1995 年全国高中数学联赛（学生版）

books/national_high_school_league_students/papers/competition-national-league-1995.pdf · HS-MATH-1024-v2.1-solution-aware

16 题0 个小问/题组

一试 · 一、选择题(每小题 6 分，共 36 分) · 代数

设等差数列 $\htmlData{tutor-start=0,tutor-end=2}{\{}\htmlData{tutor-start=2,tutor-end=3}{a}_{\htmlData{tutor-start=5,tutor-end=6}{n}}\htmlData{tutor-start=7,tutor-end=9}{\}}$ 满足 $\htmlData{tutor-start=0,tutor-end=1}{3}\htmlData{tutor-start=1,tutor-end=2}{a}_{\htmlData{tutor-start=4,tutor-end=5}{8}}\htmlData{tutor-start=6,tutor-end=7}{=}\htmlData{tutor-start=7,tutor-end=8}{5}\htmlData{tutor-start=8,tutor-end=9}{a}_{\htmlData{tutor-start=11,tutor-end=12}{1}\htmlData{tutor-start=12,tutor-end=13}{3}}$ 且 $\htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{>}\htmlData{tutor-start=6,tutor-end=7}{0}$ ， $\htmlData{tutor-start=0,tutor-end=1}{S}_{\htmlData{tutor-start=3,tutor-end=4}{n}}$ 为其前项之和，则 $\htmlData{tutor-start=0,tutor-end=1}{S}_{\htmlData{tutor-start=3,tutor-end=4}{n}}$ 中最大的是( ) (A) $\htmlData{tutor-start=0,tutor-end=1}{S}_{\htmlData{tutor-start=3,tutor-end=4}{1}\htmlData{tutor-start=4,tutor-end=5}{0}}$ (B) $\htmlData{tutor-start=0,tutor-end=1}{S}_{\htmlData{tutor-start=3,tutor-end=4}{1}\htmlData{tutor-start=4,tutor-end=5}{1}}$ (C) $\htmlData{tutor-start=0,tutor-end=1}{S}_{\htmlData{tutor-start=3,tutor-end=4}{2}\htmlData{tutor-start=4,tutor-end=5}{0}}$ (D) $\htmlData{tutor-start=0,tutor-end=1}{S}_{\htmlData{tutor-start=3,tutor-end=4}{2}\htmlData{tutor-start=4,tutor-end=5}{1}}$

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。

一试 · 一、选择题(每小题 6 分，共 36 分) · 平面几何

设复平面上单位圆内接正 20 边形的 20 个顶点所对应的复数依次为 $\htmlData{tutor-start=0,tutor-end=1}{Z}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{Z}_{\htmlData{tutor-start=10,tutor-end=11}{2}}\htmlData{tutor-start=12,tutor-end=13}{,} \cdots\htmlData{tutor-start=20,tutor-end=21}{,} \htmlData{tutor-start=22,tutor-end=23}{Z}_{\htmlData{tutor-start=25,tutor-end=26}{2}\htmlData{tutor-start=26,tutor-end=27}{0}}$ ，则复数 $\htmlData{tutor-start=0,tutor-end=1}{Z}_{\htmlData{tutor-start=3,tutor-end=4}{1}}^{\htmlData{tutor-start=7,tutor-end=8}{1}\htmlData{tutor-start=8,tutor-end=9}{9}\htmlData{tutor-start=9,tutor-end=10}{9}\htmlData{tutor-start=10,tutor-end=11}{5}}\htmlData{tutor-start=12,tutor-end=13}{,} \htmlData{tutor-start=14,tutor-end=15}{Z}_{\htmlData{tutor-start=17,tutor-end=18}{2}}^{\htmlData{tutor-start=21,tutor-end=22}{1}\htmlData{tutor-start=22,tutor-end=23}{9}\htmlData{tutor-start=23,tutor-end=24}{9}\htmlData{tutor-start=24,tutor-end=25}{5}}\htmlData{tutor-start=26,tutor-end=27}{,} \cdots\htmlData{tutor-start=34,tutor-end=35}{,} \htmlData{tutor-start=36,tutor-end=37}{Z}_{\htmlData{tutor-start=39,tutor-end=40}{2}\htmlData{tutor-start=40,tutor-end=41}{0}}^{\htmlData{tutor-start=44,tutor-end=45}{1}\htmlData{tutor-start=45,tutor-end=46}{9}\htmlData{tutor-start=46,tutor-end=47}{9}\htmlData{tutor-start=47,tutor-end=48}{5}}$ 所对应的不同的点的个数是( ) (A) 4 (B) 5 (C) 10 (D) 20

