1993 年全国高中数学联赛（学生版）

books/national_high_school_league_students/papers/competition-national-league-1993.pdf · HS-MATH-1024-v2.1-solution-aware

18 题0 个小问/题组

一试 · 一、选择题（每小题 5 分，共 30 分） · 数学竞赛/待细分

若 $\htmlData{tutor-start=0,tutor-end=1}{M}\htmlData{tutor-start=1,tutor-end=2}{=}\htmlData{tutor-start=2,tutor-end=4}{\{}\htmlData{tutor-start=4,tutor-end=5}{(}\htmlData{tutor-start=5,tutor-end=6}{x}\htmlData{tutor-start=6,tutor-end=7}{,} \htmlData{tutor-start=8,tutor-end=9}{y}\htmlData{tutor-start=9,tutor-end=10}{)} \htmlData{tutor-start=11,tutor-end=16}{\mid }\htmlData{tutor-start=16,tutor-end=17}{|}\tan \htmlData{tutor-start=22,tutor-end=26}{\pi }\htmlData{tutor-start=26,tutor-end=27}{y}\htmlData{tutor-start=27,tutor-end=28}{|}\htmlData{tutor-start=28,tutor-end=29}{+}\sin^{\htmlData{tutor-start=35,tutor-end=36}{2}} \htmlData{tutor-start=38,tutor-end=42}{\pi }\htmlData{tutor-start=42,tutor-end=43}{x}\htmlData{tutor-start=43,tutor-end=44}{=}\htmlData{tutor-start=44,tutor-end=45}{0}\htmlData{tutor-start=45,tutor-end=47}{\}}$ ， $\htmlData{tutor-start=0,tutor-end=1}{N}\htmlData{tutor-start=1,tutor-end=2}{=}\htmlData{tutor-start=2,tutor-end=4}{\{}\htmlData{tutor-start=4,tutor-end=5}{(}\htmlData{tutor-start=5,tutor-end=6}{x}\htmlData{tutor-start=6,tutor-end=7}{,} \htmlData{tutor-start=8,tutor-end=9}{y}\htmlData{tutor-start=9,tutor-end=10}{)} \htmlData{tutor-start=11,tutor-end=16}{\mid }\htmlData{tutor-start=16,tutor-end=17}{x}^{\htmlData{tutor-start=19,tutor-end=20}{2}}\htmlData{tutor-start=21,tutor-end=22}{+}\htmlData{tutor-start=22,tutor-end=23}{y}^{\htmlData{tutor-start=25,tutor-end=26}{2}} \htmlData{tutor-start=28,tutor-end=38}{\leqslant }\htmlData{tutor-start=38,tutor-end=39}{2}\htmlData{tutor-start=39,tutor-end=41}{\}}$ ，则 $\htmlData{tutor-start=0,tutor-end=1}{M} \htmlData{tutor-start=2,tutor-end=7}{\cap }\htmlData{tutor-start=7,tutor-end=8}{N}$ 的元素个数是（） (A) 4 (B) 5 (C) 8 (D) 9

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 5 分，共 30 分） · 数学竞赛/待细分

已知 $\htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{x}\htmlData{tutor-start=3,tutor-end=4}{)}\htmlData{tutor-start=4,tutor-end=5}{=}\htmlData{tutor-start=5,tutor-end=6}{a}\sin \htmlData{tutor-start=11,tutor-end=12}{x}\htmlData{tutor-start=12,tutor-end=13}{+}\htmlData{tutor-start=13,tutor-end=14}{b}\sqrt[\htmlData{tutor-start=20,tutor-end=21}{3}]{\htmlData{tutor-start=23,tutor-end=24}{x}}\htmlData{tutor-start=25,tutor-end=26}{+}\htmlData{tutor-start=26,tutor-end=27}{4}$ （ $\htmlData{tutor-start=0,tutor-end=1}{a}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{b}$ 为实数），且 $\htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\lg \log_{\htmlData{tutor-start=12,tutor-end=13}{3}} \htmlData{tutor-start=15,tutor-end=16}{1}\htmlData{tutor-start=16,tutor-end=17}{0}\htmlData{tutor-start=17,tutor-end=18}{)}\htmlData{tutor-start=18,tutor-end=19}{=}\htmlData{tutor-start=19,tutor-end=20}{5}$ ，则 $\htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\lg \lg \htmlData{tutor-start=10,tutor-end=11}{3}\htmlData{tutor-start=11,tutor-end=12}{)}$ 的值是（） (A) -5 (B) -3 (C) 3 (D) 随 $\htmlData{tutor-start=0,tutor-end=1}{a}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{b}$ 取不同值而取不同值

