2011 年全国高中数学联赛 A 卷（学生版）

books/national_high_school_league_students/papers/competition-national-league-2011-a.pdf · HS-MATH-1024-v2.1-solution-aware

15 题0 个小问/题组

一试（A 卷） · 一、填空题（本大题共 8 小题，每小题 8 分，共 64 分） · 组合数学

设集合 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{=}\htmlData{tutor-start=2,tutor-end=4}{\{}\htmlData{tutor-start=4,tutor-end=5}{a}_{\htmlData{tutor-start=7,tutor-end=8}{1}}\htmlData{tutor-start=9,tutor-end=10}{,}\htmlData{tutor-start=10,tutor-end=11}{a}_{\htmlData{tutor-start=13,tutor-end=14}{2}}\htmlData{tutor-start=15,tutor-end=16}{,}\htmlData{tutor-start=16,tutor-end=17}{a}_{\htmlData{tutor-start=19,tutor-end=20}{3}}\htmlData{tutor-start=21,tutor-end=22}{,}\htmlData{tutor-start=22,tutor-end=23}{a}_{\htmlData{tutor-start=25,tutor-end=26}{4}}\htmlData{tutor-start=27,tutor-end=29}{\}}$ ，若 $\htmlData{tutor-start=0,tutor-end=1}{A}$ 中所有三元子集的三个元素之和组成的集合为 $\htmlData{tutor-start=0,tutor-end=1}{B}\htmlData{tutor-start=1,tutor-end=2}{=}\htmlData{tutor-start=2,tutor-end=4}{\{}\htmlData{tutor-start=4,tutor-end=5}{-}\htmlData{tutor-start=5,tutor-end=6}{1}\htmlData{tutor-start=6,tutor-end=7}{,}\htmlData{tutor-start=7,tutor-end=8}{3}\htmlData{tutor-start=8,tutor-end=9}{,}\htmlData{tutor-start=9,tutor-end=10}{5}\htmlData{tutor-start=10,tutor-end=11}{,}\htmlData{tutor-start=11,tutor-end=12}{8}\htmlData{tutor-start=12,tutor-end=14}{\}}$ ，则集合 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{=}$ \underline{\hspace{3cm}}.

题解状态：标准答案与规范题解待补充

题目标签：FIRST-A-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。

一试（A 卷） · 一、填空题（本大题共 8 小题，每小题 8 分，共 64 分） · 代数

函数 $\htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{x}\htmlData{tutor-start=3,tutor-end=4}{)}\htmlData{tutor-start=4,tutor-end=5}{=}\frac{\sqrt{\htmlData{tutor-start=17,tutor-end=18}{x}^{\htmlData{tutor-start=20,tutor-end=21}{2}}\htmlData{tutor-start=22,tutor-end=23}{+}\htmlData{tutor-start=23,tutor-end=24}{1}}}{\htmlData{tutor-start=27,tutor-end=28}{x}\htmlData{tutor-start=28,tutor-end=29}{-}\htmlData{tutor-start=29,tutor-end=30}{1}}$ 的值域为\underline{\hspace{3cm}}.

