2009 年全国高中数学联赛 A 卷（学生版）

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

15 题0 个小问/题组

一试（A 卷） · 一试 · 代数

若函数 $\htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{x}\htmlData{tutor-start=3,tutor-end=4}{)}\htmlData{tutor-start=4,tutor-end=5}{=}\frac{\htmlData{tutor-start=11,tutor-end=12}{x}}{\sqrt{\htmlData{tutor-start=20,tutor-end=21}{1}\htmlData{tutor-start=21,tutor-end=22}{+}\htmlData{tutor-start=22,tutor-end=23}{x}^{\htmlData{tutor-start=25,tutor-end=26}{2}}}}$ 且 $\htmlData{tutor-start=0,tutor-end=1}{f}^{\htmlData{tutor-start=3,tutor-end=4}{(}\htmlData{tutor-start=4,tutor-end=5}{n}\htmlData{tutor-start=5,tutor-end=6}{)}}\htmlData{tutor-start=7,tutor-end=8}{(}\htmlData{tutor-start=8,tutor-end=9}{x}\htmlData{tutor-start=9,tutor-end=10}{)}\htmlData{tutor-start=10,tutor-end=11}{=}\htmlData{tutor-start=11,tutor-end=12}{f}\htmlData{tutor-start=12,tutor-end=13}{[}\underbrace{\htmlData{tutor-start=25,tutor-end=26}{f}\htmlData{tutor-start=26,tutor-end=27}{[}\htmlData{tutor-start=27,tutor-end=28}{f}\cdots \htmlData{tutor-start=35,tutor-end=36}{f}\htmlData{tutor-start=36,tutor-end=37}{(}\htmlData{tutor-start=37,tutor-end=38}{x}\htmlData{tutor-start=38,tutor-end=39}{)}\htmlData{tutor-start=39,tutor-end=40}{]}}_{\htmlData{tutor-start=43,tutor-end=44}{n}}\htmlData{tutor-start=45,tutor-end=46}{]}$ ，则 $\htmlData{tutor-start=0,tutor-end=1}{f}^{\htmlData{tutor-start=3,tutor-end=4}{(}\htmlData{tutor-start=4,tutor-end=5}{9}\htmlData{tutor-start=5,tutor-end=6}{9}\htmlData{tutor-start=6,tutor-end=7}{)}}\htmlData{tutor-start=8,tutor-end=9}{(}\htmlData{tutor-start=9,tutor-end=10}{1}\htmlData{tutor-start=10,tutor-end=11}{)}\htmlData{tutor-start=11,tutor-end=12}{=}$ ______.

题解状态：标准答案与规范题解待补充

题目标签：FIRST-A-PROBLEM-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。

一试（A 卷） · 一试 · 平面几何

已知直线 $\htmlData{tutor-start=0,tutor-end=1}{L}\htmlData{tutor-start=1,tutor-end=2}{:}\htmlData{tutor-start=2,tutor-end=3}{x}\htmlData{tutor-start=3,tutor-end=4}{+}\htmlData{tutor-start=4,tutor-end=5}{y}\htmlData{tutor-start=5,tutor-end=6}{-}\htmlData{tutor-start=6,tutor-end=7}{9}\htmlData{tutor-start=7,tutor-end=8}{=}\htmlData{tutor-start=8,tutor-end=9}{0}$ 和圆 $\htmlData{tutor-start=0,tutor-end=1}{M}\htmlData{tutor-start=1,tutor-end=2}{:}\htmlData{tutor-start=2,tutor-end=3}{2}\htmlData{tutor-start=3,tutor-end=4}{x}^{\htmlData{tutor-start=6,tutor-end=7}{2}}\htmlData{tutor-start=8,tutor-end=9}{+}\htmlData{tutor-start=9,tutor-end=10}{2}\htmlData{tutor-start=10,tutor-end=11}{y}^{\htmlData{tutor-start=13,tutor-end=14}{2}}\htmlData{tutor-start=15,tutor-end=16}{-}\htmlData{tutor-start=16,tutor-end=17}{8}\htmlData{tutor-start=17,tutor-end=18}{x}\htmlData{tutor-start=18,tutor-end=19}{-}\htmlData{tutor-start=19,tutor-end=20}{8}\htmlData{tutor-start=20,tutor-end=21}{y}\htmlData{tutor-start=21,tutor-end=22}{-}\htmlData{tutor-start=22,tutor-end=23}{1}\htmlData{tutor-start=23,tutor-end=24}{=}\htmlData{tutor-start=24,tutor-end=25}{0}$ ，点 $\htmlData{tutor-start=0,tutor-end=1}{A}$ 在直线 $\htmlData{tutor-start=0,tutor-end=1}{L}$ 上， $\htmlData{tutor-start=0,tutor-end=1}{B}$ ， $\htmlData{tutor-start=0,tutor-end=1}{C}$ 为圆 $\htmlData{tutor-start=0,tutor-end=1}{M}$ 上两点，在 $\htmlData{tutor-start=0,tutor-end=10}{\triangle }\htmlData{tutor-start=10,tutor-end=11}{A}\htmlData{tutor-start=11,tutor-end=12}{B}\htmlData{tutor-start=12,tutor-end=13}{C}$ 中， $\angle BAC=45^\circ$ ， $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{B}$ 过圆心 $\htmlData{tutor-start=0,tutor-end=1}{M}$ ，则点 $\htmlData{tutor-start=0,tutor-end=1}{A}$ 横坐标范围为______.