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 5 分，共 30 分） · 组合数学

集合 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{B}$ 的并集 $\htmlData{tutor-start=0,tutor-end=1}{A} \htmlData{tutor-start=2,tutor-end=7}{\cup }\htmlData{tutor-start=7,tutor-end=8}{B}\htmlData{tutor-start=8,tutor-end=9}{=}\htmlData{tutor-start=9,tutor-end=11}{\{}\htmlData{tutor-start=11,tutor-end=12}{a}_{\htmlData{tutor-start=14,tutor-end=15}{1}}\htmlData{tutor-start=16,tutor-end=17}{,} \htmlData{tutor-start=18,tutor-end=19}{a}_{\htmlData{tutor-start=21,tutor-end=22}{2}}\htmlData{tutor-start=23,tutor-end=24}{,} \htmlData{tutor-start=25,tutor-end=26}{a}_{\htmlData{tutor-start=28,tutor-end=29}{3}}\htmlData{tutor-start=30,tutor-end=32}{\}}$ ，当 $\htmlData{tutor-start=0,tutor-end=1}{A} \neq \htmlData{tutor-start=7,tutor-end=8}{B}$ 时， $\htmlData{tutor-start=0,tutor-end=1}{(}\htmlData{tutor-start=1,tutor-end=2}{A}\htmlData{tutor-start=2,tutor-end=3}{,} \htmlData{tutor-start=4,tutor-end=5}{B}\htmlData{tutor-start=5,tutor-end=6}{)}$ 与 $\htmlData{tutor-start=0,tutor-end=1}{(}\htmlData{tutor-start=1,tutor-end=2}{B}\htmlData{tutor-start=2,tutor-end=3}{,} \htmlData{tutor-start=4,tutor-end=5}{A}\htmlData{tutor-start=5,tutor-end=6}{)}$ 视为不同的对，则这样的 $\htmlData{tutor-start=0,tutor-end=1}{(}\htmlData{tutor-start=1,tutor-end=2}{A}\htmlData{tutor-start=2,tutor-end=3}{,} \htmlData{tutor-start=4,tutor-end=5}{B}\htmlData{tutor-start=5,tutor-end=6}{)}$ 对的个数是（） (A) 8 (B) 9 (C) 26 (D) 27

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 5 分，共 30 分） · 平面几何

若直线 $\htmlData{tutor-start=0,tutor-end=1}{x}\htmlData{tutor-start=1,tutor-end=2}{=}\frac{\htmlData{tutor-start=8,tutor-end=11}{\pi}}{\htmlData{tutor-start=13,tutor-end=14}{4}}$ 被曲线 $\htmlData{tutor-start=0,tutor-end=1}{C}\htmlData{tutor-start=1,tutor-end=2}{:} \htmlData{tutor-start=3,tutor-end=4}{(}\htmlData{tutor-start=4,tutor-end=5}{x}\htmlData{tutor-start=5,tutor-end=6}{-}\arcsin \htmlData{tutor-start=14,tutor-end=15}{a}\htmlData{tutor-start=15,tutor-end=16}{)}\htmlData{tutor-start=16,tutor-end=17}{(}\htmlData{tutor-start=17,tutor-end=18}{x}\htmlData{tutor-start=18,tutor-end=19}{-}\arccos \htmlData{tutor-start=27,tutor-end=28}{a}\htmlData{tutor-start=28,tutor-end=29}{)}\htmlData{tutor-start=29,tutor-end=30}{+}\htmlData{tutor-start=30,tutor-end=31}{(}\htmlData{tutor-start=31,tutor-end=32}{y}\htmlData{tutor-start=32,tutor-end=33}{-}\arcsin \htmlData{tutor-start=41,tutor-end=42}{a}\htmlData{tutor-start=42,tutor-end=43}{)}\htmlData{tutor-start=43,tutor-end=44}{(}\htmlData{tutor-start=44,tutor-end=45}{y}\htmlData{tutor-start=45,tutor-end=46}{+}\arccos \htmlData{tutor-start=54,tutor-end=55}{a}\htmlData{tutor-start=55,tutor-end=56}{)}\htmlData{tutor-start=56,tutor-end=57}{=}\htmlData{tutor-start=57,tutor-end=58}{0}$ 所截的弦长为 $\htmlData{tutor-start=0,tutor-end=1}{d}$ ，当 $\htmlData{tutor-start=0,tutor-end=1}{a}$ 变化时 $\htmlData{tutor-start=0,tutor-end=1}{d}$ 的最小值是（） (A) $\frac{\htmlData{tutor-start=6,tutor-end=9}{\pi}}{\htmlData{tutor-start=11,tutor-end=12}{4}}$ (B) $\frac{\htmlData{tutor-start=6,tutor-end=9}{\pi}}{\htmlData{tutor-start=11,tutor-end=12}{3}}$ (C) $\frac{\htmlData{tutor-start=6,tutor-end=9}{\pi}}{\htmlData{tutor-start=11,tutor-end=12}{2}}$ (D) $\htmlData{tutor-start=0,tutor-end=3}{\pi}$