题解状态：标准答案与规范题解待补充

题目标签：FIRST-A-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。

一试（A 卷） · 一、填空题（本大题共 8 小题，每小题 8 分，共 64 分） · 数学竞赛/待细分

设 $\htmlData{tutor-start=0,tutor-end=1}{a}\htmlData{tutor-start=1,tutor-end=2}{,}\htmlData{tutor-start=2,tutor-end=3}{b}$ 为正实数， $\frac{\htmlData{tutor-start=6,tutor-end=7}{1}}{\htmlData{tutor-start=9,tutor-end=10}{a}}\htmlData{tutor-start=11,tutor-end=12}{+}\frac{\htmlData{tutor-start=18,tutor-end=19}{1}}{\htmlData{tutor-start=21,tutor-end=22}{b}}\htmlData{tutor-start=23,tutor-end=27}{\le }\htmlData{tutor-start=27,tutor-end=28}{2}\sqrt{\htmlData{tutor-start=34,tutor-end=35}{2}}$ ， $\htmlData{tutor-start=0,tutor-end=1}{(}\htmlData{tutor-start=1,tutor-end=2}{a}\htmlData{tutor-start=2,tutor-end=3}{-}\htmlData{tutor-start=3,tutor-end=4}{b}\htmlData{tutor-start=4,tutor-end=5}{)}^{\htmlData{tutor-start=7,tutor-end=8}{2}}\htmlData{tutor-start=9,tutor-end=10}{=}\htmlData{tutor-start=10,tutor-end=11}{4}\htmlData{tutor-start=11,tutor-end=12}{(}\htmlData{tutor-start=12,tutor-end=13}{a}\htmlData{tutor-start=13,tutor-end=14}{b}\htmlData{tutor-start=14,tutor-end=15}{)}^{\htmlData{tutor-start=17,tutor-end=18}{3}}$ ，则 $\log_{\htmlData{tutor-start=6,tutor-end=7}{a}} \htmlData{tutor-start=9,tutor-end=10}{b}\htmlData{tutor-start=10,tutor-end=11}{=}$ \underline{\hspace{3cm}}.

题解状态：标准答案与规范题解待补充

题目标签：FIRST-A-FILL-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、填空题（本大题共 8 小题，每小题 8 分，共 64 分） · 数学竞赛/待细分

如果 $\cos^{\htmlData{tutor-start=6,tutor-end=7}{5}}\htmlData{tutor-start=8,tutor-end=14}{\theta}\htmlData{tutor-start=14,tutor-end=15}{-}\sin^{\htmlData{tutor-start=21,tutor-end=22}{5}}\htmlData{tutor-start=23,tutor-end=29}{\theta}\htmlData{tutor-start=29,tutor-end=30}{<}\htmlData{tutor-start=30,tutor-end=31}{7}\htmlData{tutor-start=31,tutor-end=32}{(}\sin^{\htmlData{tutor-start=38,tutor-end=39}{3}}\htmlData{tutor-start=40,tutor-end=46}{\theta}\htmlData{tutor-start=46,tutor-end=47}{-}\cos^{\htmlData{tutor-start=53,tutor-end=54}{3}}\htmlData{tutor-start=55,tutor-end=61}{\theta}\htmlData{tutor-start=61,tutor-end=62}{)}$ ， $\htmlData{tutor-start=0,tutor-end=6}{\theta}\htmlData{tutor-start=6,tutor-end=9}{\in}\htmlData{tutor-start=9,tutor-end=10}{[}\htmlData{tutor-start=10,tutor-end=11}{0}\htmlData{tutor-start=11,tutor-end=12}{,}\htmlData{tutor-start=12,tutor-end=13}{2}\htmlData{tutor-start=13,tutor-end=16}{\pi}\htmlData{tutor-start=16,tutor-end=17}{)}$ ，那么 $\htmlData{tutor-start=0,tutor-end=6}{\theta}$ 的取值范围是\underline{\hspace{3cm}}.

题解状态：标准答案与规范题解待补充

题目标签：FIRST-A-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。

一试（A 卷） · 一、填空题（本大题共 8 小题，每小题 8 分，共 64 分） · 数学竞赛/待细分

现安排 7 名同学去参加 5 个运动项目，要求甲、乙两同学不能参加同一个项目，每个项目都有人参加，每人只参加一个项目，则满足上述要求的不同安排方案数为\underline{\hspace{3cm}}.（用数字作答）

