,tutor-end=13}{=} \htmlData{tutor-start=14,tutor-end=16}{\{}\htmlData{tutor-start=16,tutor-end=17}{f}\htmlData{tutor-start=17,tutor-end=18}{(}\htmlData{tutor-start=18,tutor-end=19}{a}\htmlData{tutor-start=19,tutor-end=20}{+}\htmlData{tutor-start=20,tutor-end=21}{b}\htmlData{tutor-start=21,tutor-end=22}{)} \htmlData{tutor-start=23,tutor-end=24}{-} \htmlData{tutor-start=25,tutor-end=26}{f}\htmlData{tutor-start=26,tutor-end=27}{(}\htmlData{tutor-start=27,tutor-end=28}{a}\htmlData{tutor-start=28,tutor-end=29}{)} \htmlData{tutor-start=30,tutor-end=31}{-} \htmlData{tutor-start=32,tutor-end=33}{f}\htmlData{tutor-start=33,tutor-end=34}{(}\htmlData{tutor-start=34,tutor-end=35}{b}\htmlData{tutor-start=35,tutor-end=36}{)} \htmlData{tutor-start=37,tutor-end=42}{\mid }\htmlData{tutor-start=42,tutor-end=43}{a}\htmlData{tutor-start=43,tutor-end=44}{,} \htmlData{tutor-start=45,tutor-end=46}{b} \htmlData{tutor-start=47,tutor-end=51}{\in }\mathbb{\htmlData{tutor-start=59,tutor-end=60}{Z}}_{\htmlData{tutor-start=63,tutor-end=64}{>}\htmlData{tutor-start=64,tutor-end=65}{0}}\htmlData{tutor-start=66,tutor-end=68}{\}}\htmlData{tutor-start=68,tutor-end=69}{.} S = { f ( a + b ) − f ( a ) − f ( b ) ∣ a , b ∈ Z > 0 } . 题解状态： 标准答案与规范题解待补充
题目标签：2024 IMO Shortlist A4
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2024-A5 2024 · Algebra · IMO/代数
Find all periodic sequences a 1 , a 2 , … \htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{a}_{\htmlData{tutor-start=10,tutor-end=11}{2}}\htmlData{tutor-start=12,tutor-end=13}{,} \dots a 1 , a 2 , … n ⩾ 1 \htmlData{tutor-start=0,tutor-end=1}{n} \htmlData{tutor-start=2,tutor-end=12}{\geqslant }\htmlData{tutor-start=12,tutor-end=13}{1} n ⩾ 1 a n + 2 + a n 2 = a n + a n + 1 2 and ∣ a n + 1 − a n ∣ ⩽ 1. \htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{n}\htmlData{tutor-start=4,tutor-end=5}{+}\htmlData{tutor-start=5,tutor-end=6}{2}} \htmlData{tutor-start=8,tutor-end=9}{+} \htmlData{tutor-start=10,tutor-end=11}{a}_{\htmlData{tutor-start=13,tutor-end=14}{n}}^{\htmlData{tutor-start=17,tutor-end=18}{2}} \htmlData{tutor-start=20,tutor-end=21}{=} \htmlData{tutor-start=22,tutor-end=23}{a}_{\htmlData{tutor-start=25,tutor-end=26}{n}} \htmlData{tutor-start=28,tutor-end=29}{+} \htmlData{tutor-start=30,tutor-end=31}{a}_{\htmlData{tutor-start=33,tutor-end=34}{n}\htmlData{tutor-start=34,tutor-end=35}{+}\htmlData{tutor-start=35,tutor-end=36}{1}}^{\htmlData{tutor-start=39,tutor-end=40}{2}} \quad \text{\htmlData{tutor-start=54,tutor-end=55}{a}\htmlData{tutor-start=55,tutor-end=56}{n}\htmlData{tutor-start=56,tutor-end=57}{d}} \quad \htmlData{tutor-start=65,tutor-end=66}{|}\htmlData{tutor-start=66,tutor-end=67}{a}_{\htmlData{tutor-start=69,tutor-end=70}{n}\htmlData{tutor-start=70,tutor-end=71}{+}\htmlData{tutor-start=71,tutor-end=72}{1}} \htmlData{tutor-start=74,tutor-end=75}{-} \htmlData{tutor-start=76,tutor-end=77}{a}_{\htmlData{tutor-start=79,tutor-end=80}{n}}\htmlData{tutor-start=81,tutor-end=82}{|} \htmlData{tutor-start=83,tutor-end=93}{\leqslant }\htmlData{tutor-start=93,tutor-end=94}{1}\htmlData{tutor-start=94,tutor-end=95}{.} a n + 2 + a n 2 = a n + a n + 1 2 a n d ∣ a n + 1 − a n ∣ ⩽ 1 . 题解状态： 标准答案与规范题解待补充
题目标签：2024 IMO Shortlist A5
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2024-A6 2024 · Algebra · IMO/代数
Let a 0 , a 1 , a 2 , … \htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{0}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{a}_{\htmlData{tutor-start=10,tutor-end=11}{1}}\htmlData{tutor-start=12,tutor-end=13}{,} \htmlData{tutor-start=14,tutor-end=15}{a}_{\htmlData{tutor-start=17,tutor-end=18}{2}}\htmlData{tutor-start=19,tutor-end=20}{,} \dots a 0 , a 1 , a 2 , … n ⩾ 1 \htmlData{tutor-start=0,tutor-end=1}{n} \htmlData{tutor-start=2,tutor-end=12}{\geqslant }\htmlData{tutor-start=12,tutor-end=13}{1} n ⩾ 1 a n ∈ { a n − 1 + a n + 1 2 , a n − 1 ⋅ a n + 1 } . \htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{n}} \htmlData{tutor-start=6,tutor-end=10}{\in }\left\{ \frac{\htmlData{tutor-start=24,tutor-end=25}{a}_{\htmlData{tutor-start=27,tutor-end=28}{n}\htmlData{tutor-start=28,tutor-end=29}{-}\htmlData{tutor-start=29,tutor-end=30}{1}} \htmlData{tutor-start=32,tutor-end=33}{+} \htmlData{tutor-start=34,tutor-end=35}{a}_{\htmlData{tutor-start=37,tutor-end=38}{n}\htmlData{tutor-start=38,tutor-end=39}{+}\htmlData{tutor-start=39,tutor-end=40}{1}}}{\htmlData{tutor-start=43,tutor-end=44}{2}}\htmlData{tutor-start=45,tutor-end=46}{,} \sqrt{\htmlData{tutor-start=53,tutor-end=54}{a}_{\htmlData{tutor-start=56,tutor-end=57}{n}\htmlData{tutor-start=57,tutor-end=58}{-}\htmlData{tutor-start=58,tutor-end=59}{1}} \htmlData{tutor-start=61,tutor-end=67}{\cdot }\htmlData{tutor-start=67,tutor-end=68}{a}_{\htmlData{tutor-start=70,tutor-end=71}{n}\htmlData{tutor-start=71,tutor-end=72}{+}\htmlData{tutor-start=72,tutor-end=73}{1}}} \right\}\htmlData{tutor-start=84,tutor-end=85}{.