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 5 分，共 30 分） · 平面几何

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，角 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{B}\htmlData{tutor-start=4,tutor-end=5}{,} \htmlData{tutor-start=6,tutor-end=7}{C}$ 的对边长分别为 $\htmlData{tutor-start=0,tutor-end=1}{a}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{b}\htmlData{tutor-start=4,tutor-end=5}{,} \htmlData{tutor-start=6,tutor-end=7}{c}$ ，若 $\htmlData{tutor-start=0,tutor-end=1}{c}\htmlData{tutor-start=1,tutor-end=2}{-}\htmlData{tutor-start=2,tutor-end=3}{a}$ 等于 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{C}$ 边上的高 $\htmlData{tutor-start=0,tutor-end=1}{h}$ ，则 $\sin\frac{\htmlData{tutor-start=10,tutor-end=11}{C}\htmlData{tutor-start=11,tutor-end=12}{-}\htmlData{tutor-start=12,tutor-end=13}{A}}{\htmlData{tutor-start=15,tutor-end=16}{2}}\htmlData{tutor-start=17,tutor-end=18}{+}\cos\frac{\htmlData{tutor-start=28,tutor-end=29}{C}\htmlData{tutor-start=29,tutor-end=30}{+}\htmlData{tutor-start=30,tutor-end=31}{A}}{\htmlData{tutor-start=33,tutor-end=34}{2}}$ 的值是（） (A) 1 (B) $\frac{\htmlData{tutor-start=6,tutor-end=7}{1}}{\htmlData{tutor-start=9,tutor-end=10}{2}}$ (C) $\frac{\htmlData{tutor-start=6,tutor-end=7}{1}}{\htmlData{tutor-start=9,tutor-end=10}{3}}$ (D) -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 5 分，共 30 分） · 数学竞赛/待细分

设 $\htmlData{tutor-start=0,tutor-end=1}{m}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{n}$ 为非零实数， $\htmlData{tutor-start=0,tutor-end=1}{i}$ 为虚数单位， $\htmlData{tutor-start=0,tutor-end=1}{z} \htmlData{tutor-start=2,tutor-end=6}{\in }\htmlData{tutor-start=6,tutor-end=7}{C}$ ，则方程 $\htmlData{tutor-start=0,tutor-end=1}{|}\htmlData{tutor-start=1,tutor-end=2}{z}\htmlData{tutor-start=2,tutor-end=3}{+}\htmlData{tutor-start=3,tutor-end=4}{n}\htmlData{tutor-start=4,tutor-end=5}{i}\htmlData{tutor-start=5,tutor-end=6}{|}\htmlData{tutor-start=6,tutor-end=7}{+}\htmlData{tutor-start=7,tutor-end=8}{|}\htmlData{tutor-start=8,tutor-end=9}{z}\htmlData{tutor-start=9,tutor-end=10}{-}\htmlData{tutor-start=10,tutor-end=11}{m}\htmlData{tutor-start=11,tutor-end=12}{i}\htmlData{tutor-start=12,tutor-end=13}{|}\htmlData{tutor-start=13,tutor-end=14}{=}\htmlData{tutor-start=14,tutor-end=15}{n}$ 与 $\htmlData{tutor-start=0,tutor-end=1}{|}\htmlData{tutor-start=1,tutor-end=2}{z}\htmlData{tutor-start=2,tutor-end=3}{+}\htmlData{tutor-start=3,tutor-end=4}{n}\htmlData{tutor-start=4,tutor-end=5}{i}\htmlData{tutor-start=5,tutor-end=6}{|}\htmlData{tutor-start=6,tutor-end=7}{-}\htmlData{tutor-start=7,tutor-end=8}{|}\htmlData{tutor-start=8,tutor-end=9}{z}\htmlData{tutor-start=9,tutor-end=10}{-}\htmlData{tutor-start=10,tutor-end=11}{m}\htmlData{tutor-start=11,tutor-end=12}{i}\htmlData{tutor-start=12,tutor-end=13}{|}\htmlData{tutor-start=13,tutor-end=14}{=}\htmlData{tutor-start=14,tutor-end=15}{-}\htmlData{tutor-start=15,tutor-end=16}{m}$ 在同一复平面内的图形（ $\htmlData{tutor-start=0,tutor-end=1}{F}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{F}_{\htmlData{tutor-start=10,tutor-end=11}{2}}$ 为焦点）是（）

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

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

**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。

一试 · 二、填空题（每小题 5 分，共 30 分） · 数学竞赛/待细分

二次方程 $\htmlData{tutor-start=0,tutor-end=1}{(}\htmlData{tutor-start=1,tutor-end=2}{1}\htmlData{tutor-start=2,tutor-end=3}{-}\htmlData{tutor-start=3,tutor-end=4}{i}\htmlData{tutor-start=4,tutor-end=5}{)}\htmlData{tutor-start=5,tutor-end=6}{x}^{\htmlData{tutor-start=8,tutor-end=9}{2}}\htmlData{tutor-start=10,tutor-end=11}{+}\htmlData{tutor-start=11,tutor-end=12}{(}\htmlData{tutor-start=12,tutor-end=19}{\lambda}\htmlData{tutor-start=19,tutor-end=20}{+}\htmlData{tutor-start=20,tutor-end=21}{i}\htmlData{tutor-start=21,tutor-end=22}{)}\htmlData{tutor-start=22,tutor-end=23}{x}\htmlData{tutor-start=23,tutor-end=24}{+}\htmlData{tutor-start=24,tutor-end=25}{(}\htmlData{tutor-start=25,tutor-end=26}{1}\htmlData{tutor-start=26,tutor-end=27}{+}\htmlData{tutor-start=27,tutor-end=28}{i}\htmlData{tutor-start=28,tutor-end=35}{\lambda}\htmlData{tutor-start=35,tutor-end=36}{)}\htmlData{tutor-start=36,tutor-end=37}{=}\htmlData{tutor-start=37,tutor-end=38}{0}$ （ $\htmlData{tutor-start=0,tutor-end=1}{i}$ 为虚数单位， $\htmlData{tutor-start=0,tutor-end=8}{\lambda }\htmlData{tutor-start=8,tutor-end=12}{\in }\htmlData{tutor-start=12,tutor-end=13}{R}$ ）有两个虚根的充分必要条件件是 $\htmlData{tutor-start=0,tutor-end=7}{\lambda}$ 的取值范围为______.

