返回特征解读

1990 年全国高中数学联赛（学生版）

books/national_high_school_league_students/papers/competition-national-league-1990.pdf · HS-MATH-1024-v2.1-solution-aware

18 题0 个小问/题组
1

一试 · 一、选择题 (本题满分 30 分，每小题 5 分) · 数学竞赛/待细分

设 a∈(π4,π2)\htmlData{tutor-start=0,tutor-end=1}{a} \htmlData{tutor-start=2,tutor-end=6}{\in }\htmlData{tutor-start=6,tutor-end=7}{(}\frac{\htmlData{tutor-start=13,tutor-end=16}{\pi}}{\htmlData{tutor-start=18,tutor-end=19}{4}}\htmlData{tutor-start=20,tutor-end=21}{,} \frac{\htmlData{tutor-start=28,tutor-end=31}{\pi}}{\htmlData{tutor-start=33,tutor-end=34}{2}}\htmlData{tutor-start=35,tutor-end=36}{)}，则 (cos⁡α)cos⁡α\htmlData{tutor-start=0,tutor-end=1}{(}\cos\htmlData{tutor-start=5,tutor-end=11}{\alpha}\htmlData{tutor-start=11,tutor-end=12}{)}^{\cos\htmlData{tutor-start=18,tutor-end=24}{\alpha}}，(sin⁡α)cos⁡α\htmlData{tutor-start=0,tutor-end=1}{(}\sin\htmlData{tutor-start=5,tutor-end=11}{\alpha}\htmlData{tutor-start=11,tutor-end=12}{)}^{\cos\htmlData{tutor-start=18,tutor-end=24}{\alpha}}，(cos⁡α)sin⁡α\htmlData{tutor-start=0,tutor-end=1}{(}\cos\htmlData{tutor-start=5,tutor-end=11}{\alpha}\htmlData{tutor-start=11,tutor-end=12}{)}^{\sin\htmlData{tutor-start=18,tutor-end=24}{\alpha}} 的大小顺序是 A. (cos⁡α)cos⁡α<(sin⁡α)cos⁡α<(cos⁡α)sin⁡α\htmlData{tutor-start=0,tutor-end=1}{(}\cos\htmlData{tutor-start=5,tutor-end=11}{\alpha}\htmlData{tutor-start=11,tutor-end=12}{)}^{\cos\htmlData{tutor-start=18,tutor-end=24}{\alpha}} \htmlData{tutor-start=26,tutor-end=27}{<} \htmlData{tutor-start=28,tutor-end=29}{(}\sin\htmlData{tutor-start=33,tutor-end=39}{\alpha}\htmlData{tutor-start=39,tutor-end=40}{)}^{\cos\htmlData{tutor-start=46,tutor-end=52}{\alpha}} \htmlData{tutor-start=54,tutor-end=55}{<} \htmlData{tutor-start=56,tutor-end=57}{(}\cos\htmlData{tutor-start=61,tutor-end=67}{\alpha}\htmlData{tutor-start=67,tutor-end=68}{)}^{\sin\htmlData{tutor-start=74,tutor-end=80}{\alpha}} B. (cos⁡α)cos⁡α<(cos⁡α)sin⁡α<(sin⁡α)cos⁡α\htmlData{tutor-start=0,tutor-end=1}{(}\cos\htmlData{tutor-start=5,tutor-end=11}{\alpha}\htmlData{tutor-start=11,tutor-end=12}{)}^{\cos\htmlData{tutor-start=18,tutor-end=24}{\alpha}} \htmlData{tutor-start=26,tutor-end=27}{<} \htmlData{tutor-start=28,tutor-end=29}{(}\cos\htmlData{tutor-start=33,tutor-end=39}{\alpha}\htmlData{tutor-start=39,tutor-end=40}{)}^{\sin\htmlData{tutor-start=46,tutor-end=52}{\alpha}} \htmlData{tutor-start=54,tutor-end=55}{<} \htmlData{tutor-start=56,tutor-end=57}{(}\sin\htmlData{tutor-start=61,tutor-end=67}{\alpha}\htmlData{tutor-start=67,tutor-end=68}{)}^{\cos\htmlData{tutor-start=74,tutor-end=80}{\alpha}} C. (sin⁡α)sin⁡α<(cos⁡α)cos⁡α<(cos⁡α)sin⁡α\htmlData{tutor-start=0,tutor-end=1}{(}\sin\htmlData{tutor-start=5,tutor-end=11}{\alpha}\htmlData{tutor-start=11,tutor-end=12}{)}^{\sin\htmlData{tutor-start=18,tutor-end=24}{\alpha}} \htmlData{tutor-start=26,tutor-end=27}{<} \htmlData{tutor-start=28,tutor-end=29}{(}\cos\htmlData{tutor-start=33,tutor-end=39}{\alpha}\htmlData{tutor-start=39,tutor-end=40}{)}^{\cos\htmlData{tutor-start=46,tutor-end=52}{\alpha}} \htmlData{tutor-start=54,tutor-end=55}{<} \htmlData{tutor-start=56,tutor-end=57}{(}\cos\htmlData{tutor-start=61,tutor-end=67}{\alpha}\htmlData{tutor-start=67,tutor-end=68}{)}^{\sin\htmlData{tutor-start=74,tutor-end=80}{\alpha}} D. (cos⁡α)sin⁡α<(cos⁡α)cos⁡α<(sin⁡α)cos⁡α\htmlData{tutor-start=0,tutor-end=1}{(}\cos\htmlData{tutor-start=5,tutor-end=11}{\alpha}\htmlData{tutor-start=11,tutor-end=12}{)}^{\sin\htmlData{tutor-start=18,tutor-end=24}{\alpha}} \htmlData{tutor-start=26,tutor-end=27}{<} \htmlData{tutor-start=28,tutor-end=29}{(}\cos\htmlData{tutor-start=33,tutor-end=39}{\alpha}\htmlData{tutor-start=39,tutor-end=40}{)}^{\cos\htmlData{tutor-start=46,tutor-end=52}{\alpha}} \htmlData{tutor-start=54,tutor-end=55}{<} \htmlData{tutor-start=56,tutor-end=57}{(}\sin\htmlData{tutor-start=61,tutor-end=67}{\alpha}\htmlData{tutor-start=67,tutor-end=68}{)}^{\cos\htmlData{tutor-start=74,tutor-end=80}{\alpha}}