题解状态：标准答案与规范题解待补充

题目标签：FIRST-A-PROBLEM-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。

一试（A 卷） · 一试 · 代数

在坐标平面上有两个区域 $\htmlData{tutor-start=0,tutor-end=1}{M}$ 和 $\htmlData{tutor-start=0,tutor-end=1}{N}$ ， $\htmlData{tutor-start=0,tutor-end=1}{M}$ 为 $\begin{cases} \htmlData{tutor-start=14,tutor-end=15}{y} \htmlData{tutor-start=16,tutor-end=20}{\ge }\htmlData{tutor-start=20,tutor-end=21}{0} \\ \htmlData{tutor-start=25,tutor-end=26}{y} \htmlData{tutor-start=27,tutor-end=31}{\le }\htmlData{tutor-start=31,tutor-end=32}{x} \\ \htmlData{tutor-start=36,tutor-end=37}{y} \htmlData{tutor-start=38,tutor-end=42}{\le }\htmlData{tutor-start=42,tutor-end=43}{2}\htmlData{tutor-start=43,tutor-end=44}{-}\htmlData{tutor-start=44,tutor-end=45}{x} \end{cases}$ ， $\htmlData{tutor-start=0,tutor-end=1}{N}$ 是随 $\htmlData{tutor-start=0,tutor-end=1}{t}$ 变化的区域，它由不等式 $\htmlData{tutor-start=0,tutor-end=1}{t} \htmlData{tutor-start=2,tutor-end=6}{\le }\htmlData{tutor-start=6,tutor-end=7}{x} \htmlData{tutor-start=8,tutor-end=12}{\le }\htmlData{tutor-start=12,tutor-end=13}{t}\htmlData{tutor-start=13,tutor-end=14}{+}\htmlData{tutor-start=14,tutor-end=15}{1}$ 所确定， $\htmlData{tutor-start=0,tutor-end=1}{t}$ 的取值范围是 $\htmlData{tutor-start=0,tutor-end=1}{0} \htmlData{tutor-start=2,tutor-end=6}{\le }\htmlData{tutor-start=6,tutor-end=7}{t} \htmlData{tutor-start=8,tutor-end=12}{\le }\htmlData{tutor-start=12,tutor-end=13}{1}$ ，则 $\htmlData{tutor-start=0,tutor-end=1}{M}$ 和 $\htmlData{tutor-start=0,tutor-end=1}{N}$ 的公共面积是函数 $\htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{t}\htmlData{tutor-start=3,tutor-end=4}{)}\htmlData{tutor-start=4,tutor-end=5}{=}$ ______.

题解状态：标准答案与规范题解待补充

题目标签：FIRST-A-PROBLEM-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。

一试（A 卷） · 一试 · 数论

使不等式 $\frac{1}{n+1}+\frac{1}{n+2}+\cdots+\frac{1}{2n+1} < a-2007\frac{1}{3}$ 对一切正整数 $\htmlData{tutor-start=0,tutor-end=1}{n}$ 都成立的最小正整数 $\htmlData{tutor-start=0,tutor-end=1}{a}$ 的值为______.

题解状态：标准答案与规范题解待补充

题目标签：FIRST-A-PROBLEM-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。

一试（A 卷） · 一试 · 平面几何

椭圆 $\frac{\htmlData{tutor-start=6,tutor-end=7}{x}^{\htmlData{tutor-start=9,tutor-end=10}{2}}}{\htmlData{tutor-start=13,tutor-end=14}{a}^{\htmlData{tutor-start=16,tutor-end=17}{2}}}\htmlData{tutor-start=19,tutor-end=20}{+}\frac{\htmlData{tutor-start=26,tutor-end=27}{y}^{\htmlData{tutor-start=29,tutor-end=30}{2}}}{\htmlData{tutor-start=33,tutor-end=34}{b}^{\htmlData{tutor-start=36,tutor-end=37}{2}}}\htmlData{tutor-start=39,tutor-end=40}{=}\htmlData{tutor-start=40,tutor-end=41}{1}$ $\htmlData{tutor-start=0,tutor-end=1}{(}\htmlData{tutor-start=1,tutor-end=2}{a}\htmlData{tutor-start=2,tutor-end=3}{>}\htmlData{tutor-start=3,tutor-end=4}{b}\htmlData{tutor-start=4,tutor-end=5}{>}\htmlData{tutor-start=5,tutor-end=6}{0}\htmlData{tutor-start=6,tutor-end=7}{)}$ 上任意两点 $\htmlData{tutor-start=0,tutor-end=1}{P}$ ， $\htmlData{tutor-start=0,tutor-end=1}{Q}$ ，若 $\htmlData{tutor-start=0,tutor-end=1}{O}\htmlData{tutor-start=1,tutor-end=2}{P} \htmlData{tutor-start=3,tutor-end=9}{\perp }\htmlData{tutor-start=9,tutor-end=10}{O}\htmlData{tutor-start=10,tutor-end=11}{Q}$ ，则乘积 $\htmlData{tutor-start=0,tutor-end=1}{|}\htmlData{tutor-start=1,tutor-end=2}{O}\htmlData{tutor-start=2,tutor-end=3}{P}\htmlData{tutor-start=3,tutor-end=4}{|}\htmlData{tutor-start=4,tutor-end=9}{\cdot}\htmlData{tutor-start=9,tutor-end=10}{|}\htmlData{tutor-start=10,tutor-end=11}{O}\htmlData{tutor-start=11,tutor-end=12}{Q}\htmlData{tutor-start=12,tutor-end=13}{|}$ 的最小值为______.

题解状态：标准答案与规范题解待补充

题目标签：FIRST-A-PROBLEM-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。

一试（A 卷） · 一试 · 数学竞赛/待细分

若方程 $\lg \htmlData{tutor-start=4,tutor-end=5}{k}\htmlData{tutor-start=5,tutor-end=6}{x}\htmlData{tutor-start=6,tutor-end=7}{=}\htmlData{tutor-start=7,tutor-end=8}{2}\lg\htmlData{tutor-start=11,tutor-end=12}{(}\htmlData{tutor-start=12,tutor-end=13}{x}\htmlData{tutor-start=13,tutor-end=14}{+}\htmlData{tutor-start=14,tutor-end=15}{1}\htmlData{tutor-start=15,tutor-end=16}{)}$ 仅有一个实根，那么 $\htmlData{tutor-start=0,tutor-end=1}{k}$ 的取值范围是______.