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 · 填空题 · 数学竞赛/待细分

实数 $\htmlData{tutor-start=0,tutor-end=1}{x}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{y}$ 满足 $\htmlData{tutor-start=0,tutor-end=1}{4}\htmlData{tutor-start=1,tutor-end=2}{x}^{\htmlData{tutor-start=4,tutor-end=5}{2}}\htmlData{tutor-start=6,tutor-end=7}{-}\htmlData{tutor-start=7,tutor-end=8}{5}\htmlData{tutor-start=8,tutor-end=9}{x}\htmlData{tutor-start=9,tutor-end=10}{y}\htmlData{tutor-start=10,tutor-end=11}{+}\htmlData{tutor-start=11,tutor-end=12}{4}\htmlData{tutor-start=12,tutor-end=13}{y}^{\htmlData{tutor-start=15,tutor-end=16}{2}}\htmlData{tutor-start=17,tutor-end=18}{=}\htmlData{tutor-start=18,tutor-end=19}{5}$ ，设 $\htmlData{tutor-start=0,tutor-end=1}{S}\htmlData{tutor-start=1,tutor-end=2}{=}\htmlData{tutor-start=2,tutor-end=3}{x}^{\htmlData{tutor-start=5,tutor-end=6}{2}}\htmlData{tutor-start=7,tutor-end=8}{+}\htmlData{tutor-start=8,tutor-end=9}{y}^{\htmlData{tutor-start=11,tutor-end=12}{2}}$ ，则 $\frac{\htmlData{tutor-start=6,tutor-end=7}{1}}{\htmlData{tutor-start=9,tutor-end=10}{S}_{\max}}\htmlData{tutor-start=18,tutor-end=19}{+}\frac{\htmlData{tutor-start=25,tutor-end=26}{1}}{\htmlData{tutor-start=28,tutor-end=29}{S}_{\min}}\htmlData{tutor-start=37,tutor-end=38}{=}$ ______.

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 · 填空题 · 数学竞赛/待细分

若 $\htmlData{tutor-start=0,tutor-end=1}{z} \htmlData{tutor-start=2,tutor-end=6}{\in }\htmlData{tutor-start=6,tutor-end=7}{C}$ ， $\arg\htmlData{tutor-start=4,tutor-end=5}{(}\htmlData{tutor-start=5,tutor-end=6}{z}^{\htmlData{tutor-start=8,tutor-end=9}{2}}\htmlData{tutor-start=10,tutor-end=11}{-}\htmlData{tutor-start=11,tutor-end=12}{4}\htmlData{tutor-start=12,tutor-end=13}{)}\htmlData{tutor-start=13,tutor-end=14}{=}\frac{\htmlData{tutor-start=20,tutor-end=21}{5}\htmlData{tutor-start=21,tutor-end=24}{\pi}}{\htmlData{tutor-start=26,tutor-end=27}{6}}$ ， $\arg\htmlData{tutor-start=4,tutor-end=5}{(}\htmlData{tutor-start=5,tutor-end=6}{z}^{\htmlData{tutor-start=8,tutor-end=9}{2}}\htmlData{tutor-start=10,tutor-end=11}{+}\htmlData{tutor-start=11,tutor-end=12}{4}\htmlData{tutor-start=12,tutor-end=13}{)}\htmlData{tutor-start=13,tutor-end=14}{=}\frac{\htmlData{tutor-start=20,tutor-end=23}{\pi}}{\htmlData{tutor-start=25,tutor-end=26}{3}}$ ，则 $\htmlData{tutor-start=0,tutor-end=1}{z}$ 的值是______.

题解状态：标准答案与规范题解待补充

题目标签：FIRST-FILL-P3

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 · 填空题 · 数论

整数 $\left[\frac{\htmlData{tutor-start=12,tutor-end=13}{1}\htmlData{tutor-start=13,tutor-end=14}{0}^{\htmlData{tutor-start=16,tutor-end=17}{9}\htmlData{tutor-start=17,tutor-end=18}{3}}}{\htmlData{tutor-start=21,tutor-end=22}{1}\htmlData{tutor-start=22,tutor-end=23}{0}^{\htmlData{tutor-start=25,tutor-end=26}{3}\htmlData{tutor-start=26,tutor-end=27}{1}}\htmlData{tutor-start=28,tutor-end=29}{+}\htmlData{tutor-start=29,tutor-end=30}{3}}\right]$ 的末两位数是______.