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。

一试 · 一、选择题(每小题 6 分，共 36 分) · 数学竞赛/待细分

如果甲的身高数或体重数至少有一项比乙大，则称甲不亚于乙，在 100 个小伙子中，如果某人不亚于其他 99 人，就称他为棒小伙子，那么，100 个小伙子中的棒小伙子最多可能有( ) (A) 1 个 (B) 2 个 (C) 50 个 (D) 100 个

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。

一试 · 一、选择题(每小题 6 分，共 36 分) · 数学竞赛/待细分

已知方程 $\htmlData{tutor-start=0,tutor-end=1}{|}\htmlData{tutor-start=1,tutor-end=2}{x}\htmlData{tutor-start=2,tutor-end=3}{-}\htmlData{tutor-start=3,tutor-end=4}{2}\htmlData{tutor-start=4,tutor-end=5}{n}\htmlData{tutor-start=5,tutor-end=6}{|}\htmlData{tutor-start=6,tutor-end=7}{=}\htmlData{tutor-start=7,tutor-end=8}{k}\sqrt{\htmlData{tutor-start=14,tutor-end=15}{x}}$ ( $n \in \mathbb{N}^*$ ) 在区间 $\htmlData{tutor-start=0,tutor-end=1}{(}\htmlData{tutor-start=1,tutor-end=2}{2}\htmlData{tutor-start=2,tutor-end=3}{n}\htmlData{tutor-start=3,tutor-end=4}{-}\htmlData{tutor-start=4,tutor-end=5}{1}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{2}\htmlData{tutor-start=8,tutor-end=9}{n}\htmlData{tutor-start=9,tutor-end=10}{+}\htmlData{tutor-start=10,tutor-end=11}{1}\htmlData{tutor-start=11,tutor-end=12}{]}$ 上有两个不相等的实根，则 $\htmlData{tutor-start=0,tutor-end=1}{k}$ 的取值范围是( ) (A) $\htmlData{tutor-start=0,tutor-end=1}{k}\htmlData{tutor-start=1,tutor-end=2}{>}\htmlData{tutor-start=2,tutor-end=3}{0}$ (B) $\htmlData{tutor-start=0,tutor-end=1}{0}\htmlData{tutor-start=1,tutor-end=2}{<}\htmlData{tutor-start=2,tutor-end=3}{k} \htmlData{tutor-start=4,tutor-end=14}{\leqslant }\frac{\htmlData{tutor-start=20,tutor-end=21}{1}}{\sqrt{\htmlData{tutor-start=29,tutor-end=30}{2}\htmlData{tutor-start=30,tutor-end=31}{n}\htmlData{tutor-start=31,tutor-end=32}{+}\htmlData{tutor-start=32,tutor-end=33}{1}}}$ (C) $\frac{\htmlData{tutor-start=6,tutor-end=7}{1}}{\htmlData{tutor-start=9,tutor-end=10}{2}\htmlData{tutor-start=10,tutor-end=11}{n}\htmlData{tutor-start=11,tutor-end=12}{+}\htmlData{tutor-start=12,tutor-end=13}{1}}\htmlData{tutor-start=14,tutor-end=15}{<}\htmlData{tutor-start=15,tutor-end=16}{k} \htmlData{tutor-start=17,tutor-end=27}{\leqslant }\frac{\htmlData{tutor-start=33,tutor-end=34}{1}}{\sqrt{\htmlData{tutor-start=42,tutor-end=43}{2}\htmlData{tutor-start=43,tutor-end=44}{n}\htmlData{tutor-start=44,tutor-end=45}{+}\htmlData{tutor-start=45,tutor-end=46}{1}}}$ (D) 以上都不是