题解状态：标准答案与规范题解待补充

题目标签：FIRST-CHOICE-P1

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。

2

一试 · 一、选择题 (本题满分 30 分，每小题 5 分) · 代数

设 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}{)} 是定义在实数集上的周期为 2 的函数，且是偶函数，已知当 x∈[2,3]\htmlData{tutor-start=0,tutor-end=1}{x} \htmlData{tutor-start=2,tutor-end=6}{\in }\htmlData{tutor-start=6,tutor-end=7}{[}\htmlData{tutor-start=7,tutor-end=8}{2}\htmlData{tutor-start=8,tutor-end=9}{,} \htmlData{tutor-start=10,tutor-end=11}{3}\htmlData{tutor-start=11,tutor-end=12}{]} 时，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}，则当 x∈[−2,0]\htmlData{tutor-start=0,tutor-end=1}{x} \htmlData{tutor-start=2,tutor-end=6}{\in }\htmlData{tutor-start=6,tutor-end=7}{[}\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=11,tutor-end=12}{0}\htmlData{tutor-start=12,tutor-end=13}{]} 时，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}{)} 的解析式是 ( ) A. f(x)=x+4\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}\htmlData{tutor-start=6,tutor-end=7}{+}\htmlData{tutor-start=7,tutor-end=8}{4} B. 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=6,tutor-end=7}{-}\htmlData{tutor-start=7,tutor-end=8}{x} C. f(x)=3−∣x+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}{=}\htmlData{tutor-start=5,tutor-end=6}{3}\htmlData{tutor-start=6,tutor-end=7}{-}\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}{1}\htmlData{tutor-start=11,tutor-end=12}{|} D. f(x)=2+∣x+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}{=}\htmlData{tutor-start=5,tutor-end=6}{2}\htmlData{tutor-start=6,tutor-end=7}{+}\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}{1}\htmlData{tutor-start=11,tutor-end=12}{|}

题解状态：标准答案与规范题解待补充

题目标签：FIRST-CHOICE-P2

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。

3

一试 · 一、选择题 (本题满分 30 分，每小题 5 分) · 平面几何

设双曲线的左右焦点是 F1\htmlData{tutor-start=0,tutor-end=1}{F}_{\htmlData{tutor-start=3,tutor-end=4}{1}}、F2\htmlData{tutor-start=0,tutor-end=1}{F}_{\htmlData{tutor-start=3,tutor-end=4}{2}}，左右顶点是 M\htmlData{tutor-start=0,tutor-end=1}{M}、N\htmlData{tutor-start=0,tutor-end=1}{N}。若 △PF1F2\htmlData{tutor-start=0,tutor-end=10}{\triangle }\htmlData{tutor-start=10,tutor-end=11}{P}\htmlData{tutor-start=11,tutor-end=12}{F}_{\htmlData{tutor-start=14,tutor-end=15}{1}}\htmlData{tutor-start=16,tutor-end=17}{F}_{\htmlData{tutor-start=19,tutor-end=20}{2}} 的顶点 P\htmlData{tutor-start=0,tutor-end=1}{P} 在双曲线上，则 △PF1F2\htmlData{tutor-start=0,tutor-end=10}{\triangle }\htmlData{tutor-start=10,tutor-end=11}{P}\htmlData{tutor-start=11,tutor-end=12}{F}_{\htmlData{tutor-start=14,tutor-end=15}{1}}\htmlData{tutor-start=16,tutor-end=17}{F}_{\htmlData{tutor-start=19,tutor-end=20}{2}} 的内切圆与边 F1F2\htmlData{tutor-start=0,tutor-end=1}{F}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{F}_{\htmlData{tutor-start=8,tutor-end=9}{2}} 的切点位置是 ( ) A. 在线段 MN\htmlData{tutor-start=0,tutor-end=1}{M}\htmlData{tutor-start=1,tutor-end=2}{N} 内部 B. 在线段 F1M\htmlData{tutor-start=0,tutor-end=1}{F}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{M} 内部或在线段 NF2\htmlData{tutor-start=0,tutor-end=1}{N}\htmlData{tutor-start=1,tutor-end=2}{F}_{\htmlData{tutor-start=4,tutor-end=5}{2}} 内部 C. 点 M\htmlData{tutor-start=0,tutor-end=1}{M} 或点 N\htmlData{tutor-start=0,tutor-end=1}{N} D. 不能确定的