} a n ∈ { 2 a n − 1 + a n + 1 , a n − 1 ⋅ a n + 1 } . b 1 , b 2 , … \htmlData{tutor-start=0,tutor-end=1}{b}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{b}_{\htmlData{tutor-start=10,tutor-end=11}{2}}\htmlData{tutor-start=12,tutor-end=13}{,} \dots b 1 , b 2 , … b n = { A , if a n = 1 2 ( a n − 1 + a n + 1 ) ; G , otherwise . \htmlData{tutor-start=0,tutor-end=1}{b}_{\htmlData{tutor-start=3,tutor-end=4}{n}} \htmlData{tutor-start=6,tutor-end=7}{=} \begin{cases} \htmlData{tutor-start=22,tutor-end=23}{A}\htmlData{tutor-start=23,tutor-end=24}{,} & \text{\htmlData{tutor-start=33,tutor-end=34}{i}\htmlData{tutor-start=34,tutor-end=35}{f} } \htmlData{tutor-start=38,tutor-end=39}{a}_{\htmlData{tutor-start=41,tutor-end=42}{n}} \htmlData{tutor-start=44,tutor-end=45}{=} \frac{\htmlData{tutor-start=52,tutor-end=53}{1}}{\htmlData{tutor-start=55,tutor-end=56}{2}}\htmlData{tutor-start=57,tutor-end=58}{(}\htmlData{tutor-start=58,tutor-end=59}{a}_{\htmlData{tutor-start=61,tutor-end=62}{n}\htmlData{tutor-start=62,tutor-end=63}{-}\htmlData{tutor-start=63,tutor-end=64}{1}} \htmlData{tutor-start=66,tutor-end=67}{+} \htmlData{tutor-start=68,tutor-end=69}{a}_{\htmlData{tutor-start=71,tutor-end=72}{n}\htmlData{tutor-start=72,tutor-end=73}{+}\htmlData{tutor-start=73,tutor-end=74}{1}}\htmlData{tutor-start=75,tutor-end=76}{)}\htmlData{tutor-start=76,tutor-end=77}{;} \\ \htmlData{tutor-start=81,tutor-end=82}{G}\htmlData{tutor-start=82,tutor-end=83}{,} & \text{\htmlData{tutor-start=92,tutor-end=93}{o}\htmlData{tutor-start=93,tutor-end=94}{t}\htmlData{tutor-start=94,tutor-end=95}{h}\htmlData{tutor-start=95,tutor-end=96}{e}\htmlData{tutor-start=96,tutor-end=97}{r}\htmlData{tutor-start=97,tutor-end=98}{w}\htmlData{tutor-start=98,tutor-end=99}{i}\htmlData{tutor-start=99,tutor-end=100}{s}\htmlData{tutor-start=100,tutor-end=101}{e}}\htmlData{tutor-start=102,tutor-end=103}{.} \end{cases} b n = { A , G , i f a n = 2 1 ( a n − 1 + a n + 1 ) ; o t h e r w i s e . n 0 \htmlData{tutor-start=0,tutor-end=1}{n}_{\htmlData{tutor-start=3,tutor-end=4}{0}} n 0 d \htmlData{tutor-start=0,tutor-end=1}{d} d n ⩾ n 0 \htmlData{tutor-start=0,tutor-end=1}{n} \htmlData{tutor-start=2,tutor-end=12}{\geqslant }\htmlData{tutor-start=12,tutor-end=13}{n}_{\htmlData{tutor-start=15,tutor-end=16}{0}} n ⩾ n 0 b n + d = b n \htmlData{tutor-start=0,tutor-end=1}{b}_{\htmlData{tutor-start=3,tutor-end=4}{n}\htmlData{tutor-start=4,tutor-end=5}{+}\htmlData{tutor-start=5,tutor-end=6}{d}} \htmlData{tutor-start=8,tutor-end=9}{=} \htmlData{tutor-start=10,tutor-end=11}{b}_{\htmlData{tutor-start=13,tutor-end=14}{n}} b n + d = b n 题解状态： 标准答案与规范题解待补充
题目标签：2024 IMO Shortlist A6
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2024-A7 2024 · Algebra · IMO/代数
