2000 年全国高中数学联赛（学生版）

books/national_high_school_league_students/papers/competition-national-league-2000.pdf · HS-MATH-1024-v2.1-solution-aware

18 题0 个小问/题组

一试 · 第一试（10月15日上午8：00-9：40）选择题 · 数学竞赛/待细分

设全集是实数，若 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{=}\htmlData{tutor-start=2,tutor-end=4}{\{}\htmlData{tutor-start=4,tutor-end=5}{x}\htmlData{tutor-start=5,tutor-end=6}{|}\sqrt{\htmlData{tutor-start=12,tutor-end=13}{x}\htmlData{tutor-start=13,tutor-end=14}{-}\htmlData{tutor-start=14,tutor-end=15}{2}}\htmlData{tutor-start=16,tutor-end=26}{\leqslant }\htmlData{tutor-start=26,tutor-end=27}{0}\htmlData{tutor-start=27,tutor-end=29}{\}}$ ， $\htmlData{tutor-start=0,tutor-end=1}{B}\htmlData{tutor-start=1,tutor-end=2}{=}\htmlData{tutor-start=2,tutor-end=4}{\{}\htmlData{tutor-start=4,tutor-end=5}{x}\htmlData{tutor-start=5,tutor-end=6}{|}\htmlData{tutor-start=6,tutor-end=7}{1}\htmlData{tutor-start=7,tutor-end=8}{0}^{\htmlData{tutor-start=10,tutor-end=11}{x}^{\htmlData{tutor-start=13,tutor-end=14}{2}}\htmlData{tutor-start=15,tutor-end=16}{-}\htmlData{tutor-start=16,tutor-end=17}{2}}\htmlData{tutor-start=18,tutor-end=19}{=}\htmlData{tutor-start=19,tutor-end=20}{1}\htmlData{tutor-start=20,tutor-end=21}{0}^{\htmlData{tutor-start=23,tutor-end=24}{x}}\htmlData{tutor-start=25,tutor-end=27}{\}}$ ，则 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=6}{\cap }\overline{\htmlData{tutor-start=16,tutor-end=17}{B}}$ 是 ( ) (A) $\htmlData{tutor-start=0,tutor-end=2}{\{}\htmlData{tutor-start=2,tutor-end=3}{2}\htmlData{tutor-start=3,tutor-end=5}{\}}$ (B) $\htmlData{tutor-start=0,tutor-end=2}{\{}\htmlData{tutor-start=2,tutor-end=3}{-}\htmlData{tutor-start=3,tutor-end=4}{1}\htmlData{tutor-start=4,tutor-end=6}{\}}$ (C) $\htmlData{tutor-start=0,tutor-end=2}{\{}\htmlData{tutor-start=2,tutor-end=3}{x}\htmlData{tutor-start=3,tutor-end=4}{|}\htmlData{tutor-start=4,tutor-end=5}{x}\htmlData{tutor-start=5,tutor-end=15}{\leqslant }\htmlData{tutor-start=15,tutor-end=16}{2}\htmlData{tutor-start=16,tutor-end=18}{\}}$ (D) $\htmlData{tutor-start=0,tutor-end=11}{\varnothing}$

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 · 第一试（10月15日上午8：00-9：40）选择题 · 数学竞赛/待细分