题解状态：标准答案与规范题解待补充

题目标签：FIRST-CHOICE-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。

4

一试 · 一、选择题 (本题满分 30 分，每小题 5 分) · 数学竞赛/待细分

点集 {(x,y)∣lg⁡(x2+13y2+19)=lg⁡x+lg⁡y}\htmlData{tutor-start=0,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=9,tutor-end=14}{\mid }\lg\htmlData{tutor-start=17,tutor-end=18}{(}\htmlData{tutor-start=18,tutor-end=19}{x}^{\htmlData{tutor-start=21,tutor-end=22}{2}}\htmlData{tutor-start=23,tutor-end=24}{+}\frac{\htmlData{tutor-start=30,tutor-end=31}{1}}{\htmlData{tutor-start=33,tutor-end=34}{3}}\htmlData{tutor-start=35,tutor-end=36}{y}^{\htmlData{tutor-start=38,tutor-end=39}{2}}\htmlData{tutor-start=40,tutor-end=41}{+}\frac{\htmlData{tutor-start=47,tutor-end=48}{1}}{\htmlData{tutor-start=50,tutor-end=51}{9}}\htmlData{tutor-start=52,tutor-end=53}{)} \htmlData{tutor-start=54,tutor-end=55}{=} \lg \htmlData{tutor-start=60,tutor-end=61}{x} \htmlData{tutor-start=62,tutor-end=63}{+} \lg \htmlData{tutor-start=68,tutor-end=69}{y}\htmlData{tutor-start=69,tutor-end=71}{\}} 中元素个数为 ( ) A. 0 B. 1 C. 2 D. 多于 2

题解状态：标准答案与规范题解待补充

题目标签：FIRST-CHOICE-P4

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。

5

一试 · 一、选择题 (本题满分 30 分，每小题 5 分) · 数学竞赛/待细分

设非零复数 x\htmlData{tutor-start=0,tutor-end=1}{x}、y\htmlData{tutor-start=0,tutor-end=1}{y} 满足 x2+xy+y2=0\htmlData{tutor-start=0,tutor-end=1}{x}^{\htmlData{tutor-start=3,tutor-end=4}{2}}\htmlData{tutor-start=5,tutor-end=6}{+}\htmlData{tutor-start=6,tutor-end=7}{x}\htmlData{tutor-start=7,tutor-end=8}{y}\htmlData{tutor-start=8,tutor-end=9}{+}\htmlData{tutor-start=9,tutor-end=10}{y}^{\htmlData{tutor-start=12,tutor-end=13}{2}}\htmlData{tutor-start=14,tutor-end=15}{=}\htmlData{tutor-start=15,tutor-end=16}{0}，则代数式 (xx+y)1990+(yx+y)1990\left(\frac{\htmlData{tutor-start=12,tutor-end=13}{x}}{\htmlData{tutor-start=15,tutor-end=16}{x}\htmlData{tutor-start=16,tutor-end=17}{+}\htmlData{tutor-start=17,tutor-end=18}{y}}\right)^{\htmlData{tutor-start=28,tutor-end=29}{1}\htmlData{tutor-start=29,tutor-end=30}{9}\htmlData{tutor-start=30,tutor-end=31}{9}\htmlData{tutor-start=31,tutor-end=32}{0}} \htmlData{tutor-start=34,tutor-end=35}{+} \left(\frac{\htmlData{tutor-start=48,tutor-end=49}{y}}{\htmlData{tutor-start=51,tutor-end=52}{x}\htmlData{tutor-start=52,tutor-end=53}{+}\htmlData{tutor-start=53,tutor-end=54}{y}}\right)^{\htmlData{tutor-start=64,tutor-end=65}{1}\htmlData{tutor-start=65,tutor-end=66}{9}\htmlData{tutor-start=66,tutor-end=67}{9}\htmlData{tutor-start=67,tutor-end=68}{0}} 的值是 ( ) A. 2−1989\htmlData{tutor-start=0,tutor-end=1}{2}^{\htmlData{tutor-start=3,tutor-end=4}{-}\htmlData{tutor-start=4,tutor-end=5}{1}\htmlData{tutor-start=5,tutor-end=6}{9}\htmlData{tutor-start=6,tutor-end=7}{8}\htmlData{tutor-start=7,tutor-end=8}{9}} B. −1\htmlData{tutor-start=0,tutor-end=1}{-}\htmlData{tutor-start=1,tutor-end=2}{1} C. 1\htmlData{tutor-start=0,tutor-end=1}{1} D. 以上答案都不对