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。

一试 · 一、选择题(每小题 6 分，共 36 分) · 数学竞赛/待细分

$\log_{\sin \htmlData{tutor-start=11,tutor-end=12}{1}}\cos \htmlData{tutor-start=18,tutor-end=19}{1}\htmlData{tutor-start=19,tutor-end=20}{,} \log_{\sin \htmlData{tutor-start=32,tutor-end=33}{1}}\tan \htmlData{tutor-start=39,tutor-end=40}{1}\htmlData{tutor-start=40,tutor-end=41}{,} \log_{\cos \htmlData{tutor-start=53,tutor-end=54}{1}}\sin \htmlData{tutor-start=60,tutor-end=61}{1}\htmlData{tutor-start=61,tutor-end=62}{,} \log_{\cos \htmlData{tutor-start=74,tutor-end=75}{1}}\tan \htmlData{tutor-start=81,tutor-end=82}{1}$ 的大小关系是 (A) $\log_{\sin \htmlData{tutor-start=11,tutor-end=12}{1}}\cos \htmlData{tutor-start=18,tutor-end=19}{1} \htmlData{tutor-start=20,tutor-end=21}{<} \log_{\cos \htmlData{tutor-start=33,tutor-end=34}{1}}\sin \htmlData{tutor-start=40,tutor-end=41}{1} \htmlData{tutor-start=42,tutor-end=43}{<} \log_{\sin \htmlData{tutor-start=55,tutor-end=56}{1}}\tan \htmlData{tutor-start=62,tutor-end=63}{1} \htmlData{tutor-start=64,tutor-end=65}{<} \log_{\cos \htmlData{tutor-start=77,tutor-end=78}{1}}\tan \htmlData{tutor-start=84,tutor-end=85}{1}$ (B) $\log_{\cos \htmlData{tutor-start=11,tutor-end=12}{1}}\sin \htmlData{tutor-start=18,tutor-end=19}{1} \htmlData{tutor-start=20,tutor-end=21}{<} \log_{\cos \htmlData{tutor-start=33,tutor-end=34}{1}}\tan \htmlData{tutor-start=40,tutor-end=41}{1} \htmlData{tutor-start=42,tutor-end=43}{<} \log_{\sin \htmlData{tutor-start=55,tutor-end=56}{1}}\cos \htmlData{tutor-start=62,tutor-end=63}{1} \htmlData{tutor-start=64,tutor-end=65}{<} \log_{\sin \htmlData{tutor-start=77,tutor-end=78}{1}}\tan \htmlData{tutor-start=84,tutor-end=85}{1}$ (C) $\log_{\sin \htmlData{tutor-start=11,tutor-end=12}{1}}\tan \htmlData{tutor-start=18,tutor-end=19}{1} \htmlData{tutor-start=20,tutor-end=21}{<} \log_{\cos \htmlData{tutor-start=33,tutor-end=34}{1}}\tan \htmlData{tutor-start=40,tutor-end=41}{1} \htmlData{tutor-start=42,tutor-end=43}{<} \log_{\cos \htmlData{tutor-start=55,tutor-end=56}{1}}\sin \htmlData{tutor-start=62,tutor-end=63}{1} \htmlData{tutor-start=64,tutor-end=65}{<} \log_{\sin \htmlData{tutor-start=77,tutor-end=78}{1}}\cos \htmlData{tutor-start=84,tutor-end=85}{1}$ (D) $\log_{\cos \htmlData{tutor-start=11,tutor-end=12}{1}}\tan \htmlData{tutor-start=18,tutor-end=19}{1} \htmlData{tutor-start=20,tutor-end=21}{<} \log_{\sin \htmlData{tutor-start=33,tutor-end=34}{1}}\tan \htmlData{tutor-start=40,tutor-end=41}{1} \htmlData{tutor-start=42,tutor-end=43}{<} \log_{\sin \htmlData{tutor-start=55,tutor-end=56}{1}}\cos \htmlData{tutor-start=62,tutor-end=63}{1} \htmlData{tutor-start=64,tutor-end=65}{<} \log_{\cos \htmlData{tutor-start=77,tutor-end=78}{1}}\sin \htmlData{tutor-start=84,tutor-end=85}{1}$