题解状态：标准答案与规范题解待补充

题目标签：FIRST-A-PROBLEM-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。

一试（A 卷） · 一试 · 数论

一个由若干行数字组成的数表，从第二行起每一行中的数字均等于其肩上的两个数之和，最后一行仅有一个数，第一行是前 100 个正整数按从小到大排成的行，则最后一行的数是______（可以用指数表示）

题解状态：标准答案与规范题解待补充

题目标签：FIRST-A-PROBLEM-P7

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。

一试（A 卷） · 一试 · 数学竞赛/待细分

某车站每天 8:00~9:00，9:00~10:00 都恰有一辆客车到站，但到站的时刻是随机的，且两者到站的时间是相互独立的，其规律为 \begin{tabular}{|c|c|c|c|} \hline 到站时刻 & 8:10, 9:10 & 8:30, 9:30 & 8:50, 9:50 \\ \hline 概率 & $\frac{\htmlData{tutor-start=6,tutor-end=7}{1}}{\htmlData{tutor-start=9,tutor-end=10}{6}}$ & $\frac{\htmlData{tutor-start=6,tutor-end=7}{1}}{\htmlData{tutor-start=9,tutor-end=10}{2}}$ & $\frac{\htmlData{tutor-start=6,tutor-end=7}{1}}{\htmlData{tutor-start=9,tutor-end=10}{3}}$ \\ \hline \end{tabular} 一旅客 8:20 到车站，则它候车时间的数学期望为______（精确到分）.

**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：FIRST-A-PROBLEM-P8

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。

一试（A 卷） · 一试 · 平面几何

（本小题满分 14 分）设直线 $\htmlData{tutor-start=0,tutor-end=1}{l}\htmlData{tutor-start=1,tutor-end=2}{:}\htmlData{tutor-start=2,tutor-end=3}{y}\htmlData{tutor-start=3,tutor-end=4}{=}\htmlData{tutor-start=4,tutor-end=5}{k}\htmlData{tutor-start=5,tutor-end=6}{x}\htmlData{tutor-start=6,tutor-end=7}{+}\htmlData{tutor-start=7,tutor-end=8}{m}$ （其中 $\htmlData{tutor-start=0,tutor-end=1}{k}$ ， $\htmlData{tutor-start=0,tutor-end=1}{m}$ 为整数）与椭圆 $\frac{\htmlData{tutor-start=6,tutor-end=7}{x}^{\htmlData{tutor-start=9,tutor-end=10}{2}}}{\htmlData{tutor-start=13,tutor-end=14}{1}\htmlData{tutor-start=14,tutor-end=15}{6}}\htmlData{tutor-start=16,tutor-end=17}{+}\frac{\htmlData{tutor-start=23,tutor-end=24}{y}^{\htmlData{tutor-start=26,tutor-end=27}{2}}}{\htmlData{tutor-start=30,tutor-end=31}{1}\htmlData{tutor-start=31,tutor-end=32}{2}}\htmlData{tutor-start=33,tutor-end=34}{=}\htmlData{tutor-start=34,tutor-end=35}{1}$ 交于不同两点 $\htmlData{tutor-start=0,tutor-end=1}{A}$ ， $\htmlData{tutor-start=0,tutor-end=1}{B}$ ，与双曲线 $\frac{\htmlData{tutor-start=6,tutor-end=7}{x}^{\htmlData{tutor-start=9,tutor-end=10}{2}}}{\htmlData{tutor-start=13,tutor-end=14}{4}}\htmlData{tutor-start=15,tutor-end=16}{-}\frac{\htmlData{tutor-start=22,tutor-end=23}{y}^{\htmlData{tutor-start=25,tutor-end=26}{2}}}{\htmlData{tutor-start=29,tutor-end=30}{1}\htmlData{tutor-start=30,tutor-end=31}{2}}\htmlData{tutor-start=32,tutor-end=33}{=}\htmlData{tutor-start=33,tutor-end=34}{1}$ 交于不同两点 $\htmlData{tutor-start=0,tutor-end=1}{C}$ ， $\htmlData{tutor-start=0,tutor-end=1}{D}$ ，问是否存在直线 $\htmlData{tutor-start=0,tutor-end=1}{l}$ ，使得向量 $\overrightarrow{\htmlData{tutor-start=16,tutor-end=17}{A}\htmlData{tutor-start=17,tutor-end=18}{C}}\htmlData{tutor-start=19,tutor-end=20}{+}\overrightarrow{\htmlData{tutor-start=36,tutor-end=37}{B}\htmlData{tutor-start=37,tutor-end=38}{D}}\htmlData{tutor-start=39,tutor-end=40}{=}\htmlData{tutor-start=40,tutor-end=41}{0}$ ，若存在，指出这样的直线有多少条？若不存在，请说明理由.