Let Q \mathbb{\htmlData{tutor-start=8,tutor-end=9}{Q}} Q f : Q → Q \htmlData{tutor-start=0,tutor-end=1}{f} \colon \mathbb{\htmlData{tutor-start=17,tutor-end=18}{Q}} \htmlData{tutor-start=20,tutor-end=24}{\to }\mathbb{\htmlData{tutor-start=32,tutor-end=33}{Q}} f : Q → Q x , y ∈ Q \htmlData{tutor-start=0,tutor-end=1}{x}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{y} \htmlData{tutor-start=5,tutor-end=9}{\in }\mathbb{\htmlData{tutor-start=17,tutor-end=18}{Q}} x , y ∈ Q f ( x + f ( y ) ) = f ( x ) + y or f ( f ( x ) + y ) = x + f ( y ) . \htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{x} \htmlData{tutor-start=4,tutor-end=5}{+} \htmlData{tutor-start=6,tutor-end=7}{f}\htmlData{tutor-start=7,tutor-end=8}{(}\htmlData{tutor-start=8,tutor-end=9}{y}\htmlData{tutor-start=9,tutor-end=10}{)}\htmlData{tutor-start=10,tutor-end=11}{)} \htmlData{tutor-start=12,tutor-end=13}{=} \htmlData{tutor-start=14,tutor-end=15}{f}\htmlData{tutor-start=15,tutor-end=16}{(}\htmlData{tutor-start=16,tutor-end=17}{x}\htmlData{tutor-start=17,tutor-end=18}{)} \htmlData{tutor-start=19,tutor-end=20}{+} \htmlData{tutor-start=21,tutor-end=22}{y} \quad \text{\htmlData{tutor-start=35,tutor-end=36}{o}\htmlData{tutor-start=36,tutor-end=37}{r}} \quad \htmlData{tutor-start=45,tutor-end=46}{f}\htmlData{tutor-start=46,tutor-end=47}{(}\htmlData{tutor-start=47,tutor-end=48}{f}\htmlData{tutor-start=48,tutor-end=49}{(}\htmlData{tutor-start=49,tutor-end=50}{x}\htmlData{tutor-start=50,tutor-end=51}{)} \htmlData{tutor-start=52,tutor-end=53}{+} \htmlData{tutor-start=54,tutor-end=55}{y}\htmlData{tutor-start=55,tutor-end=56}{)} \htmlData{tutor-start=57,tutor-end=58}{=} \htmlData{tutor-start=59,tutor-end=60}{x} \htmlData{tutor-start=61,tutor-end=62}{+} \htmlData{tutor-start=63,tutor-end=64}{f}\htmlData{tutor-start=64,tutor-end=65}{(}\htmlData{tutor-start=65,tutor-end=66}{y}\htmlData{tutor-start=66,tutor-end=67}{)}\htmlData{tutor-start=67,tutor-end=68}{.} f ( x + f ( y ) ) = f ( x ) + y o r f ( f ( x ) + y ) = x + f ( y ) . { f ( x ) + f ( − x ) ∣ x ∈ Q } \htmlData{tutor-start=0,tutor-end=2}{\{}\htmlData{tutor-start=2,tutor-end=3}{f}\htmlData{tutor-start=3,tutor-end=4}{(}\htmlData{tutor-start=4,tutor-end=5}{x}\htmlData{tutor-start=5,tutor-end=6}{)} \htmlData{tutor-start=7,tutor-end=8}{+} \htmlData{tutor-start=9,tutor-end=10}{f}\htmlData{tutor-start=10,tutor-end=11}{(}\htmlData{tutor-start=11,tutor-end=12}{-}\htmlData{tutor-start=12,tutor-end=13}{x}\htmlData{tutor-start=13,tutor-end=14}{)} \htmlData{tutor-start=15,tutor-end=20}{\mid }\htmlData{tutor-start=20,tutor-end=21}{x} \htmlData{tutor-start=22,tutor-end=26}{\in }\mathbb{\htmlData{tutor-start=34,tutor-end=35}{Q}}\htmlData{tutor-start=36,tutor-end=38}{\}} { f ( x ) + f ( − x ) ∣ x ∈ Q } 题解状态： 标准答案与规范题解待补充
题目标签：2024 IMO 正式题 A7
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2024-A8 2024 · Algebra · IMO/代数