题解状态：标准答案与规范题解待补充

题目标签：FIRST-A-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。

一试（A 卷） · 一、填空题（本大题共 8 小题，每小题 8 分，共 64 分） · 平面几何

在四面体 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{B}\htmlData{tutor-start=2,tutor-end=3}{C}\htmlData{tutor-start=3,tutor-end=4}{D}$ 中，已知 $\angle ADB=\angle BDC=\angle CDA=60^\circ$ ， $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{D}\htmlData{tutor-start=2,tutor-end=3}{=}\htmlData{tutor-start=3,tutor-end=4}{B}\htmlData{tutor-start=4,tutor-end=5}{D}\htmlData{tutor-start=5,tutor-end=6}{=}\htmlData{tutor-start=6,tutor-end=7}{3}$ ， $\htmlData{tutor-start=0,tutor-end=1}{C}\htmlData{tutor-start=1,tutor-end=2}{D}\htmlData{tutor-start=2,tutor-end=3}{=}\htmlData{tutor-start=3,tutor-end=4}{2}$ ，则四面体 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{B}\htmlData{tutor-start=2,tutor-end=3}{C}\htmlData{tutor-start=3,tutor-end=4}{D}$ 的外接球的半径为\underline{\hspace{3cm}}.

题解状态：标准答案与规范题解待补充

题目标签：FIRST-A-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。

一试（A 卷） · 一、填空题（本大题共 8 小题，每小题 8 分，共 64 分） · 平面几何

直线 $\htmlData{tutor-start=0,tutor-end=1}{x}\htmlData{tutor-start=1,tutor-end=2}{-}\htmlData{tutor-start=2,tutor-end=3}{2}\htmlData{tutor-start=3,tutor-end=4}{y}\htmlData{tutor-start=4,tutor-end=5}{-}\htmlData{tutor-start=5,tutor-end=6}{1}\htmlData{tutor-start=6,tutor-end=7}{=}\htmlData{tutor-start=7,tutor-end=8}{0}$ 与抛物线 $\htmlData{tutor-start=0,tutor-end=1}{y}^{\htmlData{tutor-start=3,tutor-end=4}{2}}\htmlData{tutor-start=5,tutor-end=6}{=}\htmlData{tutor-start=6,tutor-end=7}{4}\htmlData{tutor-start=7,tutor-end=8}{x}$ 交于 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{,}\htmlData{tutor-start=2,tutor-end=3}{B}$ 两点， $\htmlData{tutor-start=0,tutor-end=1}{C}$ 为抛物线上的一点， $\angle ACB=90^\circ$ ，则点 $\htmlData{tutor-start=0,tutor-end=1}{C}$ 的坐标为\underline{\hspace{3cm}}.

题解状态：标准答案与规范题解待补充

题目标签：FIRST-A-FILL-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、填空题（本大题共 8 小题，每小题 8 分，共 64 分） · 数论

已知 $\htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{n}}\htmlData{tutor-start=5,tutor-end=6}{=}\htmlData{tutor-start=6,tutor-end=7}{C}_{\htmlData{tutor-start=9,tutor-end=10}{2}\htmlData{tutor-start=10,tutor-end=11}{0}\htmlData{tutor-start=11,tutor-end=12}{0}}^{\htmlData{tutor-start=15,tutor-end=16}{n}}\htmlData{tutor-start=17,tutor-end=18}{(}\sqrt[\htmlData{tutor-start=24,tutor-end=25}{3}]{\htmlData{tutor-start=27,tutor-end=28}{6}}\htmlData{tutor-start=29,tutor-end=30}{)}^{\htmlData{tutor-start=32,tutor-end=33}{2}\htmlData{tutor-start=33,tutor-end=34}{0}\htmlData{tutor-start=34,tutor-end=35}{0}\htmlData{tutor-start=35,tutor-end=36}{-}\htmlData{tutor-start=36,tutor-end=37}{n}}\htmlData{tutor-start=38,tutor-end=43}{\cdot}\left(\frac{\htmlData{tutor-start=55,tutor-end=56}{1}}{\sqrt{\htmlData{tutor-start=64,tutor-end=65}{2}}}\right)^{\htmlData{tutor-start=76,tutor-end=77}{n}}$ ( $\htmlData{tutor-start=0,tutor-end=1}{n}\htmlData{tutor-start=1,tutor-end=2}{=}\htmlData{tutor-start=2,tutor-end=3}{1}\htmlData{tutor-start=3,tutor-end=4}{,}\htmlData{tutor-start=4,tutor-end=5}{2}\htmlData{tutor-start=5,tutor-end=6}{,}\cdots\htmlData{tutor-start=12,tutor-end=13}{,}\htmlData{tutor-start=13,tutor-end=14}{9}\htmlData{tutor-start=14,tutor-end=15}{5}$ )，则数列 $\htmlData{tutor-start=0,tutor-end=2}{\{}\htmlData{tutor-start=2,tutor-end=3}{a}_{\htmlData{tutor-start=5,tutor-end=6}{n}}\htmlData{tutor-start=7,tutor-end=9}{\}}$ 中整数项的个数为\underline{\hspace{3cm}}.

