2009 年全国高中数学联赛 B 卷（学生版）

books/national_high_school_league_students/papers/competition-national-league-2009-b.pdf · HS-MATH-1024-v2.1-solution-aware

3 题0 个小问/题组

二试/加试（B 卷） · 加试 · 平面几何

如图， $\htmlData{tutor-start=0,tutor-end=1}{M}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{N}$ 分别为锐角三角形 $\htmlData{tutor-start=0,tutor-end=7}{\Delta }\htmlData{tutor-start=7,tutor-end=8}{A}\htmlData{tutor-start=8,tutor-end=9}{B}\htmlData{tutor-start=9,tutor-end=10}{C}\htmlData{tutor-start=10,tutor-end=11}{(}\htmlData{tutor-start=11,tutor-end=18}{\angle }\htmlData{tutor-start=18,tutor-end=19}{A} \htmlData{tutor-start=20,tutor-end=21}{<} \htmlData{tutor-start=22,tutor-end=29}{\angle }\htmlData{tutor-start=29,tutor-end=30}{B}\htmlData{tutor-start=30,tutor-end=31}{)}$ 的外接圆 $\htmlData{tutor-start=0,tutor-end=6}{\Gamma}$ 上弧 $\widehat{\htmlData{tutor-start=9,tutor-end=10}{B}\htmlData{tutor-start=10,tutor-end=11}{C}}\htmlData{tutor-start=12,tutor-end=13}{,} \widehat{\htmlData{tutor-start=23,tutor-end=24}{A}\htmlData{tutor-start=24,tutor-end=25}{C}}$ 的中点。过点 $\htmlData{tutor-start=0,tutor-end=1}{C}$ 作 $\htmlData{tutor-start=0,tutor-end=1}{P}\htmlData{tutor-start=1,tutor-end=2}{C} \htmlData{tutor-start=3,tutor-end=4}{/}\htmlData{tutor-start=4,tutor-end=5}{/} \htmlData{tutor-start=6,tutor-end=7}{M}\htmlData{tutor-start=7,tutor-end=8}{N}$ 交圆 $\htmlData{tutor-start=0,tutor-end=6}{\Gamma}$ 于 $\htmlData{tutor-start=0,tutor-end=1}{P}$ 点， $\htmlData{tutor-start=0,tutor-end=1}{I}$ 为 $\htmlData{tutor-start=0,tutor-end=7}{\Delta }\htmlData{tutor-start=7,tutor-end=8}{A}\htmlData{tutor-start=8,tutor-end=9}{B}\htmlData{tutor-start=9,tutor-end=10}{C}$ 的内心，连接 $\htmlData{tutor-start=0,tutor-end=1}{P}\htmlData{tutor-start=1,tutor-end=2}{I}$ 并延长交圆 $\htmlData{tutor-start=0,tutor-end=6}{\Gamma}$ 于 $\htmlData{tutor-start=0,tutor-end=1}{T}$ 。 (I) 求证： $\htmlData{tutor-start=0,tutor-end=1}{M}\htmlData{tutor-start=1,tutor-end=2}{P} \htmlData{tutor-start=3,tutor-end=9}{\cdot }\htmlData{tutor-start=9,tutor-end=10}{M}\htmlData{tutor-start=10,tutor-end=11}{T} \htmlData{tutor-start=12,tutor-end=13}{=} \htmlData{tutor-start=14,tutor-end=15}{N}\htmlData{tutor-start=15,tutor-end=16}{P} \htmlData{tutor-start=17,tutor-end=23}{\cdot }\htmlData{tutor-start=23,tutor-end=24}{N}\htmlData{tutor-start=24,tutor-end=25}{T}$ ； (II) 在弧 $\widehat{\htmlData{tutor-start=9,tutor-end=10}{A}\htmlData{tutor-start=10,tutor-end=11}{B}}$ (不含点 $\htmlData{tutor-start=0,tutor-end=1}{C}$ ) 上任取一点 $\htmlData{tutor-start=0,tutor-end=1}{Q}$ ( $\htmlData{tutor-start=0,tutor-end=1}{Q} \neq \htmlData{tutor-start=7,tutor-end=8}{A}\htmlData{tutor-start=8,tutor-end=9}{,} \htmlData{tutor-start=10,tutor-end=11}{T}\htmlData{tutor-start=11,tutor-end=12}{,} \htmlData{tutor-start=13,tutor-end=14}{B}$ )，记 $\htmlData{tutor-start=0,tutor-end=7}{\Delta }\htmlData{tutor-start=7,tutor-end=8}{A}\htmlData{tutor-start=8,tutor-end=9}{Q}\htmlData{tutor-start=9,tutor-end=10}{C}\htmlData{tutor-start=10,tutor-end=11}{,} \htmlData{tutor-start=12,tutor-end=19}{\Delta }\htmlData{tutor-start=19,tutor-end=20}{Q}\htmlData{tutor-start=20,tutor-end=21}{C}\htmlData{tutor-start=21,tutor-end=22}{B}$ 的内心分别为 $\htmlData{tutor-start=0,tutor-end=1}{I}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{I}_{\htmlData{tutor-start=10,tutor-end=11}{2}}$ ，

**原卷图示 1**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：SECOND-B-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。