题解状态：标准答案与规范题解待补充

题目标签：FIRST-A-PROBLEM-P9

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。

一试（A 卷） · 一试 · 代数

（本小题满分 15 分）已知 $\htmlData{tutor-start=0,tutor-end=1}{p}$ ， $\htmlData{tutor-start=0,tutor-end=1}{q}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{q} \neq \htmlData{tutor-start=9,tutor-end=10}{0}\htmlData{tutor-start=10,tutor-end=11}{)}$ 是实数，方程 $\htmlData{tutor-start=0,tutor-end=1}{x}^{\htmlData{tutor-start=3,tutor-end=4}{2}} \htmlData{tutor-start=6,tutor-end=7}{-} \htmlData{tutor-start=8,tutor-end=9}{p}\htmlData{tutor-start=9,tutor-end=10}{x} \htmlData{tutor-start=11,tutor-end=12}{+} \htmlData{tutor-start=13,tutor-end=14}{q} \htmlData{tutor-start=15,tutor-end=16}{=} \htmlData{tutor-start=17,tutor-end=18}{0}$ 有两个实根 $\htmlData{tutor-start=0,tutor-end=6}{\alpha}$ ， $\htmlData{tutor-start=0,tutor-end=5}{\beta}$ ，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 $\htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{1}} \htmlData{tutor-start=6,tutor-end=7}{=} \htmlData{tutor-start=8,tutor-end=9}{p}$ ， $\htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{2}} \htmlData{tutor-start=6,tutor-end=7}{=} \htmlData{tutor-start=8,tutor-end=9}{p}^{\htmlData{tutor-start=11,tutor-end=12}{2}} \htmlData{tutor-start=14,tutor-end=15}{-} \htmlData{tutor-start=16,tutor-end=17}{q}$ ， $\htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{n}} \htmlData{tutor-start=6,tutor-end=7}{=} \htmlData{tutor-start=8,tutor-end=9}{p}\htmlData{tutor-start=9,tutor-end=10}{a}_{\htmlData{tutor-start=12,tutor-end=13}{n}\htmlData{tutor-start=13,tutor-end=14}{-}\htmlData{tutor-start=14,tutor-end=15}{1}} \htmlData{tutor-start=17,tutor-end=18}{-} \htmlData{tutor-start=19,tutor-end=20}{q}\htmlData{tutor-start=20,tutor-end=21}{a}_{\htmlData{tutor-start=23,tutor-end=24}{n}\htmlData{tutor-start=24,tutor-end=25}{-}\htmlData{tutor-start=25,tutor-end=26}{2}} \htmlData{tutor-start=28,tutor-end=29}{(}\htmlData{tutor-start=29,tutor-end=30}{n}\htmlData{tutor-start=30,tutor-end=31}{=}\htmlData{tutor-start=31,tutor-end=32}{3}\htmlData{tutor-start=32,tutor-end=33}{,} \htmlData{tutor-start=34,tutor-end=35}{4}\htmlData{tutor-start=35,tutor-end=36}{,} \cdots\htmlData{tutor-start=43,tutor-end=44}{)}$ （I）求数列 $\htmlData{tutor-start=0,tutor-end=2}{\{}\htmlData{tutor-start=2,tutor-end=3}{a}_{\htmlData{tutor-start=5,tutor-end=6}{n}}\htmlData{tutor-start=7,tutor-end=9}{\}}$ 的通项公式（用 $\htmlData{tutor-start=0,tutor-end=6}{\alpha}$ ， $\htmlData{tutor-start=0,tutor-end=5}{\beta}$ 表示）；（II）若 $\htmlData{tutor-start=0,tutor-end=1}{p}\htmlData{tutor-start=1,tutor-end=2}{=}\htmlData{tutor-start=2,tutor-end=3}{1}$ ， $\htmlData{tutor-start=0,tutor-end=1}{q}\htmlData{tutor-start=1,tutor-end=2}{=}\frac{\htmlData{tutor-start=8,tutor-end=9}{1}}{\htmlData{tutor-start=11,tutor-end=12}{4}}$ ，求 $\htmlData{tutor-start=0,tutor-end=2}{\{}\htmlData{tutor-start=2,tutor-end=3}{a}_{\htmlData{tutor-start=5,tutor-end=6}{n}}\htmlData{tutor-start=7,tutor-end=9}{\}}$ 的前 $\htmlData{tutor-start=0,tutor-end=1}{n}$ 项和。

题解状态：标准答案与规范题解待补充

题目标签：FIRST-A-PROBLEM-P10

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。

一试（A 卷） · 一试 · 代数

（本小题满分 15 分）求函数 $\htmlData{tutor-start=0,tutor-end=1}{y}\htmlData{tutor-start=1,tutor-end=2}{=}\sqrt{\htmlData{tutor-start=8,tutor-end=9}{x}\htmlData{tutor-start=9,tutor-end=10}{+}\htmlData{tutor-start=10,tutor-end=11}{2}\htmlData{tutor-start=11,tutor-end=12}{7}}\htmlData{tutor-start=13,tutor-end=14}{+}\sqrt{\htmlData{tutor-start=20,tutor-end=21}{1}\htmlData{tutor-start=21,tutor-end=22}{3}\htmlData{tutor-start=22,tutor-end=23}{-}\htmlData{tutor-start=23,tutor-end=24}{x}}\htmlData{tutor-start=25,tutor-end=26}{+}\sqrt{\htmlData{tutor-start=32,tutor-end=33}{x}}$ 的最大和最小值。