设 $\sin\htmlData{tutor-start=4,tutor-end=10}{\alpha}\htmlData{tutor-start=10,tutor-end=11}{>}\htmlData{tutor-start=11,tutor-end=12}{0}\htmlData{tutor-start=12,tutor-end=13}{,} \cos\htmlData{tutor-start=18,tutor-end=24}{\alpha}\htmlData{tutor-start=24,tutor-end=25}{<}\htmlData{tutor-start=25,tutor-end=26}{0}$ ，且 $\sin\frac{\htmlData{tutor-start=10,tutor-end=16}{\alpha}}{\htmlData{tutor-start=18,tutor-end=19}{3}}\htmlData{tutor-start=20,tutor-end=21}{>}\cos\frac{\htmlData{tutor-start=31,tutor-end=37}{\alpha}}{\htmlData{tutor-start=39,tutor-end=40}{3}}$ ，则 $\frac{\htmlData{tutor-start=6,tutor-end=12}{\alpha}}{\htmlData{tutor-start=14,tutor-end=15}{3}}$ 的取值范围是 ( ) (A) $\htmlData{tutor-start=0,tutor-end=1}{(}\htmlData{tutor-start=1,tutor-end=2}{2}\htmlData{tutor-start=2,tutor-end=3}{k}\htmlData{tutor-start=3,tutor-end=6}{\pi}\htmlData{tutor-start=6,tutor-end=7}{+}\frac{\htmlData{tutor-start=13,tutor-end=16}{\pi}}{\htmlData{tutor-start=18,tutor-end=19}{6}}\htmlData{tutor-start=20,tutor-end=21}{,} \htmlData{tutor-start=22,tutor-end=23}{2}\htmlData{tutor-start=23,tutor-end=24}{k}\htmlData{tutor-start=24,tutor-end=27}{\pi}\htmlData{tutor-start=27,tutor-end=28}{+}\frac{\htmlData{tutor-start=34,tutor-end=37}{\pi}}{\htmlData{tutor-start=39,tutor-end=40}{3}}\htmlData{tutor-start=41,tutor-end=42}{)}\htmlData{tutor-start=42,tutor-end=43}{,} \htmlData{tutor-start=44,tutor-end=45}{k}\htmlData{tutor-start=45,tutor-end=49}{\in }\htmlData{tutor-start=49,tutor-end=50}{Z}$ (B) $\htmlData{tutor-start=0,tutor-end=1}{(}\frac{\htmlData{tutor-start=7,tutor-end=8}{2}\htmlData{tutor-start=8,tutor-end=9}{k}\htmlData{tutor-start=9,tutor-end=12}{\pi}}{\htmlData{tutor-start=14,tutor-end=15}{3}}\htmlData{tutor-start=16,tutor-end=17}{+}\frac{\htmlData{tutor-start=23,tutor-end=26}{\pi}}{\htmlData{tutor-start=28,tutor-end=29}{6}}\htmlData{tutor-start=30,tutor-end=31}{,} \frac{\htmlData{tutor-start=38,tutor-end=39}{2}\htmlData{tutor-start=39,tutor-end=40}{k}\htmlData{tutor-start=40,tutor-end=43}{\pi}}{\htmlData{tutor-start=45,tutor-end=46}{3}}\htmlData{tutor-start=47,tutor-end=48}{+}\frac{\htmlData{tutor-start=54,tutor-end=57}{\pi}}{\htmlData{tutor-start=59,tutor-end=60}{3}}\htmlData{tutor-start=61,tutor-end=62}{)}\htmlData{tutor-start=62,tutor-end=63}{,} \htmlData{tutor-start=64,tutor-end=65}{k}\htmlData{tutor-start=65,tutor-end=69}{\in }\htmlData{tutor-start=69,tutor-end=70}{Z}$ (C) $\htmlData{tutor-start=0,tutor-end=1}{(}\htmlData{tutor-start=1,tutor-end=2}{2}\htmlData{tutor-start=2,tutor-end=3}{k}\htmlData{tutor-start=3,tutor-end=6}{\pi}\htmlData{tutor-start=6,tutor-end=7}{+}\frac{\htmlData{tutor-start=13,tutor-end=14}{5}\htmlData{tutor-start=14,tutor-end=17}{\pi}}{\htmlData{tutor-start=19,tutor-end=20}{6}}\htmlData{tutor-start=21,tutor-end=22}{,} \htmlData{tutor-start=23,tutor-end=24}{2}\htmlData{tutor-start=24,tutor-end=25}{k}\htmlData{tutor-start=25,tutor-end=28}{\pi}\htmlData{tutor-start=28,tutor-end=29}{+}\htmlData{tutor-start=29,tutor-end=32}{\pi}\htmlData{tutor-start=32,tutor-end=33}{)}\htmlData{tutor-start=33,tutor-end=34}{,} \htmlData{tutor-start=35,tutor-end=36}{k}\htmlData{tutor-start=36,tutor-end=40}{\in }\htmlData{tutor-start=40,tutor-end=41}{Z}$ (D) $\htmlData{tutor-start=0,tutor-end=1}{(}\htmlData{tutor-start=1,tutor-end=2}{2}\htmlData{tutor-start=2,tutor-end=3}{k}\htmlData{tutor-start=3,tutor-end=6}{\pi}\htmlData{tutor-start=6,tutor-end=7}{+}\frac{\htmlData{tutor-start=13,tutor-end=16}{\pi}}{\htmlData{tutor-start=18,tutor-end=19}{4}}\htmlData{tutor-start=20,tutor-end=21}{,} \htmlData{tutor-start=22,tutor-end=23}{2}\htmlData{tutor-start=23,tutor-end=24}{k}\htmlData{tutor-start=24,tutor-end=27}{\pi}\htmlData{tutor-start=27,tutor-end=28}{+}\frac{\htmlData{tutor-start=34,tutor-end=37}{\pi}}{\htmlData{tutor-start=39,tutor-end=40}{3}}\htmlData{tutor-start=41,tutor-end=42}{)} \htmlData{tutor-start=43,tutor-end=48}{\cup }\htmlData{tutor-start=48,tutor-end=49}{(}\htmlData{tutor-start=49,tutor-end=50}{2}\htmlData{tutor-start=50,tutor-end=51}{k}\htmlData{tutor-start=51,tutor-end=54}{\pi}\htmlData{tutor-start=54,tutor-end=55}{+}\frac{\htmlData{tutor-start=61,tutor-end=62}{5}\htmlData{tutor-start=62,tutor-end=65}{\pi}}{\htmlData{tutor-start=67,tutor-end=68}{6}}\htmlData{tutor-start=69,tutor-end=70}{,} \htmlData{tutor-start=71,tutor-end=72}{2}\htmlData{tutor-start=72,tutor-end=73}{k}\htmlData{tutor-start=73,tutor-end=76}{\pi}\htmlData{tutor-start=76,tutor-end=77}{+}\htmlData{tutor-start=77,tutor-end=80}{\pi}\htmlData{tutor-start=80,tutor-end=81}{)}\htmlData{tutor-start=81,tutor-end=82}{,} \htmlData{tutor-start=83,tutor-end=84}{k}\htmlData{tutor-start=84,tutor-end=88}{\in }\htmlData{tutor-start=88,tutor-end=89}{Z}$

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 · 第一试（10月15日上午8：00-9：40）选择题 · 平面几何