二试/加试（B 卷） · 加试 · 数学竞赛/待细分

设 $\htmlData{tutor-start=0,tutor-end=1}{0} \htmlData{tutor-start=2,tutor-end=3}{<} \htmlData{tutor-start=4,tutor-end=11}{\alpha }\htmlData{tutor-start=11,tutor-end=12}{<} \frac{\htmlData{tutor-start=19,tutor-end=22}{\pi}}{\htmlData{tutor-start=24,tutor-end=25}{2}}$ ， $\sin \htmlData{tutor-start=5,tutor-end=11}{\alpha}\htmlData{tutor-start=11,tutor-end=12}{,} \cos \htmlData{tutor-start=18,tutor-end=24}{\alpha}$ 均为有理数。求证：存在 $\htmlData{tutor-start=0,tutor-end=1}{0} \htmlData{tutor-start=2,tutor-end=3}{<} \htmlData{tutor-start=4,tutor-end=10}{\alpha}_{\htmlData{tutor-start=12,tutor-end=13}{1}}\htmlData{tutor-start=14,tutor-end=15}{,} \htmlData{tutor-start=16,tutor-end=22}{\alpha}_{\htmlData{tutor-start=24,tutor-end=25}{2}} \htmlData{tutor-start=27,tutor-end=28}{<} \frac{\htmlData{tutor-start=35,tutor-end=38}{\pi}}{\htmlData{tutor-start=40,tutor-end=41}{2}}$ 满足 (I) $\htmlData{tutor-start=0,tutor-end=6}{\alpha}_{\htmlData{tutor-start=8,tutor-end=9}{1}} \htmlData{tutor-start=11,tutor-end=12}{+} \htmlData{tutor-start=13,tutor-end=19}{\alpha}_{\htmlData{tutor-start=21,tutor-end=22}{2}} \htmlData{tutor-start=24,tutor-end=25}{=} \htmlData{tutor-start=26,tutor-end=32}{\alpha}$ ; (II) $\sin \htmlData{tutor-start=5,tutor-end=11}{\alpha}_{\htmlData{tutor-start=13,tutor-end=14}{1}}\htmlData{tutor-start=15,tutor-end=16}{,} \sin \htmlData{tutor-start=22,tutor-end=28}{\alpha}_{\htmlData{tutor-start=30,tutor-end=31}{2}}\htmlData{tutor-start=32,tutor-end=33}{,} \cos \htmlData{tutor-start=39,tutor-end=45}{\alpha}_{\htmlData{tutor-start=47,tutor-end=48}{1}}\htmlData{tutor-start=49,tutor-end=50}{,} \cos \htmlData{tutor-start=56,tutor-end=62}{\alpha}_{\htmlData{tutor-start=64,tutor-end=65}{2}}$ 都是有理数。

题解状态：标准答案与规范题解待补充

题目标签：SECOND-B-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。

二试/加试（B 卷） · 二试 · 数论

已知无穷数列 $\htmlData{tutor-start=0,tutor-end=2}{\{}\htmlData{tutor-start=2,tutor-end=3}{a}_{\htmlData{tutor-start=5,tutor-end=6}{n}}\htmlData{tutor-start=7,tutor-end=9}{\}}$ 满足 $\htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{=}\htmlData{tutor-start=6,tutor-end=7}{1}\htmlData{tutor-start=7,tutor-end=8}{,} \htmlData{tutor-start=9,tutor-end=10}{a}_{\htmlData{tutor-start=12,tutor-end=13}{2}}\htmlData{tutor-start=14,tutor-end=15}{=}\htmlData{tutor-start=15,tutor-end=16}{1}\htmlData{tutor-start=16,tutor-end=17}{,} \htmlData{tutor-start=18,tutor-end=19}{a}_{\htmlData{tutor-start=21,tutor-end=22}{3}}\htmlData{tutor-start=23,tutor-end=24}{=}\htmlData{tutor-start=24,tutor-end=25}{3}\htmlData{tutor-start=25,tutor-end=26}{,} \htmlData{tutor-start=27,tutor-end=28}{a}_{\htmlData{tutor-start=30,tutor-end=31}{n}}\htmlData{tutor-start=32,tutor-end=33}{=}\htmlData{tutor-start=33,tutor-end=34}{a}_{\htmlData{tutor-start=36,tutor-end=37}{n}\htmlData{tutor-start=37,tutor-end=38}{-}\htmlData{tutor-start=38,tutor-end=39}{1}}\htmlData{tutor-start=40,tutor-end=41}{+}\htmlData{tutor-start=41,tutor-end=42}{2}\htmlData{tutor-start=42,tutor-end=43}{a}_{\htmlData{tutor-start=45,tutor-end=46}{n}\htmlData{tutor-start=46,tutor-end=47}{-}\htmlData{tutor-start=47,tutor-end=48}{2}}\htmlData{tutor-start=49,tutor-end=50}{+}\htmlData{tutor-start=50,tutor-end=51}{a}_{\htmlData{tutor-start=53,tutor-end=54}{n}\htmlData{tutor-start=54,tutor-end=55}{-}\htmlData{tutor-start=55,tutor-end=56}{3}}\htmlData{tutor-start=57,tutor-end=58}{,} \htmlData{tutor-start=59,tutor-end=60}{n}\htmlData{tutor-start=60,tutor-end=61}{=}\htmlData{tutor-start=61,tutor-end=62}{4}\htmlData{tutor-start=62,tutor-end=63}{,}\htmlData{tutor-start=63,tutor-end=64}{5}\htmlData{tutor-start=64,tutor-end=65}{,}\dots$ . 求证：对任何正常数 $\htmlData{tutor-start=0,tutor-end=1}{m}$ ，存在某个正整数 $\htmlData{tutor-start=0,tutor-end=1}{n}$ 使得 $\htmlData{tutor-start=0,tutor-end=1}{m} \htmlData{tutor-start=2,tutor-end=7}{\mid }\htmlData{tutor-start=7,tutor-end=8}{a}_{\htmlData{tutor-start=10,tutor-end=11}{n}}$ .

题解状态：标准答案与规范题解待补充

题目标签：SECOND-B-PROBLEM-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。