题解状态：标准答案与规范题解待补充

题目标签：FIRST-A-PROBLEM-P11

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（A 卷） · 二试 · 平面几何

（本题满分 50 分）如图， $\htmlData{tutor-start=0,tutor-end=1}{M}$ ， $\htmlData{tutor-start=0,tutor-end=1}{N}$ 分别为锐角三角形 $\htmlData{tutor-start=0,tutor-end=10}{\triangle }\htmlData{tutor-start=10,tutor-end=11}{A}\htmlData{tutor-start=11,tutor-end=12}{B}\htmlData{tutor-start=12,tutor-end=13}{C}$ （ $\htmlData{tutor-start=0,tutor-end=7}{\angle }\htmlData{tutor-start=7,tutor-end=8}{A} \htmlData{tutor-start=9,tutor-end=10}{<} \htmlData{tutor-start=11,tutor-end=18}{\angle }\htmlData{tutor-start=18,tutor-end=19}{B}$ ）的外接圆 $\htmlData{tutor-start=0,tutor-end=6}{\Gamma}$ 上弧 $\widehat{\htmlData{tutor-start=9,tutor-end=10}{B}\htmlData{tutor-start=10,tutor-end=11}{C}}$ 、 $\widehat{\htmlData{tutor-start=9,tutor-end=10}{A}\htmlData{tutor-start=10,tutor-end=11}{C}}$ 的中点。过点 $\htmlData{tutor-start=0,tutor-end=1}{C}$ 作 $\htmlData{tutor-start=0,tutor-end=1}{P}\htmlData{tutor-start=1,tutor-end=2}{C} \htmlData{tutor-start=3,tutor-end=13}{\parallel }\htmlData{tutor-start=13,tutor-end=14}{M}\htmlData{tutor-start=14,tutor-end=15}{N}$ 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 $\htmlData{tutor-start=0,tutor-end=10}{\triangle }\htmlData{tutor-start=10,tutor-end=11}{A}\htmlData{tutor-start=11,tutor-end=12}{B}\htmlData{tutor-start=12,tutor-end=13}{C}$ 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 $\htmlData{tutor-start=0,tutor-end=1}{T}$ 。 (1)求证： $\htmlData{tutor-start=0,tutor-end=1}{M}\htmlData{tutor-start=1,tutor-end=2}{P} \htmlData{tutor-start=3,tutor-end=9}{\cdot }\htmlData{tutor-start=9,tutor-end=10}{M}\htmlData{tutor-start=10,tutor-end=11}{T} \htmlData{tutor-start=12,tutor-end=13}{=} \htmlData{tutor-start=14,tutor-end=15}{N}\htmlData{tutor-start=15,tutor-end=16}{P} \htmlData{tutor-start=17,tutor-end=23}{\cdot }\htmlData{tutor-start=23,tutor-end=24}{N}\htmlData{tutor-start=24,tutor-end=25}{T}$ ； (2)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 $\htmlData{tutor-start=0,tutor-end=1}{Q}$ （ $\htmlData{tutor-start=0,tutor-end=1}{Q} \neq \htmlData{tutor-start=7,tutor-end=8}{A}$ ， $\htmlData{tutor-start=0,tutor-end=1}{T}$ ， $\htmlData{tutor-start=0,tutor-end=1}{B}$ ），记 $\htmlData{tutor-start=0,tutor-end=10}{\triangle }\htmlData{tutor-start=10,tutor-end=11}{A}\htmlData{tutor-start=11,tutor-end=12}{Q}\htmlData{tutor-start=12,tutor-end=13}{C}$ ， $\htmlData{tutor-start=0,tutor-end=10}{\triangle }\htmlData{tutor-start=10,tutor-end=11}{Q}\htmlData{tutor-start=11,tutor-end=12}{C}\htmlData{tutor-start=12,tutor-end=13}{B}$ 的内心分别为 $\htmlData{tutor-start=0,tutor-end=1}{I}_{\htmlData{tutor-start=3,tutor-end=4}{1}}$ ， $\htmlData{tutor-start=0,tutor-end=1}{I}_{\htmlData{tutor-start=3,tutor-end=4}{2}}$ ，求证： $\htmlData{tutor-start=0,tutor-end=1}{Q}$ ， $\htmlData{tutor-start=0,tutor-end=1}{I}_{\htmlData{tutor-start=3,tutor-end=4}{1}}$ ， $\htmlData{tutor-start=0,tutor-end=1}{I}_{\htmlData{tutor-start=3,tutor-end=4}{2}}$ ， $\htmlData{tutor-start=0,tutor-end=1}{T}$ 四点共圆。

题解状态：标准答案与规范题解待补充

题目标签：SECOND-A-PROBLEM-P12

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（A 卷） · 二试 · 代数

（本题满分 50 分）求证不等式： $\htmlData{tutor-start=0,tutor-end=1}{-}\htmlData{tutor-start=1,tutor-end=2}{1} \htmlData{tutor-start=3,tutor-end=4}{<} \left( \sum_{\htmlData{tutor-start=18,tutor-end=19}{k}\htmlData{tutor-start=19,tutor-end=20}{=}\htmlData{tutor-start=20,tutor-end=21}{1}}^{\htmlData{tutor-start=24,tutor-end=25}{n}} \frac{\htmlData{tutor-start=33,tutor-end=34}{k}}{\htmlData{tutor-start=36,tutor-end=37}{k}^{\htmlData{tutor-start=39,tutor-end=40}{2}}\htmlData{tutor-start=41,tutor-end=42}{+}\htmlData{tutor-start=42,tutor-end=43}{1}} \right) \htmlData{tutor-start=53,tutor-end=54}{-} \ln \htmlData{tutor-start=59,tutor-end=60}{n} \htmlData{tutor-start=61,tutor-end=71}{\leqslant }\frac{\htmlData{tutor-start=77,tutor-end=78}{1}}{\htmlData{tutor-start=80,tutor-end=81}{2}}\htmlData{tutor-start=82,tutor-end=83}{,} \quad \htmlData{tutor-start=90,tutor-end=91}{n}\htmlData{tutor-start=91,tutor-end=92}{=}\htmlData{tutor-start=92,tutor-end=93}{1}\htmlData{tutor-start=93,tutor-end=94}{,} \htmlData{tutor-start=95,tutor-end=96}{2}\htmlData{tutor-start=96,tutor-end=97}{,} \cdots$

题解状态：标准答案与规范题解待补充

题目标签：SECOND-A-PROBLEM-P13

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（A 卷） · 二试 · 数论

（本题满分 50 分）设 $\htmlData{tutor-start=0,tutor-end=1}{k}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{l}$ 是给定的两个正整数. 证明：有无穷多个正整数 $\htmlData{tutor-start=0,tutor-end=1}{m} \htmlData{tutor-start=2,tutor-end=12}{\geqslant }\htmlData{tutor-start=12,tutor-end=13}{k}$ ，使得 $\htmlData{tutor-start=0,tutor-end=1}{C}_{\htmlData{tutor-start=3,tutor-end=4}{m}}^{\htmlData{tutor-start=7,tutor-end=8}{k}}$ 与 $\htmlData{tutor-start=0,tutor-end=1}{l}$ 互素.