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。

一试 · 一、选择题(每小题 6 分，共 36 分) · 平面几何

设 $\htmlData{tutor-start=0,tutor-end=1}{O}$ 是正三棱锥 $\htmlData{tutor-start=0,tutor-end=1}{P}\htmlData{tutor-start=1,tutor-end=2}{-}\htmlData{tutor-start=2,tutor-end=3}{A}\htmlData{tutor-start=3,tutor-end=4}{B}\htmlData{tutor-start=4,tutor-end=5}{C}$ 底面三角形 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{B}\htmlData{tutor-start=2,tutor-end=3}{C}$ 的中心，过 $\htmlData{tutor-start=0,tutor-end=1}{O}$ 的动平面与 $\htmlData{tutor-start=0,tutor-end=1}{P}\htmlData{tutor-start=1,tutor-end=2}{C}$ 交于 $\htmlData{tutor-start=0,tutor-end=1}{S}$ ，与 $\htmlData{tutor-start=0,tutor-end=1}{P}\htmlData{tutor-start=1,tutor-end=2}{A}\htmlData{tutor-start=2,tutor-end=3}{,} \htmlData{tutor-start=4,tutor-end=5}{P}\htmlData{tutor-start=5,tutor-end=6}{B}$ 的延长线分别交于 $\htmlData{tutor-start=0,tutor-end=1}{Q}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{R}$ ，则和式 $\frac{\htmlData{tutor-start=6,tutor-end=7}{1}}{\htmlData{tutor-start=9,tutor-end=10}{P}\htmlData{tutor-start=10,tutor-end=11}{Q}}\htmlData{tutor-start=12,tutor-end=13}{+}\frac{\htmlData{tutor-start=19,tutor-end=20}{1}}{\htmlData{tutor-start=22,tutor-end=23}{P}\htmlData{tutor-start=23,tutor-end=24}{R}}\htmlData{tutor-start=25,tutor-end=26}{+}\frac{\htmlData{tutor-start=32,tutor-end=33}{1}}{\htmlData{tutor-start=35,tutor-end=36}{P}\htmlData{tutor-start=36,tutor-end=37}{S}}$ (A) 有最大值而无最小值 (B) 有最小值而无最大值 (C) 既有最大值又有最小值，两者不等 (D) 是一个与面 $\htmlData{tutor-start=0,tutor-end=1}{Q}\htmlData{tutor-start=1,tutor-end=2}{P}\htmlData{tutor-start=2,tutor-end=3}{S}$ 无关的常数