已知点 $\htmlData{tutor-start=0,tutor-end=1}{A}$ 为双曲线 $\htmlData{tutor-start=0,tutor-end=1}{x}^{\htmlData{tutor-start=3,tutor-end=4}{2}}\htmlData{tutor-start=5,tutor-end=6}{-}\htmlData{tutor-start=6,tutor-end=7}{y}^{\htmlData{tutor-start=9,tutor-end=10}{2}}\htmlData{tutor-start=11,tutor-end=12}{=}\htmlData{tutor-start=12,tutor-end=13}{1}$ 的左顶点，点 $\htmlData{tutor-start=0,tutor-end=1}{B}$ 和点 $\htmlData{tutor-start=0,tutor-end=1}{C}$ 在双曲线的右分支上， $\htmlData{tutor-start=0,tutor-end=10}{\triangle }\htmlData{tutor-start=10,tutor-end=11}{A}\htmlData{tutor-start=11,tutor-end=12}{B}\htmlData{tutor-start=12,tutor-end=13}{C}$ 是等边三角形，则 $\htmlData{tutor-start=0,tutor-end=10}{\triangle }\htmlData{tutor-start=10,tutor-end=11}{A}\htmlData{tutor-start=11,tutor-end=12}{B}\htmlData{tutor-start=12,tutor-end=13}{C}$ 的面积是 ( ) (A) $\frac{\sqrt{\htmlData{tutor-start=12,tutor-end=13}{3}}}{\htmlData{tutor-start=16,tutor-end=17}{3}}$ (B) $\frac{\htmlData{tutor-start=6,tutor-end=7}{3}\sqrt{\htmlData{tutor-start=13,tutor-end=14}{3}}}{\htmlData{tutor-start=17,tutor-end=18}{2}}$ (C) $\htmlData{tutor-start=0,tutor-end=1}{3}\sqrt{\htmlData{tutor-start=7,tutor-end=8}{3}}$ (D) $\htmlData{tutor-start=0,tutor-end=1}{6}\sqrt{\htmlData{tutor-start=7,tutor-end=8}{3}}$

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 · 第一试（10月15日上午8：00-9：40）选择题 · 代数

给定正数 $\htmlData{tutor-start=0,tutor-end=1}{p}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{q}\htmlData{tutor-start=4,tutor-end=5}{,} \htmlData{tutor-start=6,tutor-end=7}{a}\htmlData{tutor-start=7,tutor-end=8}{,} \htmlData{tutor-start=9,tutor-end=10}{b}\htmlData{tutor-start=10,tutor-end=11}{,} \htmlData{tutor-start=12,tutor-end=13}{c}$ ，其中 $\htmlData{tutor-start=0,tutor-end=1}{p}\neq \htmlData{tutor-start=6,tutor-end=7}{q}$ ，若 $\htmlData{tutor-start=0,tutor-end=1}{p}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{a}\htmlData{tutor-start=4,tutor-end=5}{,} \htmlData{tutor-start=6,tutor-end=7}{q}$ 是等比数列， $\htmlData{tutor-start=0,tutor-end=1}{p}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{b}\htmlData{tutor-start=4,tutor-end=5}{,} \htmlData{tutor-start=6,tutor-end=7}{c}\htmlData{tutor-start=7,tutor-end=8}{,} \htmlData{tutor-start=9,tutor-end=10}{q}$ 是等差数列，则一元二次方程 $\htmlData{tutor-start=0,tutor-end=1}{b}\htmlData{tutor-start=1,tutor-end=2}{x}^{\htmlData{tutor-start=4,tutor-end=5}{2}}\htmlData{tutor-start=6,tutor-end=7}{-}\htmlData{tutor-start=7,tutor-end=8}{2}\htmlData{tutor-start=8,tutor-end=9}{a}\htmlData{tutor-start=9,tutor-end=10}{x}\htmlData{tutor-start=10,tutor-end=11}{+}\htmlData{tutor-start=11,tutor-end=12}{c}\htmlData{tutor-start=12,tutor-end=13}{=}\htmlData{tutor-start=13,tutor-end=14}{0}$ ( ) (A) 无实根 (B) 有两个相等实根 (C) 有两个同号相异实根 (D) 有两个异号实根

题解状态：标准答案与规范题解待补充

题目标签：FIRST-CHOICE-P2-2

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 · 第一试（10月15日上午8：00-9：40）选择题 · 平面几何

平面上整点（纵、横坐标都是整数的点）到直线 [图] 的距离中的最小值是 (A) $\frac{\sqrt{\htmlData{tutor-start=12,tutor-end=13}{3}\htmlData{tutor-start=13,tutor-end=14}{4}}}{\htmlData{tutor-start=17,tutor-end=18}{1}\htmlData{tutor-start=18,tutor-end=19}{7}\htmlData{tutor-start=19,tutor-end=20}{0}}$ (B) $\frac{\sqrt{\htmlData{tutor-start=12,tutor-end=13}{3}\htmlData{tutor-start=13,tutor-end=14}{4}}}{\htmlData{tutor-start=17,tutor-end=18}{8}\htmlData{tutor-start=18,tutor-end=19}{5}}$ (C) $\frac{\htmlData{tutor-start=6,tutor-end=7}{1}}{\htmlData{tutor-start=9,tutor-end=10}{2}\htmlData{tutor-start=10,tutor-end=11}{0}}$ (D) $\frac{\htmlData{tutor-start=6,tutor-end=7}{1}}{\htmlData{tutor-start=9,tutor-end=10}{3}\htmlData{tutor-start=10,tutor-end=11}{0}}$ ( )

