1991 年全国高中数学联赛（学生版）

books/national_high_school_league_students/papers/competition-national-league-1991.pdf · HS-MATH-1024-v2.1-solution-aware

18 题0 个小问/题组

一试 · 一试题 · 平面几何

由一个正方体的三个顶点所能构成的正三角形的个数为( ) A. 4 B. 8 C. 12 D. 24

题解状态：标准答案与规范题解待补充

题目标签：FIRST-PROBLEM-P1

解题过程

该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。

一试 · 一试题 · 数学竞赛/待细分

设 $\htmlData{tutor-start=0,tutor-end=1}{a}$ 、 $\htmlData{tutor-start=0,tutor-end=1}{b}$ 、 $\htmlData{tutor-start=0,tutor-end=1}{c}$ 均为非零复数，且 $\frac{\htmlData{tutor-start=6,tutor-end=7}{a}}{\htmlData{tutor-start=9,tutor-end=10}{b}}\htmlData{tutor-start=11,tutor-end=12}{=}\frac{\htmlData{tutor-start=18,tutor-end=19}{b}}{\htmlData{tutor-start=21,tutor-end=22}{c}}\htmlData{tutor-start=23,tutor-end=24}{=}\frac{\htmlData{tutor-start=30,tutor-end=31}{c}}{\htmlData{tutor-start=33,tutor-end=34}{a}}$ ，则 $\frac{\htmlData{tutor-start=6,tutor-end=7}{a}\htmlData{tutor-start=7,tutor-end=8}{+}\htmlData{tutor-start=8,tutor-end=9}{b}\htmlData{tutor-start=9,tutor-end=10}{-}\htmlData{tutor-start=10,tutor-end=11}{c}}{\htmlData{tutor-start=13,tutor-end=14}{a}\htmlData{tutor-start=14,tutor-end=15}{-}\htmlData{tutor-start=15,tutor-end=16}{b}\htmlData{tutor-start=16,tutor-end=17}{+}\htmlData{tutor-start=17,tutor-end=18}{c}}$ 的值为( ) A. 1 B. $\htmlData{tutor-start=0,tutor-end=4}{\pm }\htmlData{tutor-start=4,tutor-end=10}{\omega}$ C. $\htmlData{tutor-start=0,tutor-end=1}{1}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=9}{\omega}\htmlData{tutor-start=9,tutor-end=10}{,} \htmlData{tutor-start=11,tutor-end=17}{\omega}^{\htmlData{tutor-start=19,tutor-end=20}{2}}$ D. $\htmlData{tutor-start=0,tutor-end=1}{1}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{-}\htmlData{tutor-start=4,tutor-end=10}{\omega}\htmlData{tutor-start=10,tutor-end=11}{,} \htmlData{tutor-start=12,tutor-end=13}{-}\htmlData{tutor-start=13,tutor-end=19}{\omega}^{\htmlData{tutor-start=21,tutor-end=22}{2}}$

题解状态：标准答案与规范题解待补充

题目标签：FIRST-PROBLEM-P2

解题过程

该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。

一试 · 一试题 · 数论

设 $\htmlData{tutor-start=0,tutor-end=1}{a}$ 是正整数， $\htmlData{tutor-start=0,tutor-end=1}{a}\htmlData{tutor-start=1,tutor-end=2}{<}\htmlData{tutor-start=2,tutor-end=3}{1}\htmlData{tutor-start=3,tutor-end=4}{0}\htmlData{tutor-start=4,tutor-end=5}{0}$ ，并且 $\htmlData{tutor-start=0,tutor-end=1}{a}^{\htmlData{tutor-start=3,tutor-end=4}{3}}\htmlData{tutor-start=5,tutor-end=6}{+}\htmlData{tutor-start=6,tutor-end=7}{2}\htmlData{tutor-start=7,tutor-end=8}{3}$ 能被 24 整除，那么，这样的 $\htmlData{tutor-start=0,tutor-end=1}{a}$ 的个数为( ) A. 4 B. 5 C. 9 D. 10