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。

一试 · 二、填空题(每小题 9 分，共 54 分) · 数学竞赛/待细分

设 $\htmlData{tutor-start=0,tutor-end=6}{\alpha}\htmlData{tutor-start=6,tutor-end=7}{,} \htmlData{tutor-start=8,tutor-end=13}{\beta}$ 为一对共轭复数，若 $\htmlData{tutor-start=0,tutor-end=1}{|}\htmlData{tutor-start=1,tutor-end=7}{\alpha}\htmlData{tutor-start=7,tutor-end=8}{-}\htmlData{tutor-start=8,tutor-end=13}{\beta}\htmlData{tutor-start=13,tutor-end=14}{|}\htmlData{tutor-start=14,tutor-end=15}{=}\htmlData{tutor-start=15,tutor-end=16}{2}\sqrt{\htmlData{tutor-start=22,tutor-end=23}{3}}$ ，且 $\frac{\htmlData{tutor-start=6,tutor-end=12}{\alpha}}{\htmlData{tutor-start=14,tutor-end=19}{\beta}^{\htmlData{tutor-start=21,tutor-end=22}{2}}}$ 为实数，则 $\htmlData{tutor-start=0,tutor-end=1}{|}\htmlData{tutor-start=1,tutor-end=7}{\alpha}\htmlData{tutor-start=7,tutor-end=8}{|}\htmlData{tutor-start=8,tutor-end=9}{=}$ ______.

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。

一试 · 二、填空题(每小题 9 分，共 54 分) · 平面几何

一个球的内接圆锥的最大体积与这个球的体积之比为______.

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。

一试 · 二、填空题(每小题 9 分，共 54 分) · 数论

用 $\htmlData{tutor-start=0,tutor-end=1}{[}\htmlData{tutor-start=1,tutor-end=2}{x}\htmlData{tutor-start=2,tutor-end=3}{]}$ 表示不大于实数 $\htmlData{tutor-start=0,tutor-end=1}{x}$ 的最大整数，方程 $\lg^{\htmlData{tutor-start=5,tutor-end=6}{2}} \htmlData{tutor-start=8,tutor-end=9}{x} \htmlData{tutor-start=10,tutor-end=11}{-} \htmlData{tutor-start=12,tutor-end=13}{[}\lg \htmlData{tutor-start=17,tutor-end=18}{x}\htmlData{tutor-start=18,tutor-end=19}{]} \htmlData{tutor-start=20,tutor-end=21}{-} \htmlData{tutor-start=22,tutor-end=23}{2} \htmlData{tutor-start=24,tutor-end=25}{=} \htmlData{tutor-start=26,tutor-end=27}{0}$ 的实根个数是______.

题解状态：标准答案与规范题解待补充

题目标签：FIRST-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 · 二、填空题(每小题 9 分，共 54 分) · 代数

直角坐标平面上，满足不等式组 $\begin{cases} \htmlData{tutor-start=14,tutor-end=15}{y} \htmlData{tutor-start=16,tutor-end=26}{\leqslant }\htmlData{tutor-start=26,tutor-end=27}{3}\htmlData{tutor-start=27,tutor-end=28}{x}\htmlData{tutor-start=28,tutor-end=29}{,} \\ \htmlData{tutor-start=33,tutor-end=34}{y} \htmlData{tutor-start=35,tutor-end=45}{\geqslant }\frac{\htmlData{tutor-start=51,tutor-end=52}{x}}{\htmlData{tutor-start=54,tutor-end=55}{3}}\htmlData{tutor-start=56,tutor-end=57}{,} \\ \htmlData{tutor-start=61,tutor-end=62}{x}\htmlData{tutor-start=62,tutor-end=63}{+}\htmlData{tutor-start=63,tutor-end=64}{y} \htmlData{tutor-start=65,tutor-end=75}{\leqslant }\htmlData{tutor-start=75,tutor-end=76}{1}\htmlData{tutor-start=76,tutor-end=77}{0}\htmlData{tutor-start=77,tutor-end=78}{0} \end{cases}$ 的整点个数是______.

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。

一试 · 二、填空题(每小题 9 分，共 54 分) · 组合数学

将一个四棱锥的每个顶点染上一种颜色，并使同一条棱的两端点异色，如果只有 5 种颜色可使用，那么不同的染色方法的总数是______.