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 · 填空题 · 数学竞赛/待细分

设任意实数 $\htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{>}\htmlData{tutor-start=6,tutor-end=7}{x}_{\htmlData{tutor-start=9,tutor-end=10}{2}}\htmlData{tutor-start=11,tutor-end=12}{>}\htmlData{tutor-start=12,tutor-end=13}{x}_{\htmlData{tutor-start=15,tutor-end=16}{3}}\htmlData{tutor-start=17,tutor-end=18}{>}\htmlData{tutor-start=18,tutor-end=19}{0}$ ，要使 $\log_{\htmlData{tutor-start=6,tutor-end=7}{x}_{\htmlData{tutor-start=9,tutor-end=10}{1}}}\htmlData{tutor-start=12,tutor-end=13}{1}\htmlData{tutor-start=13,tutor-end=14}{9}\htmlData{tutor-start=14,tutor-end=15}{9}\htmlData{tutor-start=15,tutor-end=16}{3}\htmlData{tutor-start=16,tutor-end=17}{+}\log_{\htmlData{tutor-start=23,tutor-end=24}{x}_{\htmlData{tutor-start=26,tutor-end=27}{2}}}\htmlData{tutor-start=29,tutor-end=30}{1}\htmlData{tutor-start=30,tutor-end=31}{9}\htmlData{tutor-start=31,tutor-end=32}{9}\htmlData{tutor-start=32,tutor-end=33}{3}\htmlData{tutor-start=33,tutor-end=34}{+}\log_{\htmlData{tutor-start=40,tutor-end=41}{x}_{\htmlData{tutor-start=43,tutor-end=44}{3}}}\htmlData{tutor-start=46,tutor-end=47}{1}\htmlData{tutor-start=47,tutor-end=48}{9}\htmlData{tutor-start=48,tutor-end=49}{9}\htmlData{tutor-start=49,tutor-end=50}{3} \htmlData{tutor-start=51,tutor-end=55}{\ge }\htmlData{tutor-start=55,tutor-end=56}{k} \htmlData{tutor-start=57,tutor-end=63}{\cdot }\log_{\htmlData{tutor-start=69,tutor-end=70}{x}_{\htmlData{tutor-start=72,tutor-end=73}{1}}}\htmlData{tutor-start=75,tutor-end=76}{1}\htmlData{tutor-start=76,tutor-end=77}{9}\htmlData{tutor-start=77,tutor-end=78}{9}\htmlData{tutor-start=78,tutor-end=79}{3}$ 恒成立，则 $\htmlData{tutor-start=0,tutor-end=1}{k}$ 的最大值是______.

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 · 填空题 · 数学竞赛/待细分

三位数 (100, 101, $\cdots$ , 999) 共 900 个，在卡片上打印这些三位数，每张卡片上打印一个三位数，有的卡片所印的，倒过来看仍为三位数，如 198 倒过来看是 861；有的卡片则不然，如 531 倒过来看是，因此，有些卡片可以一卡二用，于是至多可以少打印____张卡片.

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。

一试 · 解答题 · 平面几何

（本题满分 20 分）三棱锥 $\htmlData{tutor-start=0,tutor-end=1}{S}\htmlData{tutor-start=1,tutor-end=2}{-}\htmlData{tutor-start=2,tutor-end=3}{A}\htmlData{tutor-start=3,tutor-end=4}{B}\htmlData{tutor-start=4,tutor-end=5}{C}$ 中，侧棱 $\htmlData{tutor-start=0,tutor-end=1}{S}\htmlData{tutor-start=1,tutor-end=2}{A}\htmlData{tutor-start=2,tutor-end=3}{,} \htmlData{tutor-start=4,tutor-end=5}{S}\htmlData{tutor-start=5,tutor-end=6}{B}\htmlData{tutor-start=6,tutor-end=7}{,} \htmlData{tutor-start=8,tutor-end=9}{S}\htmlData{tutor-start=9,tutor-end=10}{C}$ 两两互相垂直， $\htmlData{tutor-start=0,tutor-end=1}{M}$ 为三角形 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{B}\htmlData{tutor-start=2,tutor-end=3}{C}$ 的重心， $\htmlData{tutor-start=0,tutor-end=1}{D}$ 为 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{B}$ 的中点，作与 $\htmlData{tutor-start=0,tutor-end=1}{S}\htmlData{tutor-start=1,tutor-end=2}{C}$ 平行的直线 $\htmlData{tutor-start=0,tutor-end=1}{D}\htmlData{tutor-start=1,tutor-end=2}{P}$ . 证明：(1) $\htmlData{tutor-start=0,tutor-end=1}{D}\htmlData{tutor-start=1,tutor-end=2}{P}$ 与 $\htmlData{tutor-start=0,tutor-end=1}{S}\htmlData{tutor-start=1,tutor-end=2}{M}$ 相交；(2) 设 $\htmlData{tutor-start=0,tutor-end=1}{D}\htmlData{tutor-start=1,tutor-end=2}{P}$ 与 $\htmlData{tutor-start=0,tutor-end=1}{S}\htmlData{tutor-start=1,tutor-end=2}{M}$ 的交点为 $\htmlData{tutor-start=0,tutor-end=1}{D}$ ，则 $\htmlData{tutor-start=0,tutor-end=1}{D}$ 为三棱锥 $\htmlData{tutor-start=0,tutor-end=1}{S}\htmlData{tutor-start=1,tutor-end=2}{-}\htmlData{tutor-start=2,tutor-end=3}{A}\htmlData{tutor-start=3,tutor-end=4}{B}\htmlData{tutor-start=4,tutor-end=5}{C}$ 的外接球球心.