题解状态：标准答案与规范题解待补充

题目标签：FIRST-CHOICE-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。

6

一试 · 一、选择题 (本题满分 30 分，每小题 5 分) · 平面几何

已知椭圆 x2a2+y2b2=1(a>b>0)\frac{\htmlData{tutor-start=6,tutor-end=7}{x}^{\htmlData{tutor-start=9,tutor-end=10}{2}}}{\htmlData{tutor-start=13,tutor-end=14}{a}^{\htmlData{tutor-start=16,tutor-end=17}{2}}}\htmlData{tutor-start=19,tutor-end=20}{+}\frac{\htmlData{tutor-start=26,tutor-end=27}{y}^{\htmlData{tutor-start=29,tutor-end=30}{2}}}{\htmlData{tutor-start=33,tutor-end=34}{b}^{\htmlData{tutor-start=36,tutor-end=37}{2}}}\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}{a}\htmlData{tutor-start=44,tutor-end=45}{>}\htmlData{tutor-start=45,tutor-end=46}{b}\htmlData{tutor-start=46,tutor-end=47}{>}\htmlData{tutor-start=47,tutor-end=48}{0}\htmlData{tutor-start=48,tutor-end=49}{)} 通过点 (2,1)\htmlData{tutor-start=0,tutor-end=1}{(}\htmlData{tutor-start=1,tutor-end=2}{2}\htmlData{tutor-start=2,tutor-end=3}{,} \htmlData{tutor-start=4,tutor-end=5}{1}\htmlData{tutor-start=5,tutor-end=6}{)}，所有这些椭圆上满足 ∣y∣>1\htmlData{tutor-start=0,tutor-end=1}{|}\htmlData{tutor-start=1,tutor-end=2}{y}\htmlData{tutor-start=2,tutor-end=3}{|}\htmlData{tutor-start=3,tutor-end=4}{>}\htmlData{tutor-start=4,tutor-end=5}{1} 的点的集合用阴影表示是下面图中的 ( )

原卷图示 1
原卷图示 1原卷第 1 页 · qwen3.7-plus_layout_detection · 需复核
原卷图示 2
原卷图示 2原卷第 1 页 · qwen3.7-plus_layout_detection · 需复核
原卷图示 3
原卷图示 3原卷第 1 页 · qwen3.7-plus_layout_detection · 需复核
原卷图示 4
原卷图示 4原卷第 1 页 · qwen3.7-plus_layout_detection · 需复核

题解状态：标准答案与规范题解待补充

题目标签：FIRST-CHOICE-P6

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。

7

一试 · 二、填空题 (本题满分 30 分，每小题 5 分) · 数学竞赛/待细分

设 n\htmlData{tutor-start=0,tutor-end=1}{n} 为自然数，a\htmlData{tutor-start=0,tutor-end=1}{a}、b\htmlData{tutor-start=0,tutor-end=1}{b} 为正实数，且满足 a+b=2\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}{2}，则 11+a2+11+b2\frac{\htmlData{tutor-start=6,tutor-end=7}{1}}{\htmlData{tutor-start=9,tutor-end=10}{1}\htmlData{tutor-start=10,tutor-end=11}{+}\htmlData{tutor-start=11,tutor-end=12}{a}^{\htmlData{tutor-start=14,tutor-end=15}{2}}} \htmlData{tutor-start=18,tutor-end=19}{+} \frac{\htmlData{tutor-start=26,tutor-end=27}{1}}{\htmlData{tutor-start=29,tutor-end=30}{1}\htmlData{tutor-start=30,tutor-end=31}{+}\htmlData{tutor-start=31,tutor-end=32}{b}^{\htmlData{tutor-start=34,tutor-end=35}{2}}} 的最小值是________。

题解状态：标准答案与规范题解待补充

题目标签：FIRST-FILL-P1

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。

8

一试 · 二、填空题 (本题满分 30 分，每小题 5 分) · 数学竞赛/待细分

设 A(2,0)\htmlData{tutor-start=0,tutor-end=1}{A}\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=6}{0}\htmlData{tutor-start=6,tutor-end=7}{)} 为平面上一定点，P(sin⁡(2t−60∘),cos⁡(2t−60∘))P(\sin(2t-60^\circ), \cos(2t-60^\circ)) 为动点，则当 t\htmlData{tutor-start=0,tutor-end=1}{t} 由 15∘15^\circ 变到 45∘45^\circ 时，线段 AP\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{P} 扫过的面积是________。