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。

一试 · 二、填空题(每小题 9 分，共 54 分) · 数学竞赛/待细分

设 $\htmlData{tutor-start=0,tutor-end=1}{M}\htmlData{tutor-start=1,tutor-end=2}{=}\htmlData{tutor-start=2,tutor-end=4}{\{}\htmlData{tutor-start=4,tutor-end=5}{1}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{2}\htmlData{tutor-start=8,tutor-end=9}{,} \htmlData{tutor-start=10,tutor-end=11}{3}\htmlData{tutor-start=11,tutor-end=12}{,} \cdots\htmlData{tutor-start=19,tutor-end=20}{,} \htmlData{tutor-start=21,tutor-end=22}{1}\htmlData{tutor-start=22,tutor-end=23}{9}\htmlData{tutor-start=23,tutor-end=24}{9}\htmlData{tutor-start=24,tutor-end=25}{5}\htmlData{tutor-start=25,tutor-end=27}{\}}$ ， $\htmlData{tutor-start=0,tutor-end=1}{A}$ 是 $\htmlData{tutor-start=0,tutor-end=1}{M}$ 的子集且满足条件：当 $\htmlData{tutor-start=0,tutor-end=1}{x} \htmlData{tutor-start=2,tutor-end=6}{\in }\htmlData{tutor-start=6,tutor-end=7}{A}$ 时， $\htmlData{tutor-start=0,tutor-end=1}{1}\htmlData{tutor-start=1,tutor-end=2}{5}\htmlData{tutor-start=2,tutor-end=3}{x} \notin \htmlData{tutor-start=11,tutor-end=12}{A}$ ，则 $\htmlData{tutor-start=0,tutor-end=1}{A}$ 中元素的个数最多是______.

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。

二试/加试 · 第二试 · 平面几何

（25 分）给定曲线族 $\htmlData{tutor-start=0,tutor-end=1}{2}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{2}\sin\htmlData{tutor-start=7,tutor-end=13}{\theta}\htmlData{tutor-start=13,tutor-end=14}{-}\cos\htmlData{tutor-start=18,tutor-end=24}{\theta}\htmlData{tutor-start=24,tutor-end=25}{+}\htmlData{tutor-start=25,tutor-end=26}{3}\htmlData{tutor-start=26,tutor-end=27}{)}\htmlData{tutor-start=27,tutor-end=28}{x}^{\htmlData{tutor-start=30,tutor-end=31}{2}}\htmlData{tutor-start=32,tutor-end=33}{-}\htmlData{tutor-start=33,tutor-end=34}{(}\htmlData{tutor-start=34,tutor-end=35}{8}\sin\htmlData{tutor-start=39,tutor-end=45}{\theta}\htmlData{tutor-start=45,tutor-end=46}{+}\cos\htmlData{tutor-start=50,tutor-end=56}{\theta}\htmlData{tutor-start=56,tutor-end=57}{+}\htmlData{tutor-start=57,tutor-end=58}{1}\htmlData{tutor-start=58,tutor-end=59}{)}\htmlData{tutor-start=59,tutor-end=60}{y}\htmlData{tutor-start=60,tutor-end=61}{=}\htmlData{tutor-start=61,tutor-end=62}{0}$ ， $\htmlData{tutor-start=0,tutor-end=6}{\theta}$ 为参数，求该曲线在直线 $\htmlData{tutor-start=0,tutor-end=1}{y}\htmlData{tutor-start=1,tutor-end=2}{=}\htmlData{tutor-start=2,tutor-end=3}{2}\htmlData{tutor-start=3,tutor-end=4}{x}$ 上所截得的弦长的最大值。