题解状态：标准答案与规范题解待补充

题目标签：FIRST-PROBLEM-P3

解题过程

该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。

一试 · 一试题 · 代数

设函数 $\htmlData{tutor-start=0,tutor-end=1}{y}\htmlData{tutor-start=1,tutor-end=2}{=}\htmlData{tutor-start=2,tutor-end=3}{f}\htmlData{tutor-start=3,tutor-end=4}{(}\htmlData{tutor-start=4,tutor-end=5}{x}\htmlData{tutor-start=5,tutor-end=6}{)}$ 对于一切实数 $\htmlData{tutor-start=0,tutor-end=1}{x}$ 满足 $\htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{3}\htmlData{tutor-start=3,tutor-end=4}{+}\htmlData{tutor-start=4,tutor-end=5}{x}\htmlData{tutor-start=5,tutor-end=6}{)}\htmlData{tutor-start=6,tutor-end=7}{=}\htmlData{tutor-start=7,tutor-end=8}{f}\htmlData{tutor-start=8,tutor-end=9}{(}\htmlData{tutor-start=9,tutor-end=10}{3}\htmlData{tutor-start=10,tutor-end=11}{-}\htmlData{tutor-start=11,tutor-end=12}{x}\htmlData{tutor-start=12,tutor-end=13}{)}$ ，且方程 $\htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{x}\htmlData{tutor-start=3,tutor-end=4}{)}\htmlData{tutor-start=4,tutor-end=5}{=}\htmlData{tutor-start=5,tutor-end=6}{0}$ 恰有 6 个不同的实数根，则这 6 个实根的和为( ) A. 18 B. 12 C. 9 D. 0