题解状态：标准答案与规范题解待补充

题目标签：FIRST-A-FILL-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） · 二、解答题（本大题共 3 个小题，共 56 分. 解答应写出文字说明、证明过程或演算步骤.） · 代数

（本题满分 16 分）设函数 $\htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{x}\htmlData{tutor-start=3,tutor-end=4}{)}\htmlData{tutor-start=4,tutor-end=5}{=}\htmlData{tutor-start=5,tutor-end=6}{|}\lg\htmlData{tutor-start=9,tutor-end=10}{(}\htmlData{tutor-start=10,tutor-end=11}{x}\htmlData{tutor-start=11,tutor-end=12}{+}\htmlData{tutor-start=12,tutor-end=13}{1}\htmlData{tutor-start=13,tutor-end=14}{)}\htmlData{tutor-start=14,tutor-end=15}{|}$ ，实数 $\htmlData{tutor-start=0,tutor-end=1}{a}\htmlData{tutor-start=1,tutor-end=2}{,}\htmlData{tutor-start=2,tutor-end=3}{b}\htmlData{tutor-start=3,tutor-end=4}{(}\htmlData{tutor-start=4,tutor-end=5}{a}\htmlData{tutor-start=5,tutor-end=6}{<}\htmlData{tutor-start=6,tutor-end=7}{b}\htmlData{tutor-start=7,tutor-end=8}{)}$ 满足 $\htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{a}\htmlData{tutor-start=3,tutor-end=4}{)}\htmlData{tutor-start=4,tutor-end=5}{=}\htmlData{tutor-start=5,tutor-end=6}{f}\left(\htmlData{tutor-start=12,tutor-end=13}{-}\frac{\htmlData{tutor-start=19,tutor-end=20}{b}\htmlData{tutor-start=20,tutor-end=21}{+}\htmlData{tutor-start=21,tutor-end=22}{1}}{\htmlData{tutor-start=24,tutor-end=25}{b}\htmlData{tutor-start=25,tutor-end=26}{+}\htmlData{tutor-start=26,tutor-end=27}{2}}\right)$ ， $\htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{1}\htmlData{tutor-start=3,tutor-end=4}{0}\htmlData{tutor-start=4,tutor-end=5}{a}\htmlData{tutor-start=5,tutor-end=6}{+}\htmlData{tutor-start=6,tutor-end=7}{6}\htmlData{tutor-start=7,tutor-end=8}{b}\htmlData{tutor-start=8,tutor-end=9}{+}\htmlData{tutor-start=9,tutor-end=10}{2}\htmlData{tutor-start=10,tutor-end=11}{1}\htmlData{tutor-start=11,tutor-end=12}{)}\htmlData{tutor-start=12,tutor-end=13}{=}\htmlData{tutor-start=13,tutor-end=14}{4}\lg \htmlData{tutor-start=18,tutor-end=19}{2}$ ，求 $\htmlData{tutor-start=0,tutor-end=1}{a}\htmlData{tutor-start=1,tutor-end=2}{,}\htmlData{tutor-start=2,tutor-end=3}{b}$ 的值.