题解状态：标准答案与规范题解待补充

题目标签：SECOND-GENERAL-PROBLEM-P一

解题过程

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二试/加试 · 第二试 · 数论

（25 分）求一切实数 $\htmlData{tutor-start=0,tutor-end=1}{p}$ ，使得三次方程 $\htmlData{tutor-start=0,tutor-end=1}{5}\htmlData{tutor-start=1,tutor-end=2}{x}^{\htmlData{tutor-start=4,tutor-end=5}{3}}\htmlData{tutor-start=6,tutor-end=7}{-}\htmlData{tutor-start=7,tutor-end=8}{5}\htmlData{tutor-start=8,tutor-end=9}{(}\htmlData{tutor-start=9,tutor-end=10}{p}\htmlData{tutor-start=10,tutor-end=11}{+}\htmlData{tutor-start=11,tutor-end=12}{1}\htmlData{tutor-start=12,tutor-end=13}{)}\htmlData{tutor-start=13,tutor-end=14}{x}^{\htmlData{tutor-start=16,tutor-end=17}{2}}\htmlData{tutor-start=18,tutor-end=19}{+}\htmlData{tutor-start=19,tutor-end=20}{(}\htmlData{tutor-start=20,tutor-end=21}{7}\htmlData{tutor-start=21,tutor-end=22}{1}\htmlData{tutor-start=22,tutor-end=23}{p}\htmlData{tutor-start=23,tutor-end=24}{-}\htmlData{tutor-start=24,tutor-end=25}{1}\htmlData{tutor-start=25,tutor-end=26}{)}\htmlData{tutor-start=26,tutor-end=27}{x}\htmlData{tutor-start=27,tutor-end=28}{+}\htmlData{tutor-start=28,tutor-end=29}{1}\htmlData{tutor-start=29,tutor-end=30}{=}\htmlData{tutor-start=30,tutor-end=31}{6}\htmlData{tutor-start=31,tutor-end=32}{6}\htmlData{tutor-start=32,tutor-end=33}{p}$ 的三个根均为正整数。

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。

二试/加试 · 第二试 · 平面几何

（35 分）如图，菱形 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{B}\htmlData{tutor-start=2,tutor-end=3}{C}\htmlData{tutor-start=3,tutor-end=4}{D}$ 的内切圆 $\htmlData{tutor-start=0,tutor-end=1}{O}$ 与各边分别切于 $\htmlData{tutor-start=0,tutor-end=1}{E}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{F}\htmlData{tutor-start=4,tutor-end=5}{,} \htmlData{tutor-start=6,tutor-end=7}{G}\htmlData{tutor-start=7,tutor-end=8}{,} \htmlData{tutor-start=9,tutor-end=10}{H}$ ，在弧 $\htmlData{tutor-start=0,tutor-end=1}{E}\htmlData{tutor-start=1,tutor-end=2}{F}$ 与 $\htmlData{tutor-start=0,tutor-end=1}{G}\htmlData{tutor-start=1,tutor-end=2}{H}$ 上分别作圆 $\htmlData{tutor-start=0,tutor-end=1}{O}$ 的切线交 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{B}$ 于 $\htmlData{tutor-start=0,tutor-end=1}{M}$ ，交 $\htmlData{tutor-start=0,tutor-end=1}{B}\htmlData{tutor-start=1,tutor-end=2}{C}$ 于 $\htmlData{tutor-start=0,tutor-end=1}{N}$ ，交 $\htmlData{tutor-start=0,tutor-end=1}{C}\htmlData{tutor-start=1,tutor-end=2}{D}$ 于 $\htmlData{tutor-start=0,tutor-end=1}{P}$ ，交 $\htmlData{tutor-start=0,tutor-end=1}{D}\htmlData{tutor-start=1,tutor-end=2}{A}$ 于 $\htmlData{tutor-start=0,tutor-end=1}{Q}$ 。求证： $\htmlData{tutor-start=0,tutor-end=1}{M}\htmlData{tutor-start=1,tutor-end=2}{Q} \htmlData{tutor-start=3,tutor-end=4}{/}\htmlData{tutor-start=4,tutor-end=5}{/} \htmlData{tutor-start=6,tutor-end=7}{N}\htmlData{tutor-start=7,tutor-end=8}{P}$ 。

**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 · 第二试 · 平面几何

（35 分）将平面上的每个点都以红，蓝两色之一着色。证明：存在这样两个相似的三角形，它们的相似比为 1995，并且每一个三角形的三个顶点同色。

题解状态：标准答案与规范题解待补充

题目标签：SECOND-GENERAL-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。