Let p ≠ q \htmlData{tutor-start=0,tutor-end=1}{p} \neq \htmlData{tutor-start=7,tutor-end=8}{q} p = q a 1 , a 2 , … \htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{a}_{\htmlData{tutor-start=10,tutor-end=11}{2}}\htmlData{tutor-start=12,tutor-end=13}{,} \htmlData{tutor-start=14,tutor-end=20}{\ldots} a 1 , a 2 , … n ⩾ 1 \htmlData{tutor-start=0,tutor-end=1}{n} \htmlData{tutor-start=2,tutor-end=12}{\geqslant }\htmlData{tutor-start=12,tutor-end=13}{1} n ⩾ 1 max ( a n , a n + 1 , … , a n + p ) − min ( a n , a n + 1 , … , a n + p ) = p and max ( a n , a n + 1 , … , a n + q ) − min ( a n , a n + 1 , … , a n + q ) = q . \begin{aligned}
\max\htmlData{tutor-start=20,tutor-end=21}{(}\htmlData{tutor-start=21,tutor-end=22}{a}_{\htmlData{tutor-start=24,tutor-end=25}{n}}\htmlData{tutor-start=26,tutor-end=27}{,} \htmlData{tutor-start=28,tutor-end=29}{a}_{\htmlData{tutor-start=31,tutor-end=32}{n}\htmlData{tutor-start=32,tutor-end=33}{+}\htmlData{tutor-start=33,tutor-end=34}{1}}\htmlData{tutor-start=35,tutor-end=36}{,} \htmlData{tutor-start=37,tutor-end=43}{\ldots}\htmlData{tutor-start=43,tutor-end=44}{,} \htmlData{tutor-start=45,tutor-end=46}{a}_{\htmlData{tutor-start=48,tutor-end=49}{n}\htmlData{tutor-start=49,tutor-end=50}{+}\htmlData{tutor-start=50,tutor-end=51}{p}}\htmlData{tutor-start=52,tutor-end=53}{)} \htmlData{tutor-start=54,tutor-end=55}{-} \min\htmlData{tutor-start=60,tutor-end=61}{(}\htmlData{tutor-start=61,tutor-end=62}{a}_{\htmlData{tutor-start=64,tutor-end=65}{n}}\htmlData{tutor-start=66,tutor-end=67}{,} \htmlData{tutor-start=68,tutor-end=69}{a}_{\htmlData{tutor-start=71,tutor-end=72}{n}\htmlData{tutor-start=72,tutor-end=73}{+}\htmlData{tutor-start=73,tutor-end=74}{1}}\htmlData{tutor-start=75,tutor-end=76}{,} \htmlData{tutor-start=77,tutor-end=83}{\ldots}\htmlData{tutor-start=83,tutor-end=84}{,} \htmlData{tutor-start=85,tutor-end=86}{a}_{\htmlData{tutor-start=88,tutor-end=89}{n}\htmlData{tutor-start=89,tutor-end=90}{+}\htmlData{tutor-start=90,tutor-end=91}{p}}\htmlData{tutor-start=92,tutor-end=93}{)} &\htmlData{tutor-start=95,tutor-end=96}{=} \htmlData{tutor-start=97,tutor-end=98}{p} \quad \text{ \htmlData{tutor-start=112,tutor-end=113}{a}\htmlData{tutor-start=113,tutor-end=114}{n}\htmlData{tutor-start=114,tutor-end=115}{d}} \\
\max\htmlData{tutor-start=124,tutor-end=125}{(}\htmlData{tutor-start=125,tutor-end=126}{a}_{\htmlData{tutor-start=128,tutor-end=129}{n}}\htmlData{tutor-start=130,tutor-end=131}{,} \htmlData{tutor-start=132,tutor-end=133}{a}_{\htmlData{tutor-start=135,tutor-end=136}{n}\htmlData{tutor-start=136,tutor-end=137}{+}\htmlData{tutor-start=137,tutor-end=138}{1}}\htmlData{tutor-start=139,tutor-end=140}{,} \htmlData{tutor-start=141