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。

一试 · 解答题 · 平面几何

（本题满分 20 分）设 $\htmlData{tutor-start=0,tutor-end=1}{0}\htmlData{tutor-start=1,tutor-end=2}{<}\htmlData{tutor-start=2,tutor-end=3}{a}\htmlData{tutor-start=3,tutor-end=4}{<}\htmlData{tutor-start=4,tutor-end=5}{b}$ ，过两定点 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{a}\htmlData{tutor-start=3,tutor-end=4}{,} \htmlData{tutor-start=5,tutor-end=6}{0}\htmlData{tutor-start=6,tutor-end=7}{)}$ 和 $\htmlData{tutor-start=0,tutor-end=1}{B}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{b}\htmlData{tutor-start=3,tutor-end=4}{,} \htmlData{tutor-start=5,tutor-end=6}{0}\htmlData{tutor-start=6,tutor-end=7}{)}$ 分别引直线 $\htmlData{tutor-start=0,tutor-end=1}{l}$ 和 $\htmlData{tutor-start=0,tutor-end=1}{m}$ ，使与抛物线 $\htmlData{tutor-start=0,tutor-end=1}{y}^{\htmlData{tutor-start=3,tutor-end=4}{2}}\htmlData{tutor-start=5,tutor-end=6}{=}\htmlData{tutor-start=6,tutor-end=7}{x}$ 有四个不同的交点，当这四点共圆时，求这种直线 $\htmlData{tutor-start=0,tutor-end=1}{l}$ 与 $\htmlData{tutor-start=0,tutor-end=1}{m}$ 的交点 $\htmlData{tutor-start=0,tutor-end=1}{P}$ 的轨迹.

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。

一试 · 解答题 · 代数

（本题满分 20 分）设正数列 $\htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{0}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{a}_{\htmlData{tutor-start=10,tutor-end=11}{1}}\htmlData{tutor-start=12,tutor-end=13}{,} \htmlData{tutor-start=14,tutor-end=15}{a}_{\htmlData{tutor-start=17,tutor-end=18}{2}}\htmlData{tutor-start=19,tutor-end=20}{,} \cdots\htmlData{tutor-start=27,tutor-end=28}{,} \htmlData{tutor-start=29,tutor-end=30}{a}_{\htmlData{tutor-start=32,tutor-end=33}{n}}\htmlData{tutor-start=34,tutor-end=35}{,} \cdots$ 满足 $\sqrt{\htmlData{tutor-start=6,tutor-end=7}{a}_{\htmlData{tutor-start=9,tutor-end=10}{n}} \htmlData{tutor-start=12,tutor-end=13}{a}_{\htmlData{tutor-start=15,tutor-end=16}{n}\htmlData{tutor-start=16,tutor-end=17}{-}\htmlData{tutor-start=17,tutor-end=18}{2}}} \htmlData{tutor-start=21,tutor-end=22}{-} \sqrt{\htmlData{tutor-start=29,tutor-end=30}{a}_{\htmlData{tutor-start=32,tutor-end=33}{n}\htmlData{tutor-start=33,tutor-end=34}{-}\htmlData{tutor-start=34,tutor-end=35}{1}} \htmlData{tutor-start=37,tutor-end=38}{a}_{\htmlData{tutor-start=40,tutor-end=41}{n}\htmlData{tutor-start=41,tutor-end=42}{-}\htmlData{tutor-start=42,tutor-end=43}{2}}} \htmlData{tutor-start=46,tutor-end=47}{=} \htmlData{tutor-start=48,tutor-end=49}{2}\htmlData{tutor-start=49,tutor-end=50}{a}_{\htmlData{tutor-start=52,tutor-end=53}{n}\htmlData{tutor-start=53,tutor-end=54}{-}\htmlData{tutor-start=54,tutor-end=55}{1}}$ ， $\htmlData{tutor-start=0,tutor-end=1}{(}\htmlData{tutor-start=1,tutor-end=2}{n} \htmlData{tutor-start=3,tutor-end=7}{\ge }\htmlData{tutor-start=7,tutor-end=8}{2}\htmlData{tutor-start=8,tutor-end=9}{)}$ 且 $\htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{0}}\htmlData{tutor-start=5,tutor-end=6}{=}\htmlData{tutor-start=6,tutor-end=7}{a}_{\htmlData{tutor-start=9,tutor-end=10}{1}}\htmlData{tutor-start=11,tutor-end=12}{=}\htmlData{tutor-start=12,tutor-end=13}{1}$ ，求 $\htmlData{tutor-start=0,tutor-end=2}{\{}\htmlData{tutor-start=2,tutor-end=3}{a}_{\htmlData{tutor-start=5,tutor-end=6}{n}}\htmlData{tutor-start=7,tutor-end=9}{\}}$ 的通项公式.