**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-CHOICE-P3-2

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 · 第一试（10月15日上午8：00-9：40）选择题 · 数学竞赛/待细分

设 $\htmlData{tutor-start=0,tutor-end=6}{\omega}\htmlData{tutor-start=6,tutor-end=7}{=}\cos\frac{\htmlData{tutor-start=17,tutor-end=20}{\pi}}{\htmlData{tutor-start=22,tutor-end=23}{5}}\htmlData{tutor-start=24,tutor-end=25}{+}\htmlData{tutor-start=25,tutor-end=26}{i}\sin\frac{\htmlData{tutor-start=36,tutor-end=39}{\pi}}{\htmlData{tutor-start=41,tutor-end=42}{5}}$ ，则以 $\htmlData{tutor-start=0,tutor-end=6}{\omega}\htmlData{tutor-start=6,tutor-end=7}{,} \htmlData{tutor-start=8,tutor-end=14}{\omega}^{\htmlData{tutor-start=16,tutor-end=17}{3}}\htmlData{tutor-start=18,tutor-end=19}{,} \htmlData{tutor-start=20,tutor-end=26}{\omega}^{\htmlData{tutor-start=28,tutor-end=29}{7}}\htmlData{tutor-start=30,tutor-end=31}{,} \htmlData{tutor-start=32,tutor-end=38}{\omega}^{\htmlData{tutor-start=40,tutor-end=41}{9}}$ 为根的方程是 ( ) (A) $\htmlData{tutor-start=0,tutor-end=1}{x}^{\htmlData{tutor-start=3,tutor-end=4}{4}}\htmlData{tutor-start=5,tutor-end=6}{+}\htmlData{tutor-start=6,tutor-end=7}{x}^{\htmlData{tutor-start=9,tutor-end=10}{3}}\htmlData{tutor-start=11,tutor-end=12}{+}\htmlData{tutor-start=12,tutor-end=13}{x}^{\htmlData{tutor-start=15,tutor-end=16}{2}}\htmlData{tutor-start=17,tutor-end=18}{+}\htmlData{tutor-start=18,tutor-end=19}{x}\htmlData{tutor-start=19,tutor-end=20}{+}\htmlData{tutor-start=20,tutor-end=21}{1}\htmlData{tutor-start=21,tutor-end=22}{=}\htmlData{tutor-start=22,tutor-end=23}{0}$ (B) $\htmlData{tutor-start=0,tutor-end=1}{x}^{\htmlData{tutor-start=3,tutor-end=4}{4}}\htmlData{tutor-start=5,tutor-end=6}{-}\htmlData{tutor-start=6,tutor-end=7}{x}^{\htmlData{tutor-start=9,tutor-end=10}{3}}\htmlData{tutor-start=11,tutor-end=12}{+}\htmlData{tutor-start=12,tutor-end=13}{x}^{\htmlData{tutor-start=15,tutor-end=16}{2}}\htmlData{tutor-start=17,tutor-end=18}{-}\htmlData{tutor-start=18,tutor-end=19}{x}\htmlData{tutor-start=19,tutor-end=20}{+}\htmlData{tutor-start=20,tutor-end=21}{1}\htmlData{tutor-start=21,tutor-end=22}{=}\htmlData{tutor-start=22,tutor-end=23}{0}$ (C) $\htmlData{tutor-start=0,tutor-end=1}{x}^{\htmlData{tutor-start=3,tutor-end=4}{4}}\htmlData{tutor-start=5,tutor-end=6}{-}\htmlData{tutor-start=6,tutor-end=7}{x}^{\htmlData{tutor-start=9,tutor-end=10}{3}}\htmlData{tutor-start=11,tutor-end=12}{-}\htmlData{tutor-start=12,tutor-end=13}{x}^{\htmlData{tutor-start=15,tutor-end=16}{2}}\htmlData{tutor-start=17,tutor-end=18}{+}\htmlData{tutor-start=18,tutor-end=19}{x}\htmlData{tutor-start=19,tutor-end=20}{+}\htmlData{tutor-start=20,tutor-end=21}{1}\htmlData{tutor-start=21,tutor-end=22}{=}\htmlData{tutor-start=22,tutor-end=23}{0}$ (D) $\htmlData{tutor-start=0,tutor-end=1}{x}^{\htmlData{tutor-start=3,tutor-end=4}{4}}\htmlData{tutor-start=5,tutor-end=6}{+}\htmlData{tutor-start=6,tutor-end=7}{x}^{\htmlData{tutor-start=9,tutor-end=10}{3}}\htmlData{tutor-start=11,tutor-end=12}{+}\htmlData{tutor-start=12,tutor-end=13}{x}^{\htmlData{tutor-start=15,tutor-end=16}{2}}\htmlData{tutor-start=17,tutor-end=18}{-}\htmlData{tutor-start=18,tutor-end=19}{x}\htmlData{tutor-start=19,tutor-end=20}{-}\htmlData{tutor-start=20,tutor-end=21}{1}\htmlData{tutor-start=21,tutor-end=22}{=}\htmlData{tutor-start=22,tutor-end=23}{0}$

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 · 二、填空题（本题满分 54 分，每小题 9 分）本题共有 6 小题，要求直接将答案写在横线上。 · 数学竞赛/待细分

$\arcsin(\sin 2000^\circ)=$ __________.