,tutor-end=147}{\ldots}\htmlData{tutor-start=147,tutor-end=148}{,} \htmlData{tutor-start=149,tutor-end=150}{a}_{\htmlData{tutor-start=152,tutor-end=153}{n}\htmlData{tutor-start=153,tutor-end=154}{+}\htmlData{tutor-start=154,tutor-end=155}{q}}\htmlData{tutor-start=156,tutor-end=157}{)} \htmlData{tutor-start=158,tutor-end=159}{-} \min\htmlData{tutor-start=164,tutor-end=165}{(}\htmlData{tutor-start=165,tutor-end=166}{a}_{\htmlData{tutor-start=168,tutor-end=169}{n}}\htmlData{tutor-start=170,tutor-end=171}{,} \htmlData{tutor-start=172,tutor-end=173}{a}_{\htmlData{tutor-start=175,tutor-end=176}{n}\htmlData{tutor-start=176,tutor-end=177}{+}\htmlData{tutor-start=177,tutor-end=178}{1}}\htmlData{tutor-start=179,tutor-end=180}{,} \htmlData{tutor-start=181,tutor-end=187}{\ldots}\htmlData{tutor-start=187,tutor-end=188}{,} \htmlData{tutor-start=189,tutor-end=190}{a}_{\htmlData{tutor-start=192,tutor-end=193}{n}\htmlData{tutor-start=193,tutor-end=194}{+}\htmlData{tutor-start=194,tutor-end=195}{q}}\htmlData{tutor-start=196,tutor-end=197}{)} &\htmlData{tutor-start=199,tutor-end=200}{=} \htmlData{tutor-start=201,tutor-end=202}{q}\htmlData{tutor-start=202,tutor-end=203}{.}
\end{aligned} max ( a n , a n + 1 , … , a n + p ) − min ( a n , a n + 1 , … , a n + p ) max ( a n , a n + 1 , … , a n + q ) − min ( a n , a n + 1 , … , a n + q ) = p a n d = q . 题解状态： 标准答案与规范题解待补充
题目标签：2024 IMO Shortlist A8
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2025-P3 2025 · Algebra · IMO/代数
A function f : N → N \htmlData{tutor-start=0,tutor-end=1}{f}\colon \mathbb{\htmlData{tutor-start=16,tutor-end=17}{N}}\htmlData{tutor-start=18,tutor-end=22}{\to }\mathbb{\htmlData{tutor-start=30,tutor-end=31}{N}} f : N → N f ( a ) d i v i d e s b a − f ( b ) f ( a ) \htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{a}\htmlData{tutor-start=3,tutor-end=4}{)}\quad \mathrm{\htmlData{tutor-start=18,tutor-end=19}{d}\htmlData{tutor-start=19,tutor-end=20}{i}\htmlData{tutor-start=20,tutor-end=21}{v}\htmlData{tutor-start=21,tutor-end=22}{i}\htmlData{tutor-start=22,tutor-end=23}{d}\htmlData{tutor-start=23,tutor-end=24}{e}\htmlData{tutor-start=24,tutor-end=25}{s}}\quad \htmlData{tutor-start=32,tutor-end=33}{b}^{\htmlData{tutor-start=35,tutor-end=36}{a}} \htmlData{tutor-start=38,tutor-end=39}{-} \htmlData{tutor-start=40,tutor-end=41}{f}\htmlData{tutor-start=41,tutor-end=42}{(}\htmlData{tutor-start=42,tutor-end=43}{b}\htmlData{tutor-start=43,tutor-end=44}{)}^{\htmlData{tutor-start=46,tutor-end=47}{f}\htmlData{tutor-start=47,tutor-end=48}{(}\htmlData{tutor-start=48,tutor-end=49}{a}\htmlData{tutor-start=49,tutor-end=50}{)}} f ( a ) d i v i d e s b