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。

二试/加试 · 一、（35 分） · 平面几何

设一凸四边形 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{B}\htmlData{tutor-start=2,tutor-end=3}{C}\htmlData{tutor-start=3,tutor-end=4}{D}$ ，它的内角中仅有 $\htmlData{tutor-start=0,tutor-end=7}{\angle }\htmlData{tutor-start=7,tutor-end=8}{D}$ 是钝角，用一些直线段将该凸四边形分割成 $\htmlData{tutor-start=0,tutor-end=1}{n}$ 个钝角三角形，但除去 $\htmlData{tutor-start=0,tutor-end=1}{A}$ 、 $\htmlData{tutor-start=0,tutor-end=1}{B}$ 、 $\htmlData{tutor-start=0,tutor-end=1}{C}$ 、 $\htmlData{tutor-start=0,tutor-end=1}{D}$ 外，在该四边形的周界上，不含分割出的钝角三角形顶点. 试证 $\htmlData{tutor-start=0,tutor-end=1}{n}$ 应满足的充分必要条件是 $\htmlData{tutor-start=0,tutor-end=1}{n} \htmlData{tutor-start=2,tutor-end=12}{\geqslant }\htmlData{tutor-start=12,tutor-end=13}{4}$ .

题解状态：标准答案与规范题解待补充

题目标签：SECOND-GENERAL-SOLUTION-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 · 二、（35 分） · 组合数学

设 $\htmlData{tutor-start=0,tutor-end=1}{A}$ 是一个有 $\htmlData{tutor-start=0,tutor-end=1}{n}$ 个元素的集合， $\htmlData{tutor-start=0,tutor-end=1}{A}$ 的 $\htmlData{tutor-start=0,tutor-end=1}{m}$ 个子集 $\htmlData{tutor-start=0,tutor-end=1}{A}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{A}_{\htmlData{tutor-start=10,tutor-end=11}{2}}\htmlData{tutor-start=12,tutor-end=13}{,} \cdots\htmlData{tutor-start=20,tutor-end=21}{,} \htmlData{tutor-start=22,tutor-end=23}{A}_{\htmlData{tutor-start=25,tutor-end=26}{m}}$ 两两互不包含. 试证：(1) $\sum_{\htmlData{tutor-start=6,tutor-end=7}{i}\htmlData{tutor-start=7,tutor-end=8}{=}\htmlData{tutor-start=8,tutor-end=9}{1}}^{\htmlData{tutor-start=12,tutor-end=13}{m}} \frac{\htmlData{tutor-start=21,tutor-end=22}{1}}{\htmlData{tutor-start=24,tutor-end=25}{C}_{\htmlData{tutor-start=27,tutor-end=28}{n}}^{\htmlData{tutor-start=31,tutor-end=32}{|}\htmlData{tutor-start=32,tutor-end=33}{A}_{\htmlData{tutor-start=35,tutor-end=36}{i}}\htmlData{tutor-start=37,tutor-end=38}{|}}} \htmlData{tutor-start=41,tutor-end=51}{\leqslant }\htmlData{tutor-start=51,tutor-end=52}{1}$ ; (2) $\sum_{\htmlData{tutor-start=6,tutor-end=7}{i}\htmlData{tutor-start=7,tutor-end=8}{=}\htmlData{tutor-start=8,tutor-end=9}{1}}^{\htmlData{tutor-start=12,tutor-end=13}{m}} \htmlData{tutor-start=15,tutor-end=16}{C}_{\htmlData{tutor-start=18,tutor-end=19}{n}}^{\htmlData{tutor-start=22,tutor-end=23}{|}\htmlData{tutor-start=23,tutor-end=24}{A}_{\htmlData{tutor-start=26,tutor-end=27}{i}}\htmlData{tutor-start=28,tutor-end=29}{|}} \htmlData{tutor-start=31,tutor-end=41}{\geqslant }\htmlData{tutor-start=41,tutor-end=42}{m}$ . 其中 $\htmlData{tutor-start=0,tutor-end=1}{|}\htmlData{tutor-start=1,tutor-end=2}{A}_{\htmlData{tutor-start=4,tutor-end=5}{i}}\htmlData{tutor-start=6,tutor-end=7}{|}$ 表示 $\htmlData{tutor-start=0,tutor-end=1}{A}_{\htmlData{tutor-start=3,tutor-end=4}{i}}$ 所含元素的个数， $\htmlData{tutor-start=0,tutor-end=1}{C}_{\htmlData{tutor-start=3,tutor-end=4}{n}}^{\htmlData{tutor-start=7,tutor-end=8}{|}\htmlData{tutor-start=8,tutor-end=9}{A}_{\htmlData{tutor-start=11,tutor-end=12}{i}}\htmlData{tutor-start=13,tutor-end=14}{|}}$ 表示 $\htmlData{tutor-start=0,tutor-end=1}{n}$ 个不同元素取 $\htmlData{tutor-start=0,tutor-end=1}{|}\htmlData{tutor-start=1,tutor-end=2}{A}_{\htmlData{tutor-start=4,tutor-end=5}{i}}\htmlData{tutor-start=6,tutor-end=7}{|}$ 个的组合数.