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（本题满分 54 分，每小题 9 分）本题共有 6 小题，要求直接将答案写在横线上。 · 数学竞赛/待细分

设 $\htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{n}}$ 是 $\htmlData{tutor-start=0,tutor-end=1}{(}\htmlData{tutor-start=1,tutor-end=2}{3}\htmlData{tutor-start=2,tutor-end=3}{-}\sqrt{\htmlData{tutor-start=9,tutor-end=10}{x}}\htmlData{tutor-start=11,tutor-end=12}{)}^{\htmlData{tutor-start=14,tutor-end=15}{n}}$ 的展开式中 $\htmlData{tutor-start=0,tutor-end=1}{x}$ 项的系数（ $\htmlData{tutor-start=0,tutor-end=1}{n}\htmlData{tutor-start=1,tutor-end=2}{=}\htmlData{tutor-start=2,tutor-end=3}{2}\htmlData{tutor-start=3,tutor-end=4}{,} \htmlData{tutor-start=5,tutor-end=6}{3}\htmlData{tutor-start=6,tutor-end=7}{,} \htmlData{tutor-start=8,tutor-end=9}{4}\htmlData{tutor-start=9,tutor-end=10}{,} \cdots$ ），则 $\htmlData{tutor-start=0,tutor-end=7}{\lim_{n} \htmlData{tutor-start=8,tutor-end=12}{\to }\htmlData{tutor-start=12,tutor-end=18}{\infty}} \htmlData{tutor-start=20,tutor-end=21}{(}\frac{\htmlData{tutor-start=27,tutor-end=28}{3}^{\htmlData{tutor-start=30,tutor-end=31}{2}}}{\htmlData{tutor-start=34,tutor-end=35}{a}_{\htmlData{tutor-start=37,tutor-end=38}{2}}} \htmlData{tutor-start=41,tutor-end=42}{+} \frac{\htmlData{tutor-start=49,tutor-end=50}{3}^{\htmlData{tutor-start=52,tutor-end=53}{3}}}{\htmlData{tutor-start=56,tutor-end=57}{a}_{\htmlData{tutor-start=59,tutor-end=60}{3}}} \htmlData{tutor-start=63,tutor-end=64}{+} \cdots \htmlData{tutor-start=72,tutor-end=73}{+} \frac{\htmlData{tutor-start=80,tutor-end=81}{3}^{\htmlData{tutor-start=83,tutor-end=84}{n}}}{\htmlData{tutor-start=87,tutor-end=88}{a}_{\htmlData{tutor-start=90,tutor-end=91}{n}}}\htmlData{tutor-start=93,tutor-end=94}{)} \htmlData{tutor-start=95,tutor-end=96}{=} \underline{\hspace{2cm}}\htmlData{tutor-start=121,tutor-end=122}{.}$

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 · 填空题 · 代数

等比数列 $\htmlData{tutor-start=0,tutor-end=1}{a}\htmlData{tutor-start=1,tutor-end=2}{+}\log_{\htmlData{tutor-start=8,tutor-end=9}{2}} \htmlData{tutor-start=11,tutor-end=12}{3}\htmlData{tutor-start=12,tutor-end=13}{,} \htmlData{tutor-start=14,tutor-end=15}{a}\htmlData{tutor-start=15,tutor-end=16}{+}\log_{\htmlData{tutor-start=22,tutor-end=23}{4}} \htmlData{tutor-start=25,tutor-end=26}{3}\htmlData{tutor-start=26,tutor-end=27}{,} \htmlData{tutor-start=28,tutor-end=29}{a}\htmlData{tutor-start=29,tutor-end=30}{+}\log_{\htmlData{tutor-start=36,tutor-end=37}{8}} \htmlData{tutor-start=39,tutor-end=40}{3}$ 的公比是 $\underline{\hspace{2cm}}$ .