题解状态：标准答案与规范题解待补充

题目标签：FIRST-A-SOLUTION-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） · 二、解答题（本大题共 3 个小题，共 56 分. 解答应写出文字说明、证明过程或演算步骤.） · 代数

（本题满分 20 分）已知数列 $\htmlData{tutor-start=0,tutor-end=2}{\{}\htmlData{tutor-start=2,tutor-end=3}{a}_{\htmlData{tutor-start=5,tutor-end=6}{n}}\htmlData{tutor-start=7,tutor-end=9}{\}}$ 满足： $\htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{=}\htmlData{tutor-start=6,tutor-end=7}{2}\htmlData{tutor-start=7,tutor-end=8}{t}\htmlData{tutor-start=8,tutor-end=9}{-}\htmlData{tutor-start=9,tutor-end=10}{3}$ ( $\htmlData{tutor-start=0,tutor-end=1}{t}\htmlData{tutor-start=1,tutor-end=4}{\in}\mathbb{\htmlData{tutor-start=12,tutor-end=13}{R}}$ 且 $\htmlData{tutor-start=0,tutor-end=1}{t}\ne\htmlData{tutor-start=4,tutor-end=8}{\pm }\htmlData{tutor-start=8,tutor-end=9}{1}$ )， $a_{n+1}=\frac{(2t^{n+1}-3)a_{n}+2(t-1)t^{n}-1}{a_{n}+2t^{n}-1} \quad (n\in\mathbb{N}^*).$ （1）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；（2）若 $\htmlData{tutor-start=0,tutor-end=1}{t}\htmlData{tutor-start=1,tutor-end=2}{>}\htmlData{tutor-start=2,tutor-end=3}{0}$ ，试比较 $\htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{n}\htmlData{tutor-start=4,tutor-end=5}{+}\htmlData{tutor-start=5,tutor-end=6}{1}}$ 与 $\htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{n}}$ 的大小.

题解状态：标准答案与规范题解待补充

题目标签：FIRST-A-SOLUTION-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） · 二、解答题（本大题共 3 个小题，共 56 分. 解答应写出文字说明、证明过程或演算步骤.） · 平面几何

（本题满分 20 分）作斜率为 $\frac{\htmlData{tutor-start=6,tutor-end=7}{1}}{\htmlData{tutor-start=9,tutor-end=10}{3}}$ 的直线 $\htmlData{tutor-start=0,tutor-end=1}{l}$ 与椭圆 $\htmlData{tutor-start=0,tutor-end=1}{C}$ ： $\frac{\htmlData{tutor-start=6,tutor-end=7}{x}^{\htmlData{tutor-start=9,tutor-end=10}{2}}}{\htmlData{tutor-start=13,tutor-end=14}{3}\htmlData{tutor-start=14,tutor-end=15}{6}}\htmlData{tutor-start=16,tutor-end=17}{+}\frac{\htmlData{tutor-start=23,tutor-end=24}{y}^{\htmlData{tutor-start=26,tutor-end=27}{2}}}{\htmlData{tutor-start=30,tutor-end=31}{4}}\htmlData{tutor-start=32,tutor-end=33}{=}\htmlData{tutor-start=33,tutor-end=34}{1}$ 交于 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{,}\htmlData{tutor-start=2,tutor-end=3}{B}$ 两点（如图所示），且 $\htmlData{tutor-start=0,tutor-end=1}{P}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{3}\sqrt{\htmlData{tutor-start=9,tutor-end=10}{2}}\htmlData{tutor-start=11,tutor-end=12}{,}\sqrt{\htmlData{tutor-start=18,tutor-end=19}{2}}\htmlData{tutor-start=20,tutor-end=21}{)}$ 在直线 $\htmlData{tutor-start=0,tutor-end=1}{l}$ 的左上方. （1）证明： $\htmlData{tutor-start=0,tutor-end=10}{\triangle }\htmlData{tutor-start=10,tutor-end=11}{P}\htmlData{tutor-start=11,tutor-end=12}{A}\htmlData{tutor-start=12,tutor-end=13}{B}$ 的内切圆的圆心在一条定直线上；（2）若 $\angle APB=60^\circ$ ，求 $\htmlData{tutor-start=0,tutor-end=10}{\triangle }\htmlData{tutor-start=10,tutor-end=11}{P}\htmlData{tutor-start=11,tutor-end=12}{A}\htmlData{tutor-start=12,tutor-end=13}{B}$ 的面积.