题解状态：标准答案与规范题解待补充

题目标签：SECOND-GENERAL-SOLUTION-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 · 三、（35 分） · 平面几何

水平直线 $\htmlData{tutor-start=0,tutor-end=1}{m}$ 通过圆 $\htmlData{tutor-start=0,tutor-end=1}{O}$ 的中心，直线 $\htmlData{tutor-start=0,tutor-end=1}{l} \htmlData{tutor-start=2,tutor-end=8}{\perp }\htmlData{tutor-start=8,tutor-end=9}{m}$ ， $\htmlData{tutor-start=0,tutor-end=1}{l}$ 与 $\htmlData{tutor-start=0,tutor-end=1}{m}$ 相交于 $\htmlData{tutor-start=0,tutor-end=1}{M}$ ，点 $\htmlData{tutor-start=0,tutor-end=1}{M}$ 在圆心的右侧，直线 $\htmlData{tutor-start=0,tutor-end=1}{l}$ 上不同的三点 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{B}\htmlData{tutor-start=4,tutor-end=5}{,} \htmlData{tutor-start=6,tutor-end=7}{C}$ 在圆外，且位于直线 $\htmlData{tutor-start=0,tutor-end=1}{m}$ 上方， $\htmlData{tutor-start=0,tutor-end=1}{A}$ 点离 $\htmlData{tutor-start=0,tutor-end=1}{M}$ 点最远， $\htmlData{tutor-start=0,tutor-end=1}{C}$ 点离 $\htmlData{tutor-start=0,tutor-end=1}{M}$ 点最近， $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{P}\htmlData{tutor-start=2,tutor-end=3}{,} \htmlData{tutor-start=4,tutor-end=5}{B}\htmlData{tutor-start=5,tutor-end=6}{Q}\htmlData{tutor-start=6,tutor-end=7}{,} \htmlData{tutor-start=8,tutor-end=9}{C}\htmlData{tutor-start=9,tutor-end=10}{R}$ 为圆 $\htmlData{tutor-start=0,tutor-end=1}{O}$ 的三条切线， $\htmlData{tutor-start=0,tutor-end=1}{P}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{Q}\htmlData{tutor-start=4,tutor-end=5}{,} \htmlData{tutor-start=6,tutor-end=7}{R}$ 为切点. 试证：(1) $\htmlData{tutor-start=0,tutor-end=1}{l}$ 与圆 $\htmlData{tutor-start=0,tutor-end=1}{O}$ 相切时， $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{B} \htmlData{tutor-start=3,tutor-end=10}{\times }\htmlData{tutor-start=10,tutor-end=11}{C}\htmlData{tutor-start=11,tutor-end=12}{R} \htmlData{tutor-start=13,tutor-end=14}{+} \htmlData{tutor-start=15,tutor-end=16}{B}\htmlData{tutor-start=16,tutor-end=17}{C} \htmlData{tutor-start=18,tutor-end=25}{\times }\htmlData{tutor-start=25,tutor-end=26}{A}\htmlData{tutor-start=26,tutor-end=27}{P} \htmlData{tutor-start=28,tutor-end=29}{=} \htmlData{tutor-start=30,tutor-end=31}{A}\htmlData{tutor-start=31,tutor-end=32}{C} \htmlData{tutor-start=33,tutor-end=40}{\times }\htmlData{tutor-start=40,tutor-end=41}{B}\htmlData{tutor-start=41,tutor-end=42}{Q}$ ; (2) $\htmlData{tutor-start=0,tutor-end=1}{l}$ 与圆 $\htmlData{tutor-start=0,tutor-end=1}{O}$ 相交时， $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{B} \htmlData{tutor-start=3,tutor-end=10}{\times }\htmlData{tutor-start=10,tutor-end=11}{C}\htmlData{tutor-start=11,tutor-end=12}{R} \htmlData{tutor-start=13,tutor-end=14}{+} \htmlData{tutor-start=15,tutor-end=16}{B}\htmlData{tutor-start=16,tutor-end=17}{C} \htmlData{tutor-start=18,tutor-end=25}{\times }\htmlData{tutor-start=25,tutor-end=26}{A}\htmlData{tutor-start=26,tutor-end=27}{P} \htmlData{tutor-start=28,tutor-end=29}{<} \htmlData{tutor-start=30,tutor-end=31}{A}\htmlData{tutor-start=31,tutor-end=32}{C} \htmlData{tutor-start=33,tutor-end=40}{\times }\htmlData{tutor-start=40,tutor-end=41}{B}\htmlData{tutor-start=41,tutor-end=42}{Q}$ ; (3) $\htmlData{tutor-start=0,tutor-end=1}{l}$ 与圆 $\htmlData{tutor-start=0,tutor-end=1}{O}$ 相离时， $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{B} \htmlData{tutor-start=3,tutor-end=10}{\times }\htmlData{tutor-start=10,tutor-end=11}{C}\htmlData{tutor-start=11,tutor-end=12}{R} \htmlData{tutor-start=13,tutor-end=14}{+} \htmlData{tutor-start=15,tutor-end=16}{B}\htmlData{tutor-start=16,tutor-end=17}{C} \htmlData{tutor-start=18,tutor-end=25}{\times }\htmlData{tutor-start=25,tutor-end=26}{A}\htmlData{tutor-start=26,tutor-end=27}{P} \htmlData{tutor-start=28,tutor-end=29}{>} \htmlData{tutor-start=30,tutor-end=31}{A}\htmlData{tutor-start=31,tutor-end=32}{C} \htmlData{tutor-start=33,tutor-end=40}{\times }\htmlData{tutor-start=40,tutor-end=41}{B}\htmlData{tutor-start=41,tutor-end=42}{Q}$ .

题解状态：标准答案与规范题解待补充

题目标签：SECOND-GENERAL-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。