题解状态：标准答案与规范题解待补充

题目标签：FIRST-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 · 填空题 · 平面几何

在椭圆 $\frac{\htmlData{tutor-start=6,tutor-end=7}{x}^{\htmlData{tutor-start=9,tutor-end=10}{2}}}{\htmlData{tutor-start=13,tutor-end=14}{a}^{\htmlData{tutor-start=16,tutor-end=17}{2}}} \htmlData{tutor-start=20,tutor-end=21}{+} \frac{\htmlData{tutor-start=28,tutor-end=29}{y}^{\htmlData{tutor-start=31,tutor-end=32}{2}}}{\htmlData{tutor-start=35,tutor-end=36}{b}^{\htmlData{tutor-start=38,tutor-end=39}{2}}} \htmlData{tutor-start=42,tutor-end=43}{=} \htmlData{tutor-start=44,tutor-end=45}{1} \htmlData{tutor-start=46,tutor-end=47}{(}\htmlData{tutor-start=47,tutor-end=48}{a}\htmlData{tutor-start=48,tutor-end=49}{>}\htmlData{tutor-start=49,tutor-end=50}{b}\htmlData{tutor-start=50,tutor-end=51}{>}\htmlData{tutor-start=51,tutor-end=52}{0}\htmlData{tutor-start=52,tutor-end=53}{)}$ 中，记左焦点为 $\htmlData{tutor-start=0,tutor-end=1}{F}$ ，右顶点为 $\htmlData{tutor-start=0,tutor-end=1}{A}$ ，短轴上方的端点为 $\htmlData{tutor-start=0,tutor-end=1}{B}$ . 若该椭圆的离心率是 $\frac{\sqrt{\htmlData{tutor-start=12,tutor-end=13}{5}}\htmlData{tutor-start=14,tutor-end=15}{-}\htmlData{tutor-start=15,tutor-end=16}{1}}{\htmlData{tutor-start=18,tutor-end=19}{2}}$ ，则 $\htmlData{tutor-start=0,tutor-end=7}{\angle }\htmlData{tutor-start=7,tutor-end=8}{A}\htmlData{tutor-start=8,tutor-end=9}{B}\htmlData{tutor-start=9,tutor-end=10}{F} \htmlData{tutor-start=11,tutor-end=12}{=} \underline{\hspace{2cm}}$ .

题解状态：标准答案与规范题解待补充

题目标签：FIRST-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 · 填空题 · 数学竞赛/待细分

一个球与正四面体的六条棱都相切，若正四面体的棱长为 $\htmlData{tutor-start=0,tutor-end=1}{a}$ ，则这个球的体积是 $\underline{\hspace{2cm}}$ .

题解状态：标准答案与规范题解待补充

题目标签：FIRST-FILL-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 · 填空题 · 数学竞赛/待细分

如果：(1) $\htmlData{tutor-start=0,tutor-end=1}{a}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{b}\htmlData{tutor-start=4,tutor-end=5}{,} \htmlData{tutor-start=6,tutor-end=7}{c}\htmlData{tutor-start=7,tutor-end=8}{,} \htmlData{tutor-start=9,tutor-end=10}{d}$ 都属于 $\htmlData{tutor-start=0,tutor-end=2}{\{}\htmlData{tutor-start=2,tutor-end=3}{1}\htmlData{tutor-start=3,tutor-end=4}{,} \htmlData{tutor-start=5,tutor-end=6}{2}\htmlData{tutor-start=6,tutor-end=7}{,} \htmlData{tutor-start=8,tutor-end=9}{3}\htmlData{tutor-start=9,tutor-end=10}{,} \htmlData{tutor-start=11,tutor-end=12}{4}\htmlData{tutor-start=12,tutor-end=14}{\}}$ ；(2) $\htmlData{tutor-start=0,tutor-end=1}{a} \neq \htmlData{tutor-start=7,tutor-end=8}{b}\htmlData{tutor-start=8,tutor-end=9}{,} \htmlData{tutor-start=10,tutor-end=11}{b} \neq \htmlData{tutor-start=17,tutor-end=18}{c}\htmlData{tutor-start=18,tutor-end=19}{,} \htmlData{tutor-start=20,tutor-end=21}{c} \neq \htmlData{tutor-start=27,tutor-end=28}{d}\htmlData{tutor-start=28,tutor-end=29}{,} \htmlData{tutor-start=30,tutor-end=31}{d}\htmlData{tutor-start=31,tutor-end=36}{ \neq} \htmlData{tutor-start=37,tutor-end=38}{a}$ ；(3) $\htmlData{tutor-start=0,tutor-end=1}{a}$ 是 $\htmlData{tutor-start=0,tutor-end=1}{a}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{b}\htmlData{tutor-start=4,tutor-end=5}{,} \htmlData{tutor-start=6,tutor-end=7}{c}\htmlData{tutor-start=7,tutor-end=8}{,} \htmlData{tutor-start=9,tutor-end=10}{d}$ 中的最小值，那么，可以组成的不同的四位数 $\overline{\htmlData{tutor-start=10,tutor-end=11}{a}\htmlData{tutor-start=11,tutor-end=12}{b}\htmlData{tutor-start=12,tutor-end=13}{c}\htmlData{tutor-start=13,tutor-end=14}{d}}$ 的个数是 $\underline{\hspace{2cm}}$ .

题解状态：标准答案与规范题解待补充

题目标签：FIRST-FILL-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 · 三、解答题 (本题满分 60 分，每小题 20 分) · 数学竞赛/待细分