题解状态：标准答案与规范题解待补充

题目标签：FIRST-FILL-P2

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。

9

一试 · 二、填空题 (本题满分 30 分，每小题 5 分) · 数学竞赛/待细分

设 n\htmlData{tutor-start=0,tutor-end=1}{n} 为自然数，对于任意实数 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}，恒有 (x2+y2+z2)2≤n(x4+y4+z4)\htmlData{tutor-start=0,tutor-end=1}{(}\htmlData{tutor-start=1,tutor-end=2}{x}^{\htmlData{tutor-start=4,tutor-end=5}{2}}\htmlData{tutor-start=6,tutor-end=7}{+}\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=13,tutor-end=14}{z}^{\htmlData{tutor-start=16,tutor-end=17}{2}}\htmlData{tutor-start=18,tutor-end=19}{)}^{\htmlData{tutor-start=21,tutor-end=22}{2}} \htmlData{tutor-start=24,tutor-end=28}{\le }\htmlData{tutor-start=28,tutor-end=29}{n}\htmlData{tutor-start=29,tutor-end=30}{(}\htmlData{tutor-start=30,tutor-end=31}{x}^{\htmlData{tutor-start=33,tutor-end=34}{4}}\htmlData{tutor-start=35,tutor-end=36}{+}\htmlData{tutor-start=36,tutor-end=37}{y}^{\htmlData{tutor-start=39,tutor-end=40}{4}}\htmlData{tutor-start=41,tutor-end=42}{+}\htmlData{tutor-start=42,tutor-end=43}{z}^{\htmlData{tutor-start=45,tutor-end=46}{4}}\htmlData{tutor-start=47,tutor-end=48}{)} 成立，则 n\htmlData{tutor-start=0,tutor-end=1}{n} 的最小值是________。

题解状态：标准答案与规范题解待补充

题目标签：FIRST-FILL-P3

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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。

10

一试 · 二、填空题 (本题满分 30 分，每小题 5 分) · 数论

对任意正整数 n\htmlData{tutor-start=0,tutor-end=1}{n}，连结原点 O\htmlData{tutor-start=0,tutor-end=1}{O} 与点 An(n,n+3)\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}{n}\htmlData{tutor-start=7,tutor-end=8}{,} \htmlData{tutor-start=9,tutor-end=10}{n}\htmlData{tutor-start=10,tutor-end=11}{+}\htmlData{tutor-start=11,tutor-end=12}{3}\htmlData{tutor-start=12,tutor-end=13}{)}，用 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}{)} 表示线段 OAn\htmlData{tutor-start=0,tutor-end=1}{O}\htmlData{tutor-start=1,tutor-end=2}{A}_{\htmlData{tutor-start=4,tutor-end=5}{n}} 上的整点个数 (不计端点)，试求 f(1)+f(2)+⋯+f(1990)\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}{+}\htmlData{tutor-start=5,tutor-end=6}{f}\htmlData{tutor-start=6,tutor-end=7}{(}\htmlData{tutor-start=7,tutor-end=8}{2}\htmlData{tutor-start=8,tutor-end=9}{)}\htmlData{tutor-start=9,tutor-end=10}{+}\cdots\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}{1}\htmlData{tutor-start=20,tutor-end=21}{9}\htmlData{tutor-start=21,tutor-end=22}{9}\htmlData{tutor-start=22,tutor-end=23}{0}\htmlData{tutor-start=23,tutor-end=24}{)}。

题解状态：标准答案与规范题解待补充

题目标签：FIRST-FILL-P4

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。

11

一试 · 二、填空题 (本题满分 30 分，每小题 5 分) · 数学竞赛/待细分

设 n=1990\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}{9}\htmlData{tutor-start=4,tutor-end=5}{9}\htmlData{tutor-start=5,tutor-end=6}{0}，则 12n(1−3Cn2+32Cn4−33Cn6+⋯+3994Cn1998−3995Cn1990=\frac{1}{2^{n}}(1-3C_{n}^{2}+3^{2}C_{n}^{4}-3^{3}C_{n}^{6}+\cdots+3^{994}C_{n}^{1998}-3^{995}C_{n}^{1990}= ______.

题解状态：标准答案与规范题解待补充

题目标签：FIRST-FILL-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。

12

一试 · 二、填空题 (本题满分 30 分，每小题 5 分) · 平面几何

8 个女孩与 25 个男孩围成一圈，任何两个女孩之间至少站两个男孩，则共有______种不同和排列方法.（只要把圆旋转一下就能重合的排法认为是相同的）.