题解状态：标准答案与规范题解待补充

题目标签：SECOND-A-PROBLEM-P14

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（A 卷） · 二试 · 数学竞赛/待细分

（本题满分 50 分）在非负数构成的 $\htmlData{tutor-start=0,tutor-end=1}{3} \htmlData{tutor-start=2,tutor-end=9}{\times }\htmlData{tutor-start=9,tutor-end=10}{9}$ 数表 $\htmlData{tutor-start=0,tutor-end=1}{P} \htmlData{tutor-start=2,tutor-end=3}{=} \begin{pmatrix} \htmlData{tutor-start=20,tutor-end=21}{x}_{\htmlData{tutor-start=23,tutor-end=24}{1}\htmlData{tutor-start=24,tutor-end=25}{1}} & \htmlData{tutor-start=29,tutor-end=30}{x}_{\htmlData{tutor-start=32,tutor-end=33}{1}\htmlData{tutor-start=33,tutor-end=34}{2}} & \htmlData{tutor-start=38,tutor-end=39}{x}_{\htmlData{tutor-start=41,tutor-end=42}{1}\htmlData{tutor-start=42,tutor-end=43}{3}} & \htmlData{tutor-start=47,tutor-end=48}{x}_{\htmlData{tutor-start=50,tutor-end=51}{1}\htmlData{tutor-start=51,tutor-end=52}{4}} & \htmlData{tutor-start=56,tutor-end=57}{x}_{\htmlData{tutor-start=59,tutor-end=60}{1}\htmlData{tutor-start=60,tutor-end=61}{5}} & \htmlData{tutor-start=65,tutor-end=66}{x}_{\htmlData{tutor-start=68,tutor-end=69}{1}\htmlData{tutor-start=69,tutor-end=70}{6}} & \htmlData{tutor-start=74,tutor-end=75}{x}_{\htmlData{tutor-start=77,tutor-end=78}{1}\htmlData{tutor-start=78,tutor-end=79}{7}} & \htmlData{tutor-start=83,tutor-end=84}{x}_{\htmlData{tutor-start=86,tutor-end=87}{1}\htmlData{tutor-start=87,tutor-end=88}{8}} & \htmlData{tutor-start=92,tutor-end=93}{x}_{\htmlData{tutor-start=95,tutor-end=96}{1}\htmlData{tutor-start=96,tutor-end=97}{9}} \\ \htmlData{tutor-start=102,tutor-end=103}{x}_{\htmlData{tutor-start=105,tutor-end=106}{2}\htmlData{tutor-start=106,tutor-end=107}{1}} & \htmlData{tutor-start=111,tutor-end=112}{x}_{\htmlData{tutor-start=114,tutor-end=115}{2}\htmlData{tutor-start=115,tutor-end=116}{2}} & \htmlData{tutor-start=120,tutor-end=121}{x}_{\htmlData{tutor-start=123,tutor-end=124}{2}\htmlData{tutor-start=124,tutor-end=125}{3}} & \htmlData{tutor-start=129,tutor-end=130}{x}_{\htmlData{tutor-start=132,tutor-end=133}{2}\htmlData{tutor-start=133,tutor-end=134}{4}} & \htmlData{tutor-start=138,tutor-end=139}{x}_{\htmlData{tutor-start=141,tutor-end=142}{2}\htmlData{tutor-start=142,tutor-end=143}{5}} & \htmlData{tutor-start=147,tutor-end=148}{x}_{\htmlData{tutor-start=150,tutor-end=151}{2}\htmlData{tutor-start=151,tutor-end=152}{6}} & \htmlData{tutor-start=156,tutor-end=157}{x}_{\htmlData{tutor-start=159,tutor-end=160}{2}\htmlData{tutor-start=160,tutor-end=161}{7}} & \htmlData{tutor-start=165,tutor-end=166}{x}_{\htmlData{tutor-start=168,tutor-end=169}{2}\htmlData{tutor-start=169,tutor-end=170}{8}} & \htmlData{tutor-start=174,tutor-end=175}{x}_{\htmlData{tutor-start=177,tutor-end=178}{2}\htmlData{tutor-start=178,tutor-end=179}{9}} \\ \htmlData{tutor-start=184,tutor-end=185}{x}_{\htmlData{tutor-start=187,tutor-end=188}{3}\htmlData{tutor-start=188,tutor-end=189}{1}} & \htmlData{tutor-start=193,tutor-end=194}{x}_{\htmlData{tutor-start=196,tutor-end=197}{3}\htmlData{tutor-start=197,tutor-end=198}{2}} & \htmlData{tutor-start=202,tutor-end=203}{x}_{\htmlData{tutor-start=205,tutor-end=206}{3}\htmlData{tutor-start=206,tutor-end=207}{3}} & \htmlData{tutor-start=211,tutor-end=212}{x}_{\htmlData{tutor-start=214,tutor-end=215}{3}\htmlData{tutor-start=215,tutor-end=216}{4}} & \htmlData{tutor-start=220,tutor-end=221}{x}_{\htmlData{tutor-start=223,tutor-end=224}{3}\htmlData{tutor-start=224,tutor-end=225}{5}} & \htmlData{tutor-start=229,tutor-end=230}{x}_{\htmlData{tutor-start=232,tutor-end=233}{3}\htmlData{tutor-start=233,tutor-end=234}{6}} & \htmlData{tutor-start=238,tutor-end=239}{x}_{\htmlData{tutor-start=241,tutor-end=242}{3}\htmlData{tutor-start=242,tutor-end=243}{7}} & \htmlData{tutor-start=247,tutor-end=248}{x}_{\htmlData{tutor-start=250,tutor-end=251}{3}\htmlData{tutor-start=251,tutor-end=252}{8}} & \htmlData{tutor-start=256,tutor-end=257}{x}_{\htmlData{tutor-start=259,tutor-end=260}{3}\htmlData{tutor-start=260,tutor-end=261}{9}} \end{pmatrix}$ 