设 $\htmlData{tutor-start=0,tutor-end=1}{S}_{\htmlData{tutor-start=3,tutor-end=4}{n}} \htmlData{tutor-start=6,tutor-end=7}{=} \htmlData{tutor-start=8,tutor-end=9}{1}\htmlData{tutor-start=9,tutor-end=10}{+}\htmlData{tutor-start=10,tutor-end=11}{2}\htmlData{tutor-start=11,tutor-end=12}{+}\htmlData{tutor-start=12,tutor-end=13}{3}\htmlData{tutor-start=13,tutor-end=14}{+}\cdots\htmlData{tutor-start=20,tutor-end=21}{+}\htmlData{tutor-start=21,tutor-end=22}{n}\htmlData{tutor-start=22,tutor-end=23}{,} \htmlData{tutor-start=24,tutor-end=25}{n} \htmlData{tutor-start=26,tutor-end=30}{\in }\mathbb{\htmlData{tutor-start=38,tutor-end=39}{N}}$ . 求 $\htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{n}\htmlData{tutor-start=3,tutor-end=4}{)} \htmlData{tutor-start=5,tutor-end=6}{=} \frac{\htmlData{tutor-start=13,tutor-end=14}{S}_{\htmlData{tutor-start=16,tutor-end=17}{n}}}{\htmlData{tutor-start=20,tutor-end=21}{(}\htmlData{tutor-start=21,tutor-end=22}{n}\htmlData{tutor-start=22,tutor-end=23}{+}\htmlData{tutor-start=23,tutor-end=24}{3}\htmlData{tutor-start=24,tutor-end=25}{2}\htmlData{tutor-start=25,tutor-end=26}{)}\htmlData{tutor-start=26,tutor-end=27}{S}_{\htmlData{tutor-start=29,tutor-end=30}{n}\htmlData{tutor-start=30,tutor-end=31}{+}\htmlData{tutor-start=31,tutor-end=32}{1}}}$ 的最大值.

题解状态：标准答案与规范题解待补充

题目标签：FIRST-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。

一试 · 三、解答题 (本题满分 60 分，每小题 20 分) · 代数

若函数 $\htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{x}\htmlData{tutor-start=3,tutor-end=4}{)} \htmlData{tutor-start=5,tutor-end=6}{=} \htmlData{tutor-start=7,tutor-end=8}{-}\frac{\htmlData{tutor-start=14,tutor-end=15}{1}}{\htmlData{tutor-start=17,tutor-end=18}{2}}\htmlData{tutor-start=19,tutor-end=20}{x}^{\htmlData{tutor-start=22,tutor-end=23}{2}} \htmlData{tutor-start=25,tutor-end=26}{+} \frac{\htmlData{tutor-start=33,tutor-end=34}{1}\htmlData{tutor-start=34,tutor-end=35}{3}}{\htmlData{tutor-start=37,tutor-end=38}{2}}$ 在区间 $\htmlData{tutor-start=0,tutor-end=1}{[}\htmlData{tutor-start=1,tutor-end=2}{a}\htmlData{tutor-start=2,tutor-end=3}{,} \htmlData{tutor-start=4,tutor-end=5}{b}\htmlData{tutor-start=5,tutor-end=6}{]}$ 上的最小值为 $\htmlData{tutor-start=0,tutor-end=1}{2}\htmlData{tutor-start=1,tutor-end=2}{a}$ ，最大值为 $\htmlData{tutor-start=0,tutor-end=1}{2}\htmlData{tutor-start=1,tutor-end=2}{b}$ ，求 $\htmlData{tutor-start=0,tutor-end=1}{[}\htmlData{tutor-start=1,tutor-end=2}{a}\htmlData{tutor-start=2,tutor-end=3}{,} \htmlData{tutor-start=4,tutor-end=5}{b}\htmlData{tutor-start=5,tutor-end=6}{]}$ .

题解状态：标准答案与规范题解待补充

题目标签：FIRST-SOLUTION-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。

一试 · 三、解答题 (本题满分 60 分，每小题 20 分) · 平面几何

已知 $\htmlData{tutor-start=0,tutor-end=1}{C}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{:} \htmlData{tutor-start=7,tutor-end=8}{x}^{\htmlData{tutor-start=10,tutor-end=11}{2}}\htmlData{tutor-start=12,tutor-end=13}{+}\htmlData{tutor-start=13,tutor-end=14}{y}^{\htmlData{tutor-start=16,tutor-end=17}{2}}\htmlData{tutor-start=18,tutor-end=19}{=}\htmlData{tutor-start=19,tutor-end=20}{1}$ 和 $\htmlData{tutor-start=0,tutor-end=1}{C}_{\htmlData{tutor-start=3,tutor-end=4}{2}}\htmlData{tutor-start=5,tutor-end=6}{:} \frac{\htmlData{tutor-start=13,tutor-end=14}{x}^{\htmlData{tutor-start=16,tutor-end=17}{2}}}{\htmlData{tutor-start=20,tutor-end=21}{a}^{\htmlData{tutor-start=23,tutor-end=24}{2}}} \htmlData{tutor-start=27,tutor-end=28}{+} \frac{\htmlData{tutor-start=35,tutor-end=36}{y}^{\htmlData{tutor-start=38,tutor-end=39}{2}}}{\htmlData{tutor-start=42,tutor-end=43}{b}^{\htmlData{tutor-start=45,tutor-end=46}{2}}} \htmlData{tutor-start=49,tutor-end=50}{=} \htmlData{tutor-start=51,tutor-end=52}{1} \htmlData{tutor-start=53,tutor-end=54}{(}\htmlData{tutor-start=54,tutor-end=55}{a}\htmlData{tutor-start=55,tutor-end=56}{>}\htmlData{tutor-start=56,tutor-end=57}{b}\htmlData{tutor-start=57,tutor-end=58}{>}\htmlData{tutor-start=58,tutor-end=59}{0}\htmlData{tutor-start=59,tutor-end=60}{)}$ . 试问：当且仅当 $\htmlData{tutor-start=0,tutor-end=1}{a}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{b}$ 满足什么条件时，对 $\htmlData{tutor-start=0,tutor-end=1}{C}_{\htmlData{tutor-start=3,tutor-end=4}{2}}$ 上任意一点 $\htmlData{tutor-start=0,tutor-end=1}{P}$ ，均存在以 $\htmlData{tutor-start=0,tutor-end=1}{P}$ 为顶点，与 $\htmlData{tutor-start=0,tutor-end=1}{C}_{\htmlData{tutor-start=3,tutor-end=4}{1}}$ 外切，与 $\htmlData{tutor-start=0,tutor-end=1}{C}_{\htmlData{tutor-start=3,tutor-end=4}{2}}$ 内接的平行四边形？并证明你的结论.