题解状态：标准答案与规范题解待补充

题目标签：FIRST-FILL-P6

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。

13

一试 · 三. （本题满分 20 分） · 数论

（本题满分 20 分） 已知 a\htmlData{tutor-start=0,tutor-end=1}{a}，b\htmlData{tutor-start=0,tutor-end=1}{b} 均为正整数，且 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}，sin⁡θ=2aba2+b2\sin\htmlData{tutor-start=4,tutor-end=10}{\theta}\htmlData{tutor-start=10,tutor-end=11}{=}\frac{\htmlData{tutor-start=17,tutor-end=18}{2}\htmlData{tutor-start=18,tutor-end=19}{a}\htmlData{tutor-start=19,tutor-end=20}{b}}{\htmlData{tutor-start=22,tutor-end=23}{a}^{\htmlData{tutor-start=25,tutor-end=26}{2}}\htmlData{tutor-start=27,tutor-end=28}{+}\htmlData{tutor-start=28,tutor-end=29}{b}^{\htmlData{tutor-start=31,tutor-end=32}{2}}}，（其中 0<θ<π2\htmlData{tutor-start=0,tutor-end=1}{0}\htmlData{tutor-start=1,tutor-end=2}{<}\htmlData{tutor-start=2,tutor-end=8}{\theta}\htmlData{tutor-start=8,tutor-end=9}{<}\frac{\htmlData{tutor-start=15,tutor-end=18}{\pi}}{\htmlData{tutor-start=20,tutor-end=21}{2}}），An=(a2+b2)nsin⁡nθ\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}{(}\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=13,tutor-end=14}{b}^{\htmlData{tutor-start=16,tutor-end=17}{2}}\htmlData{tutor-start=18,tutor-end=19}{)}^{\htmlData{tutor-start=21,tutor-end=22}{n}}\sin \htmlData{tutor-start=28,tutor-end=29}{n}\htmlData{tutor-start=29,tutor-end=35}{\theta}. 求证：对于一切自然数 n\htmlData{tutor-start=0,tutor-end=1}{n}，An\htmlData{tutor-start=0,tutor-end=1}{A}_{\htmlData{tutor-start=3,tutor-end=4}{n}} 均为整数.

题解状态：标准答案与规范题解待补充

题目标签：FIRST-SOLUTION-P三

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。

14

一试 · 四. · 代数

n2\htmlData{tutor-start=0,tutor-end=1}{n}^{\htmlData{tutor-start=3,tutor-end=4}{2}} 个正数排成 n\htmlData{tutor-start=0,tutor-end=1}{n} 行 n\htmlData{tutor-start=0,tutor-end=1}{n} 列 [图示：矩阵 aij\htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{i}\htmlData{tutor-start=4,tutor-end=5}{j}}] 其中每一行的数成等差数列，每一列的数成等比数列，并且所有公比相等. 已知 a11=1\htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{1}\htmlData{tutor-start=4,tutor-end=5}{1}}\htmlData{tutor-start=6,tutor-end=7}{=}\htmlData{tutor-start=7,tutor-end=8}{1}，a12=18\htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{1}\htmlData{tutor-start=4,tutor-end=5}{2}}\htmlData{tutor-start=6,tutor-end=7}{=}\frac{\htmlData{tutor-start=13,tutor-end=14}{1}}{\htmlData{tutor-start=16,tutor-end=17}{8}}，a13=316\htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{1}\htmlData{tutor-start=4,tutor-end=5}{3}}\htmlData{tutor-start=6,tutor-end=7}{=}\frac{\htmlData{tutor-start=13,tutor-end=14}{3}}{\htmlData{tutor-start=16,tutor-end=17}{1}\htmlData{tutor-start=17,tutor-end=18}{6}}，求 a11+a22+⋯+ann\htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{1}\htmlData{tutor-start=4,tutor-end=5}{1}}\htmlData{tutor-start=6,tutor-end=7}{+}\htmlData{tutor-start=7,tutor-end=8}{a}_{\htmlData{tutor-start=10,tutor-end=11}{2}\htmlData{tutor-start=11,tutor-end=12}{2}}\htmlData{tutor-start=13,tutor-end=14}{+}\cdots\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=25,tutor-end=26}{n}}.