**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-A-SOLUTION-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 40 分）如图， $\htmlData{tutor-start=0,tutor-end=1}{P}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{Q}$ 分别是圆内接四边形 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{B}\htmlData{tutor-start=2,tutor-end=3}{C}\htmlData{tutor-start=3,tutor-end=4}{D}$ 的对角线 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{C}\htmlData{tutor-start=2,tutor-end=3}{,} \htmlData{tutor-start=4,tutor-end=5}{B}\htmlData{tutor-start=5,tutor-end=6}{D}$ 的中点. 若 $\htmlData{tutor-start=0,tutor-end=7}{\angle }\htmlData{tutor-start=7,tutor-end=8}{B}\htmlData{tutor-start=8,tutor-end=9}{P}\htmlData{tutor-start=9,tutor-end=10}{A} \htmlData{tutor-start=11,tutor-end=12}{=} \htmlData{tutor-start=13,tutor-end=20}{\angle }\htmlData{tutor-start=20,tutor-end=21}{D}\htmlData{tutor-start=21,tutor-end=22}{P}\htmlData{tutor-start=22,tutor-end=23}{A}$ ，证明： $\htmlData{tutor-start=0,tutor-end=7}{\angle }\htmlData{tutor-start=7,tutor-end=8}{A}\htmlData{tutor-start=8,tutor-end=9}{Q}\htmlData{tutor-start=9,tutor-end=10}{B} \htmlData{tutor-start=11,tutor-end=12}{=} \htmlData{tutor-start=13,tutor-end=20}{\angle }\htmlData{tutor-start=20,tutor-end=21}{C}\htmlData{tutor-start=21,tutor-end=22}{Q}\htmlData{tutor-start=22,tutor-end=23}{B}$ .

**原卷图示 2**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：SECOND-A-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（A 卷） · 二试 · 数论