中每行的数互不相同，前 6 列中每列的三数之和为 1， $\htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{1}\htmlData{tutor-start=4,tutor-end=5}{7}}\htmlData{tutor-start=6,tutor-end=7}{=}\htmlData{tutor-start=7,tutor-end=8}{x}_{\htmlData{tutor-start=10,tutor-end=11}{2}\htmlData{tutor-start=11,tutor-end=12}{8}}\htmlData{tutor-start=13,tutor-end=14}{=}\htmlData{tutor-start=14,tutor-end=15}{x}_{\htmlData{tutor-start=17,tutor-end=18}{3}\htmlData{tutor-start=18,tutor-end=19}{9}}\htmlData{tutor-start=20,tutor-end=21}{=}\htmlData{tutor-start=21,tutor-end=22}{0}$ ， $\htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{2}\htmlData{tutor-start=4,tutor-end=5}{7}}\htmlData{tutor-start=6,tutor-end=7}{,} \htmlData{tutor-start=8,tutor-end=9}{x}_{\htmlData{tutor-start=11,tutor-end=12}{3}\htmlData{tutor-start=12,tutor-end=13}{7}}\htmlData{tutor-start=14,tutor-end=15}{,} \htmlData{tutor-start=16,tutor-end=17}{x}_{\htmlData{tutor-start=19,tutor-end=20}{1}\htmlData{tutor-start=20,tutor-end=21}{8}}\htmlData{tutor-start=22,tutor-end=23}{,} \htmlData{tutor-start=24,tutor-end=25}{x}_{\htmlData{tutor-start=27,tutor-end=28}{3}\htmlData{tutor-start=28,tutor-end=29}{8}}\htmlData{tutor-start=30,tutor-end=31}{,} \htmlData{tutor-start=32,tutor-end=33}{x}_{\htmlData{tutor-start=35,tutor-end=36}{1}\htmlData{tutor-start=36,tutor-end=37}{9}}\htmlData{tutor-start=38,tutor-end=39}{,} \htmlData{tutor-start=40,tutor-end=41}{x}_{\htmlData{tutor-start=43,tutor-end=44}{2}\htmlData{tutor-start=44,tutor-end=45}{9}}$ 均大于．如果 $\htmlData{tutor-start=0,tutor-end=1}{P}$ 的前三列构成的数表 $\htmlData{tutor-start=0,tutor-end=1}{S} \htmlData{tutor-start=2,tutor-end=3}{=} \begin{pmatrix} \htmlData{tutor-start=20,tutor-end=21}{x}_{\htmlData{tutor-start=23,tutor-end=24}{1}\htmlData{tutor-start=24,tutor-end=25}{1}} & \htmlData{tutor-start=29,tutor-end=30}{x}_{\htmlData{tutor-start=32,tutor-end=33}{1}\htmlData{tutor-start=33,tutor-end=34}{2}} & \htmlData{tutor-start=38,tutor-end=39}{x}_{\htmlData{tutor-start=41,tutor-end=42}{1}\htmlData{tutor-start=42,tutor-end=43}{3}} \\ \htmlData{tutor-start=48,tutor-end=49}{x}_{\htmlData{tutor-start=51,tutor-end=52}{2}\htmlData{tutor-start=52,tutor-end=53}{1}} & \htmlData{tutor-start=57,tutor-end=58}{x}_{\htmlData{tutor-start=60,tutor-end=61}{2}\htmlData{tutor-start=61,tutor-end=62}{2}} & \htmlData{tutor-start=66,tutor-end=67}{x}_{\htmlData{tutor-start=69,tutor-end=70}{2}\htmlData{tutor-start=70,tutor-end=71}{3}} \\ \htmlData{tutor-start=76,tutor-end=77}{x}_{\htmlData{tutor-start=79,tutor-end=80}{3}\htmlData{tutor-start=80,tutor-end=81}{1}} & \htmlData{tutor-start=85,tutor-end=86}{x}_{\htmlData{tutor-start=88,tutor-end=89}{3}\htmlData{tutor-start=89,tutor-end=90}{2}} & \htmlData{tutor-start=94,tutor-end=95}{x}_{\htmlData{tutor-start=97,tutor-end=98}{3}\htmlData{tutor-start=98,tutor-end=99}{3}} \end{pmatrix}$ 满足下面的性质 $\htmlData{tutor-start=0,tutor-end=1}{(}\htmlData{tutor-start=1,tutor-end=2}{O}\htmlData{tutor-start=2,tutor-end=3}{)}$ ：对于数表 $\htmlData{tutor-start=0,tutor-end=1}{P}$ 中的任意一列 $\begin{pmatrix} \htmlData{tutor-start=16,tutor-end=17}{x}_{\htmlData{tutor-start=19,tutor-end=20}{1}\htmlData{tutor-start=20,tutor-end=21}{k}} \\ \htmlData{tutor-start=26,tutor-end=27}{x}_{\htmlData{tutor-start=29,tutor-end=30}{2}\htmlData{tutor-start=30,tutor-end=31}{k}} \\ \htmlData{tutor-start=36,tutor-end=37}{x}_{\htmlData{tutor-start=39,tutor-end=40}{3}\htmlData{tutor-start=40,tutor-end=41}{k}} \end{pmatrix}$ （ $\htmlData{tutor-start=0,tutor-end=1}{k}\htmlData{tutor-start=1,tutor-end=2}{=}\htmlData{tutor-start=2,tutor-end=3}{1}\htmlData{tutor-start=3,tutor-end=4}{,} \htmlData{tutor-start=5,tutor-end=6}{2}\htmlData{tutor-start=6,tutor-end=7}{,} \cdots\htmlData{tutor-start=14,tutor-end=15}{,} \htmlData{tutor-start=16,tutor-end=17}{9}$ ）均存在某个 $\htmlData{tutor-start=0,tutor-end=1}{i} \htmlData{tutor-start=2,tutor-end=6}{\in }\htmlData{tutor-start=6,tutor-end=8}{\{}\htmlData{tutor-start=8,tutor-end=9}{1}\htmlData{tutor-start=9,tutor-end=10}{,} \htmlData{tutor-start=11,tutor-end=12}{2}\htmlData{tutor-start=12,tutor-end=13}{,} \htmlData{tutor-start=14,tutor-end=15}{3}\htmlData{tutor-start=15,tutor-end=17}{\}}$ 使得 $\htmlData{tutor-start=0,tutor-end=1}{(}\htmlData{tutor-start=1,tutor-end=2}{3}\htmlData{tutor-start=2,tutor-end=3}{)} \htmlData{tutor-start=4,tutor-end=5}{x}_{\htmlData{tutor-start=7,tutor-end=8}{i}\htmlData{tutor-start=8,tutor-end=9}{k}} \htmlData{tutor-start=11,tutor-end=21}{\leqslant }\htmlData{tutor-start=21,tutor-end=22}{u}_{\htmlData{tutor-start=24,tutor-end=25}{i}} \htmlData{tutor-start=27,tutor-end=28}{=} \min\htmlData{tutor-start=33,tutor-end=35}{\{}\htmlData{tutor-start=35,tutor-end=36}{x}_{\htmlData{tutor-start=38,tutor-end=39}{i}\htmlData{tutor-start=39,tutor-end=40}{1}}\htmlData{tutor-start=41,tutor-end=42}{,} \htmlData{tutor-start=43,tutor-end=44}{x}_{\htmlData{tutor-start=46,tutor-end=47}{i}\htmlData{tutor-start=47,tutor-end=48}{2}}\htmlData{tutor-start=49,tutor-end=50}{,} \htmlData{tutor-start=51,tutor-end=52}{x}_{\htmlData{tutor-start=54,tutor-end=55}{i}\htmlData{tutor-start=55,tutor-end=56}{3}}\htmlData{tutor-start=57,tutor-end=59}{\}}$ ．求证：（i）最小值 $\htmlData{tutor-start=0,tutor-end=1}{u}_{\htmlData{tutor-start=3,tutor-end=4}{i}} \htmlData{tutor-start=6,tutor-end=7}{=} \min\htmlData{tutor-start=12,tutor-end=14}{\{}\htmlData{tutor-start=14,tutor-end=15}{x}_{\htmlData{tutor-start=17,tutor-end=18}{i}\htmlData{tutor-start=18,tutor-end=19}{1}}\htmlData{tutor-start=20,tutor-end=21}{,} \htmlData{tutor-start=22,tutor-end=23}{x}_{\htmlData{tutor-start=25,tutor-end=26}{i}\htmlData{tutor-start=26,tutor-end=27}{2}}\htmlData{tutor-start=28,tutor-end=29}{,} \htmlData{tutor-start=30,tutor-end=31}{x}_{\htmlData{tutor-start=33,tutor-end=34}{i}\htmlData{tutor-start=34,tutor-end=35}{3}}\htmlData{tutor-start=36,tutor-end=38}{\}}$ ， $\htmlData{tutor-start=0,tutor-end=1}{i}\htmlData{tutor-start=1,tutor-end=2}{=}\htmlData{tutor-start=2,tutor-end=3}{1}\htmlData{tutor-start=3,tutor-end=4}{,} \htmlData{tutor-start=5,tutor-end=6}{2}\htmlData{tutor-start=6,tutor-end=7}{,} \htmlData{tutor-start=8,tutor-end=9}{3}$ 一定自数表 $\htmlData{tutor-start=0,tutor-end=1}{S}$ 的不同列．（ii）存在数表 $\htmlData{tutor-start=0,tutor-end=1}{P}$ 中唯一的一列 $\begin{pmatrix} x_{1k^} \\ x_{2k^} \\ x_{3k^} \end{pmatrix}$ ， $k^ \neq 1, 2, 3$ 使得 $\htmlData{tutor-start=0,tutor-end=1}{3} \htmlData{tutor-start=2,tutor-end=9}{\times }\htmlData{tutor-start=9,tutor-end=10}{3}$ 数表 $S' = \begin{pmatrix} x_{11} & x_{12} & x_{1k^} \\ x_{21} & x_{22} & x_{2k^} \\ x_{31} & x_{32} & x_{3k^*} \end{pmatrix}$ 仍然具有性质 $\htmlData{tutor-start=0,tutor-end=1}{(}\htmlData{tutor-start=1,tutor-end=2}{O}\htmlData{tutor-start=2,tutor-end=3}{)}$ ．

题解状态：标准答案与规范题解待补充

题目标签：SECOND-A-PROBLEM-P15

解题过程

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