原卷图示 1
原卷图示 1原卷第 2 页 · qwen3.7-plus_layout_detection · 需复核

题解状态：标准答案与规范题解待补充

题目标签：FIRST-SOLUTION-P四

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。

15

一试 · 五. · 平面几何

设棱锥 M−ABCD\htmlData{tutor-start=0,tutor-end=1}{M}\htmlData{tutor-start=1,tutor-end=2}{-}\htmlData{tutor-start=2,tutor-end=3}{A}\htmlData{tutor-start=3,tutor-end=4}{B}\htmlData{tutor-start=4,tutor-end=5}{C}\htmlData{tutor-start=5,tutor-end=6}{D} 的底面为正方形，且 MA=MD\htmlData{tutor-start=0,tutor-end=1}{M}\htmlData{tutor-start=1,tutor-end=2}{A}\htmlData{tutor-start=2,tutor-end=3}{=}\htmlData{tutor-start=3,tutor-end=4}{M}\htmlData{tutor-start=4,tutor-end=5}{D}，MA⊥AB\htmlData{tutor-start=0,tutor-end=1}{M}\htmlData{tutor-start=1,tutor-end=2}{A}\htmlData{tutor-start=2,tutor-end=8}{\perp }\htmlData{tutor-start=8,tutor-end=9}{A}\htmlData{tutor-start=9,tutor-end=10}{B}. 如果 △AMD\htmlData{tutor-start=0,tutor-end=10}{\triangle }\htmlData{tutor-start=10,tutor-end=11}{A}\htmlData{tutor-start=11,tutor-end=12}{M}\htmlData{tutor-start=12,tutor-end=13}{D} 的面积为 1，试求能够放入这个棱锥的最大球的半径. [图示：棱锥 M−ABCD\htmlData{tutor-start=0,tutor-end=1}{M}\htmlData{tutor-start=1,tutor-end=2}{-}\htmlData{tutor-start=2,tutor-end=3}{A}\htmlData{tutor-start=3,tutor-end=4}{B}\htmlData{tutor-start=4,tutor-end=5}{C}\htmlData{tutor-start=5,tutor-end=6}{D}]

原卷图示 1
原卷图示 1原卷第 2 页 · qwen3.7-plus_layout_detection · 需复核

题解状态：标准答案与规范题解待补充

题目标签：FIRST-SOLUTION-P五

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。

16

二试/加试 · 第二试 · 平面几何

(本题满分 35 分) 四边形 ABCD\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{B}\htmlData{tutor-start=2,tutor-end=3}{C}\htmlData{tutor-start=3,tutor-end=4}{D} 内接于圆 O\htmlData{tutor-start=0,tutor-end=1}{O}，对角线 AC\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{C} 与 BD\htmlData{tutor-start=0,tutor-end=1}{B}\htmlData{tutor-start=1,tutor-end=2}{D} 相交于 P\htmlData{tutor-start=0,tutor-end=1}{P}，设三角形 ABP\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{B}\htmlData{tutor-start=2,tutor-end=3}{P}、BCP\htmlData{tutor-start=0,tutor-end=1}{B}\htmlData{tutor-start=1,tutor-end=2}{C}\htmlData{tutor-start=2,tutor-end=3}{P}、CDP\htmlData{tutor-start=0,tutor-end=1}{C}\htmlData{tutor-start=1,tutor-end=2}{D}\htmlData{tutor-start=2,tutor-end=3}{P} 和 DAP\htmlData{tutor-start=0,tutor-end=1}{D}\htmlData{tutor-start=1,tutor-end=2}{A}\htmlData{tutor-start=2,tutor-end=3}{P} 的外接圆圆心分别是 O1\htmlData{tutor-start=0,tutor-end=1}{O}_{\htmlData{tutor-start=3,tutor-end=4}{1}}、O2\htmlData{tutor-start=0,tutor-end=1}{O}_{\htmlData{tutor-start=3,tutor-end=4}{2}}、O3\htmlData{tutor-start=0,tutor-end=1}{O}_{\htmlData{tutor-start=3,tutor-end=4}{3}}、O4\htmlData{tutor-start=0,tutor-end=1}{O}_{\htmlData{tutor-start=3,tutor-end=4}{4}}. 求证 OP\htmlData{tutor-start=0,tutor-end=1}{O}\htmlData{tutor-start=1,tutor-end=2}{P}、O1O3\htmlData{tutor-start=0,tutor-end=1}{O}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{O}_{\htmlData{tutor-start=8,tutor-end=9}{3}}、O2O4\htmlData{tutor-start=0,tutor-end=1}{O}_{\htmlData{tutor-start=3,tutor-end=4}{2}}\htmlData{tutor-start=5,tutor-end=6}{O}_{\htmlData{tutor-start=8,tutor-end=9}{4}} 三直线共点.