（本题满分 40 分）证明：对任意整数 $\htmlData{tutor-start=0,tutor-end=1}{n} \htmlData{tutor-start=2,tutor-end=6}{\ge }\htmlData{tutor-start=6,tutor-end=7}{4}$ ，存在一个 $\htmlData{tutor-start=0,tutor-end=1}{n}$ 次多项式 $\htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{x}\htmlData{tutor-start=3,tutor-end=4}{)} \htmlData{tutor-start=5,tutor-end=6}{=} \htmlData{tutor-start=7,tutor-end=8}{x}^{\htmlData{tutor-start=10,tutor-end=11}{n}} \htmlData{tutor-start=13,tutor-end=14}{+} \htmlData{tutor-start=15,tutor-end=16}{a}_{\htmlData{tutor-start=18,tutor-end=19}{n}\htmlData{tutor-start=19,tutor-end=20}{-}\htmlData{tutor-start=20,tutor-end=21}{1}}\htmlData{tutor-start=22,tutor-end=23}{x}^{\htmlData{tutor-start=25,tutor-end=26}{n}\htmlData{tutor-start=26,tutor-end=27}{-}\htmlData{tutor-start=27,tutor-end=28}{1}} \htmlData{tutor-start=30,tutor-end=31}{+} \cdots \htmlData{tutor-start=39,tutor-end=40}{+} \htmlData{tutor-start=41,tutor-end=42}{a}_{\htmlData{tutor-start=44,tutor-end=45}{1}}\htmlData{tutor-start=46,tutor-end=47}{x} \htmlData{tutor-start=48,tutor-end=49}{+} \htmlData{tutor-start=50,tutor-end=51}{a}_{\htmlData{tutor-start=53,tutor-end=54}{0}}$ 具有如下性质：（1） $\htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{0}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{a}_{\htmlData{tutor-start=10,tutor-end=11}{1}}\htmlData{tutor-start=12,tutor-end=13}{,} \cdots\htmlData{tutor-start=20,tutor-end=21}{,} \htmlData{tutor-start=22,tutor-end=23}{a}_{\htmlData{tutor-start=25,tutor-end=26}{n}\htmlData{tutor-start=26,tutor-end=27}{-}\htmlData{tutor-start=27,tutor-end=28}{1}}$ 均为正整数；（2）对任意正整数 $\htmlData{tutor-start=0,tutor-end=1}{m}$ ，及任意 $\htmlData{tutor-start=0,tutor-end=1}{k}$ ( $\htmlData{tutor-start=0,tutor-end=1}{k} \htmlData{tutor-start=2,tutor-end=6}{\ge }\htmlData{tutor-start=6,tutor-end=7}{2}$ ) 个互不相同的正整数 $\htmlData{tutor-start=0,tutor-end=1}{r}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{r}_{\htmlData{tutor-start=10,tutor-end=11}{2}}\htmlData{tutor-start=12,tutor-end=13}{,} \cdots\htmlData{tutor-start=20,tutor-end=21}{,} \htmlData{tutor-start=22,tutor-end=23}{r}_{\htmlData{tutor-start=25,tutor-end=26}{k}}$ ，均有 $f(m) \neq f(r_{1})f(r_{2})\cdots f(r_{k}).$

题解状态：标准答案与规范题解待补充

题目标签：SECOND-A-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（A 卷） · 二试 · 数学竞赛/待细分

（本题满分 50 分）设 $\htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{a}_{\htmlData{tutor-start=10,tutor-end=11}{2}}\htmlData{tutor-start=12,tutor-end=13}{,} \cdots\htmlData{tutor-start=20,tutor-end=21}{,} \htmlData{tutor-start=22,tutor-end=23}{a}_{\htmlData{tutor-start=25,tutor-end=26}{n}}$ ( $\htmlData{tutor-start=0,tutor-end=1}{n} \htmlData{tutor-start=2,tutor-end=6}{\ge }\htmlData{tutor-start=6,tutor-end=7}{4}$ ) 是给定的正实数， $\htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{1}} \htmlData{tutor-start=6,tutor-end=7}{<} \htmlData{tutor-start=8,tutor-end=9}{a}_{\htmlData{tutor-start=11,tutor-end=12}{2}} \htmlData{tutor-start=14,tutor-end=15}{<} \cdots \htmlData{tutor-start=23,tutor-end=24}{<} \htmlData{tutor-start=25,tutor-end=26}{a}_{\htmlData{tutor-start=28,tutor-end=29}{n}}$ . 对任意正实数 $\htmlData{tutor-start=0,tutor-end=1}{r}$ ，满足 $\frac{\htmlData{tutor-start=6,tutor-end=7}{a}_{\htmlData{tutor-start=9,tutor-end=10}{j}} \htmlData{tutor-start=12,tutor-end=13}{-} \htmlData{tutor-start=14,tutor-end=15}{a}_{\htmlData{tutor-start=17,tutor-end=18}{i}}}{\htmlData{tutor-start=21,tutor-end=22}{a}_{\htmlData{tutor-start=24,tutor-end=25}{k}} \htmlData{tutor-start=27,tutor-end=28}{-} \htmlData{tutor-start=29,tutor-end=30}{a}_{\htmlData{tutor-start=32,tutor-end=33}{j}}} \htmlData{tutor-start=36,tutor-end=37}{=} \htmlData{tutor-start=38,tutor-end=39}{r} \htmlData{tutor-start=40,tutor-end=42}{\ }\htmlData{tutor-start=42,tutor-end=43}{(}\htmlData{tutor-start=43,tutor-end=44}{1} \htmlData{tutor-start=45,tutor-end=49}{\le }\htmlData{tutor-start=49,tutor-end=50}{i} \htmlData{tutor-start=51,tutor-end=52}{<} \htmlData{tutor-start=53,tutor-end=54}{j} \htmlData{tutor-start=55,tutor-end=56}{<} \htmlData{tutor-start=57,tutor-end=58}{k} \htmlData{tutor-start=59,tutor-end=63}{\le }\htmlData{tutor-start=63,tutor-end=64}{n}\htmlData{tutor-start=64,tutor-end=65}{)}$ 的三元数组 $\htmlData{tutor-start=0,tutor-end=1}{(}\htmlData{tutor-start=1,tutor-end=2}{i}\htmlData{tutor-start=2,tutor-end=3}{,} \htmlData{tutor-start=4,tutor-end=5}{j}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{k}\htmlData{tutor-start=8,tutor-end=9}{)}$ 的个数记为 $\htmlData{tutor-start=0,tutor-end=1}{f}_{\htmlData{tutor-start=3,tutor-end=4}{n}}\htmlData{tutor-start=5,tutor-end=6}{(}\htmlData{tutor-start=6,tutor-end=7}{r}\htmlData{tutor-start=7,tutor-end=8}{)}$ . 证明： $\htmlData{tutor-start=0,tutor-end=1}{f}_{\htmlData{tutor-start=3,tutor-end=4}{n}}\htmlData{tutor-start=5,tutor-end=6}{(}\htmlData{tutor-start=6,tutor-end=7}{r}\htmlData{tutor-start=7,tutor-end=8}{)} \htmlData{tutor-start=9,tutor-end=10}{<} \frac{\htmlData{tutor-start=17,tutor-end=18}{n}^{\htmlData{tutor-start=20,tutor-end=21}{2}}}{\htmlData{tutor-start=24,tutor-end=25}{4}}$ .