题解状态：标准答案与规范题解待补充

题目标签：FIRST-SOLUTION-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 · 【加试】 (10 月 15 日上午 10:00-12:00) · 平面几何

（本题满分 50 分）如图，在锐角三角形 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{B}\htmlData{tutor-start=2,tutor-end=3}{C}$ 的 $\htmlData{tutor-start=0,tutor-end=1}{B}\htmlData{tutor-start=1,tutor-end=2}{C}$ 边上有两点 $\htmlData{tutor-start=0,tutor-end=1}{E}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{F}$ ，满足 $\htmlData{tutor-start=0,tutor-end=7}{\angle }\htmlData{tutor-start=7,tutor-end=8}{B}\htmlData{tutor-start=8,tutor-end=9}{A}\htmlData{tutor-start=9,tutor-end=10}{E} \htmlData{tutor-start=11,tutor-end=12}{=} \htmlData{tutor-start=13,tutor-end=20}{\angle }\htmlData{tutor-start=20,tutor-end=21}{C}\htmlData{tutor-start=21,tutor-end=22}{A}\htmlData{tutor-start=22,tutor-end=23}{F}$ ，作 $\htmlData{tutor-start=0,tutor-end=1}{F}\htmlData{tutor-start=1,tutor-end=2}{M} \htmlData{tutor-start=3,tutor-end=9}{\perp }\htmlData{tutor-start=9,tutor-end=10}{A}\htmlData{tutor-start=10,tutor-end=11}{B}\htmlData{tutor-start=11,tutor-end=12}{,} \htmlData{tutor-start=13,tutor-end=14}{F}\htmlData{tutor-start=14,tutor-end=15}{N} \htmlData{tutor-start=16,tutor-end=22}{\perp }\htmlData{tutor-start=22,tutor-end=23}{A}\htmlData{tutor-start=23,tutor-end=24}{C}$ ( $\htmlData{tutor-start=0,tutor-end=1}{M}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{N}$ 是垂足)，延长 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{E}$ 交三角形 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{B}\htmlData{tutor-start=2,tutor-end=3}{C}$ 的外接圆于 $\htmlData{tutor-start=0,tutor-end=1}{D}$ . 证明：四边形 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{M}\htmlData{tutor-start=2,tutor-end=3}{D}\htmlData{tutor-start=3,tutor-end=4}{N}$ 与三角形 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{B}\htmlData{tutor-start=2,tutor-end=3}{C}$ 的面积相等.

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

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 · 【加试】 (10 月 15 日上午 10:00-12:00) · 代数

（本题满分 50 分）设数列 $\htmlData{tutor-start=0,tutor-end=2}{\{}\htmlData{tutor-start=2,tutor-end=3}{a}_{\htmlData{tutor-start=5,tutor-end=6}{n}}\htmlData{tutor-start=7,tutor-end=9}{\}}$ 和 $\htmlData{tutor-start=0,tutor-end=2}{\{}\htmlData{tutor-start=2,tutor-end=3}{b}_{\htmlData{tutor-start=5,tutor-end=6}{n}}\htmlData{tutor-start=7,tutor-end=9}{\}}$ 满足，且 $\begin{cases} a_{n+1} = 7a_{n} + 6b_{n} - 3 \\ b_{n+1} = 8a_{n} + 7b_{n} - 4 \end{cases} \quad n=0,1,2,\cdots$ 证明 $\htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{n}}$ ( $\htmlData{tutor-start=0,tutor-end=1}{n}\htmlData{tutor-start=1,tutor-end=2}{=}\htmlData{tutor-start=2,tutor-end=3}{0}\htmlData{tutor-start=3,tutor-end=4}{,} \htmlData{tutor-start=5,tutor-end=6}{1}\htmlData{tutor-start=6,tutor-end=7}{,} \htmlData{tutor-start=8,tutor-end=9}{2}\htmlData{tutor-start=9,tutor-end=10}{,} \cdots$ ) 是完全平方数.

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 · 【加试】 (10 月 15 日上午 10:00-12:00) · 数学竞赛/待细分

（本题满分 50 分）有 $\htmlData{tutor-start=0,tutor-end=1}{n}$ 个人，已知他们中的任意两人至多通电话一次，他们中的任意 $\htmlData{tutor-start=0,tutor-end=1}{n}\htmlData{tutor-start=1,tutor-end=2}{-}\htmlData{tutor-start=2,tutor-end=3}{2}$ 个人之间通电话的次数相等，都是 $\htmlData{tutor-start=0,tutor-end=1}{3}^{\htmlData{tutor-start=3,tutor-end=4}{k}}$ 次，其中 $\htmlData{tutor-start=0,tutor-end=1}{k}$ 是自然数，求 $\htmlData{tutor-start=0,tutor-end=1}{n}$ 的所有可能值.

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。