原卷图示 1
原卷图示 1原卷第 3 页 · qwen3.7-plus_layout_detection · 需复核

题解状态：标准答案与规范题解待补充

题目标签：SECOND-GENERAL-PROBLEM-P一

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。

17

二试/加试 · 第二试 · 组合数学

(本题满分 35 分) 设 E={1,2,3,⋯⋯,200}\htmlData{tutor-start=0,tutor-end=1}{E}\htmlData{tutor-start=1,tutor-end=2}{=}\htmlData{tutor-start=2,tutor-end=4}{\{}\htmlData{tutor-start=4,tutor-end=5}{1}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{2}\htmlData{tutor-start=8,tutor-end=9}{,} \htmlData{tutor-start=10,tutor-end=11}{3}\htmlData{tutor-start=11,tutor-end=12}{,} \cdots\cdots\htmlData{tutor-start=25,tutor-end=26}{,} \htmlData{tutor-start=27,tutor-end=28}{2}\htmlData{tutor-start=28,tutor-end=29}{0}\htmlData{tutor-start=29,tutor-end=30}{0}\htmlData{tutor-start=30,tutor-end=32}{\}}， G={a1,a2,⋯⋯,a100}⫋E\htmlData{tutor-start=0,tutor-end=1}{G}\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=7,tutor-end=8}{1}}\htmlData{tutor-start=9,tutor-end=10}{,} \htmlData{tutor-start=11,tutor-end=12}{a}_{\htmlData{tutor-start=14,tutor-end=15}{2}}\htmlData{tutor-start=16,tutor-end=17}{,} \cdots\cdots\htmlData{tutor-start=30,tutor-end=31}{,} \htmlData{tutor-start=32,tutor-end=33}{a}_{\htmlData{tutor-start=35,tutor-end=36}{1}\htmlData{tutor-start=36,tutor-end=37}{0}\htmlData{tutor-start=37,tutor-end=38}{0}}\htmlData{tutor-start=39,tutor-end=41}{\}} \htmlData{tutor-start=42,tutor-end=54}{\subsetneqq }\htmlData{tutor-start=54,tutor-end=55}{E}. 且 G\htmlData{tutor-start=0,tutor-end=1}{G} 具有下列两条性质： (1) 对任何 1⩽i<j⩽100\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=14,tutor-end=15}{<} \htmlData{tutor-start=16,tutor-end=17}{j} \htmlData{tutor-start=18,tutor-end=28}{\leqslant }\htmlData{tutor-start=28,tutor-end=29}{1}\htmlData{tutor-start=29,tutor-end=30}{0}\htmlData{tutor-start=30,tutor-end=31}{0}，恒有 ai+aj≠201\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}} \neq \htmlData{tutor-start=17,tutor-end=18}{2}\htmlData{tutor-start=18,tutor-end=19}{0}\htmlData{tutor-start=19,tutor-end=20}{1}; (2) ∑i=1100ai=10080\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}{1}\htmlData{tutor-start=13,tutor-end=14}{0}\htmlData{tutor-start=14,tutor-end=15}{0}} \htmlData{tutor-start=17,tutor-end=18}{a}_{\htmlData{tutor-start=20,tutor-end=21}{i}}\htmlData{tutor-start=22,tutor-end=23}{=}\htmlData{tutor-start=23,tutor-end=24}{1}\htmlData{tutor-start=24,tutor-end=25}{0}\htmlData{tutor-start=25,tutor-end=26}{0}\htmlData{tutor-start=26,tutor-end=27}{8}\htmlData{tutor-start=27,tutor-end=28}{0}. 试证明：G\htmlData{tutor-start=0,tutor-end=1}{G} 中的奇数的个数是 4 的倍数. 且 G\htmlData{tutor-start=0,tutor-end=1}{G} 中所有数字的平方和为一个定数.

题解状态：标准答案与规范题解待补充

题目标签：SECOND-GENERAL-PROBLEM-P二

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。

18

二试/加试 · 第二试 · 数学竞赛/待细分

(本题满分 35 分) 某市有 n\htmlData{tutor-start=0,tutor-end=1}{n} 所中学第 i\htmlData{tutor-start=0,tutor-end=1}{i} 所中学派出 Ci\htmlData{tutor-start=0,tutor-end=1}{C}_{\htmlData{tutor-start=3,tutor-end=4}{i}} 名代表 (1⩽Ci⩽39\htmlData{tutor-start=0,tutor-end=1}{1} \htmlData{tutor-start=2,tutor-end=12}{\leqslant }\htmlData{tutor-start=12,tutor-end=13}{C}_{\htmlData{tutor-start=15,tutor-end=16}{i}} \htmlData{tutor-start=18,tutor-end=28}{\leqslant }\htmlData{tutor-start=28,tutor-end=29}{3}\htmlData{tutor-start=29,tutor-end=30}{9}, 1⩽i⩽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=14,tutor-end=24}{\leqslant }\htmlData{tutor-start=24,tutor-end=25}{n}) 来到体育馆观看球赛，全部学生总数为 ∑i=1nCi=1990\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}} \htmlData{tutor-start=15,tutor-end=16}{C}_{\htmlData{tutor-start=18,tutor-end=19}{i}}\htmlData{tutor-start=20,tutor-end=21}{=}\htmlData{tutor-start=21,tutor-end=22}{1}\htmlData{tutor-start=22,tutor-end=23}{9}\htmlData{tutor-start=23,tutor-end=24}{9}\htmlData{tutor-start=24,tutor-end=25}{0}. 看台上每一横排有 199 个座位，要求同一学校的学生必须坐在同一横排，问体育馆最少要安排多少横排才能够保证全部学生都能坐下.

题解状态：标准答案与规范题解待补充

题目标签：SECOND-GENERAL-PROBLEM-P三

解题过程

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