a − f ( b ) f ( a ) a \htmlData{tutor-start=0,tutor-end=1}{a} a b \htmlData{tutor-start=0,tutor-end=1}{b} b c \htmlData{tutor-start=0,tutor-end=1}{c} c f ( n ) ≤ c n \htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{n}\htmlData{tutor-start=3,tutor-end=4}{)}\htmlData{tutor-start=4,tutor-end=9}{\leq }\htmlData{tutor-start=9,tutor-end=10}{c}\htmlData{tutor-start=10,tutor-end=11}{n} f ( n ) ≤ c n f \htmlData{tutor-start=0,tutor-end=1}{f} f n \htmlData{tutor-start=0,tutor-end=1}{n} n 题解状态： 标准答案与规范题解待补充
题目标签：2025 IMO 正式题 P3
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。
IMO2025-P5 2025 · Algebra · IMO/代数
Alice and Bazz a are playing the inekoaty game, a two- player game whose
rules depend on a positive real number λ \htmlData{tutor-start=0,tutor-end=7}{\lambda} λ n = 1 \htmlData{tutor-start=0,tutor-end=1}{n} \htmlData{tutor-start=2,tutor-end=3}{=} \htmlData{tutor-start=4,tutor-end=5}{1} n = 1 n \htmlData{tutor-start=0,tutor-end=1}{n} n x n \htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{n}} x n x 1 + x 2 + … + x n ≤ λ n . \htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{1}} \htmlData{tutor-start=6,tutor-end=7}{+} \htmlData{tutor-start=8,tutor-end=9}{x}_{\htmlData{tutor-start=11,tutor-end=12}{2}} \htmlData{tutor-start=14,tutor-end=15}{+} \dots \htmlData{tutor-start=22,tutor-end=23}{+}\htmlData{tutor-start=23,tutor-end=24}{x}_{\htmlData{tutor-start=26,tutor-end=27}{n}}\htmlData{tutor-start=28,tutor-end=33}{\leq }\htmlData{tutor-start=33,tutor-end=41}{\lambda }\htmlData{tutor-start=41,tutor-end=42}{n}\htmlData{tutor-start=42,tutor-end=43}{.} x 1 + x 2 + … + x n ≤ λ n . n \htmlData{tutor-start=0,tutor-end=1}{n} n x n \htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{n}} x n x 1 2 + x 2 2 + … + x n 2 ≤ n . \htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{1}}^{\htmlData{tutor-start=7,tutor-end=8}{2}} \htmlData{tutor-start=10,tutor-end=11}{+} \htmlData{tutor-start=12,tutor-end=13}{x}_{\htmlData{tutor-start=15,tutor-end=16}{2}}^{\htmlData{tutor-start=19,tutor-end=20}{2}} \htmlData{tutor-start=22,tutor-end=23}{+} \dots \htmlData{tutor-start=30,tutor-end=31}{+}\htmlData{tutor-start=31,tutor-end=32}{x}_{\htmlData{tutor-start=34,tutor-end=35}{n}}^{\htmlData{tutor-start=38,tutor-end=39}{2}}\htmlData{tutor-start=40,tutor-end=45}{\leq }\htmlData{tutor-start=45,tutor-end=46}{n}\htmlData{tutor-start=46,tutor-end=47}{.} x 1 2 + x 2 2 + … + x n 2 ≤ n . x n \htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{n}} x n λ \htmlData{tutor-start=0,tutor-end=7}{\lambda} λ 题解状态： 标准答案与规范题解待补充
题目标签：2025 IMO 正式题 P5
解题过程 该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。