题解状态：标准答案与规范题解待补充

题目标签：SECOND-A-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。

二试/加试（A 卷） · 二试 · 数论

（本题满分 50 分）设 A 是一个 $\htmlData{tutor-start=0,tutor-end=1}{3} \htmlData{tutor-start=2,tutor-end=9}{\times }\htmlData{tutor-start=9,tutor-end=10}{9}$ 的方格表，在每一个小方格内各填一个正整数. 称 A 中的一个 $\htmlData{tutor-start=0,tutor-end=1}{m} \htmlData{tutor-start=2,tutor-end=9}{\times }\htmlData{tutor-start=9,tutor-end=10}{n}$ ( $\htmlData{tutor-start=0,tutor-end=1}{1} \htmlData{tutor-start=2,tutor-end=6}{\le }\htmlData{tutor-start=6,tutor-end=7}{m} \htmlData{tutor-start=8,tutor-end=12}{\le }\htmlData{tutor-start=12,tutor-end=13}{3}\htmlData{tutor-start=13,tutor-end=14}{,} \htmlData{tutor-start=15,tutor-end=16}{1} \htmlData{tutor-start=17,tutor-end=21}{\le }\htmlData{tutor-start=21,tutor-end=22}{n} \htmlData{tutor-start=23,tutor-end=27}{\le }\htmlData{tutor-start=27,tutor-end=28}{9}$ ) 方格表为“好矩形”，若它的所有数的和为 10 的倍数. 称 A 中的一个 $\htmlData{tutor-start=0,tutor-end=1}{1} \htmlData{tutor-start=2,tutor-end=9}{\times }\htmlData{tutor-start=9,tutor-end=10}{1}$ 的小方格为“坏格”，若它不包含于任何一个“好矩形”．求 A 中“坏格”个数的最大值．

题解状态：标准答案与规范题解待补充

题目标签：SECOND-A-PROBLEM-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。