题解状态：标准答案与规范题解待补充

题目标签：FIRST-PROBLEM-P4

解题过程

该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。

一试 · 一试题 · 组合数学

设 $\htmlData{tutor-start=0,tutor-end=1}{S}\htmlData{tutor-start=1,tutor-end=2}{=}\htmlData{tutor-start=2,tutor-end=4}{\{}\htmlData{tutor-start=4,tutor-end=5}{(}\htmlData{tutor-start=5,tutor-end=6}{x}\htmlData{tutor-start=6,tutor-end=7}{,} \htmlData{tutor-start=8,tutor-end=9}{y}\htmlData{tutor-start=9,tutor-end=10}{)} \htmlData{tutor-start=11,tutor-end=16}{\mid }\htmlData{tutor-start=16,tutor-end=17}{x}^{\htmlData{tutor-start=19,tutor-end=20}{2}}\htmlData{tutor-start=21,tutor-end=22}{-}\htmlData{tutor-start=22,tutor-end=23}{y}^{\htmlData{tutor-start=25,tutor-end=26}{2}}\htmlData{tutor-start=27,tutor-end=28}{=}\text{\htmlData{tutor-start=34,tutor-end=35}{奇}\htmlData{tutor-start=35,tutor-end=36}{数}}\htmlData{tutor-start=37,tutor-end=38}{,} \htmlData{tutor-start=39,tutor-end=40}{x}\htmlData{tutor-start=40,tutor-end=41}{,} \htmlData{tutor-start=42,tutor-end=43}{y} \htmlData{tutor-start=44,tutor-end=48}{\in }\mathbb{\htmlData{tutor-start=56,tutor-end=57}{R}}\htmlData{tutor-start=58,tutor-end=60}{\}}$ ， $\htmlData{tutor-start=0,tutor-end=1}{T}\htmlData{tutor-start=1,tutor-end=2}{=}\htmlData{tutor-start=2,tutor-end=4}{\{}\htmlData{tutor-start=4,tutor-end=5}{(}\htmlData{tutor-start=5,tutor-end=6}{x}\htmlData{tutor-start=6,tutor-end=7}{,} \htmlData{tutor-start=8,tutor-end=9}{y}\htmlData{tutor-start=9,tutor-end=10}{)} \htmlData{tutor-start=11,tutor-end=16}{\mid }\sin\htmlData{tutor-start=20,tutor-end=21}{(}\htmlData{tutor-start=21,tutor-end=22}{2}\htmlData{tutor-start=22,tutor-end=26}{\pi }\htmlData{tutor-start=26,tutor-end=27}{x}^{\htmlData{tutor-start=29,tutor-end=30}{2}}\htmlData{tutor-start=31,tutor-end=32}{)}\htmlData{tutor-start=32,tutor-end=33}{-}\sin\htmlData{tutor-start=37,tutor-end=38}{(}\htmlData{tutor-start=38,tutor-end=39}{2}\htmlData{tutor-start=39,tutor-end=43}{\pi }\htmlData{tutor-start=43,tutor-end=44}{y}^{\htmlData{tutor-start=46,tutor-end=47}{2}}\htmlData{tutor-start=48,tutor-end=49}{)}\htmlData{tutor-start=49,tutor-end=50}{=}\cos\htmlData{tutor-start=54,tutor-end=55}{(}\htmlData{tutor-start=55,tutor-end=56}{2}\htmlData{tutor-start=56,tutor-end=60}{\pi }\htmlData{tutor-start=60,tutor-end=61}{x}^{\htmlData{tutor-start=63,tutor-end=64}{2}}\htmlData{tutor-start=65,tutor-end=66}{)}\htmlData{tutor-start=66,tutor-end=67}{-}\cos\htmlData{tutor-start=71,tutor-end=72}{(}\htmlData{tutor-start=72,tutor-end=73}{2}\htmlData{tutor-start=73,tutor-end=77}{\pi }\htmlData{tutor-start=77,tutor-end=78}{y}^{\htmlData{tutor-start=80,tutor-end=81}{2}}\htmlData{tutor-start=82,tutor-end=83}{)}\htmlData{tutor-start=83,tutor-end=84}{,} \htmlData{tutor-start=85,tutor-end=86}{x}\htmlData{tutor-start=86,tutor-end=87}{,} \htmlData{tutor-start=88,tutor-end=89}{y} \htmlData{tutor-start=90,tutor-end=94}{\in }\mathbb{\htmlData{tutor-start=102,tutor-end=103}{R}}\htmlData{tutor-start=104,tutor-end=106}{\}}$ ，则( ) A. $\htmlData{tutor-start=0,tutor-end=1}{S} \htmlData{tutor-start=2,tutor-end=13}{\subsetneq }\htmlData{tutor-start=13,tutor-end=14}{T}$ B. $\htmlData{tutor-start=0,tutor-end=1}{T} \htmlData{tutor-start=2,tutor-end=13}{\subsetneq }\htmlData{tutor-start=13,tutor-end=14}{S}$ C. $\htmlData{tutor-start=0,tutor-end=1}{S}\htmlData{tutor-start=1,tutor-end=2}{=}\htmlData{tutor-start=2,tutor-end=3}{T}$ D. $\htmlData{tutor-start=0,tutor-end=1}{S} \htmlData{tutor-start=2,tutor-end=7}{\cap }\htmlData{tutor-start=7,tutor-end=8}{T}\htmlData{tutor-start=8,tutor-end=9}{=}\htmlData{tutor-start=9,tutor-end=20}{\varnothing}$

题解状态：标准答案与规范题解待补充

题目标签：FIRST-PROBLEM-P5

解题过程

该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。

一试 · 一试题 · 数学竞赛/待细分

方程 $\htmlData{tutor-start=0,tutor-end=1}{|}\htmlData{tutor-start=1,tutor-end=2}{x}\htmlData{tutor-start=2,tutor-end=3}{-}\htmlData{tutor-start=3,tutor-end=4}{y}^{\htmlData{tutor-start=6,tutor-end=7}{2}}\htmlData{tutor-start=8,tutor-end=9}{|}\htmlData{tutor-start=9,tutor-end=10}{=}\htmlData{tutor-start=10,tutor-end=11}{1}\htmlData{tutor-start=11,tutor-end=12}{-}\htmlData{tutor-start=12,tutor-end=13}{|}\htmlData{tutor-start=13,tutor-end=14}{x}\htmlData{tutor-start=14,tutor-end=15}{|}$ 的图象为( )

**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-PROBLEM-P6

解题过程

该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。

一试 · 一试题 · 数学竞赛/待细分

$\cos^{2} 10^\circ + \cos^{2} 50^\circ - \sin 40^\circ \sin 80^\circ =$ ________.

题解状态：标准答案与规范题解待补充

题目标签：FIRST-PROBLEM-P1-2

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 · 一试题 · 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，已知三个角 $\htmlData{tutor-start=0,tutor-end=1}{A}$ 、 $\htmlData{tutor-start=0,tutor-end=1}{B}$ 、 $\htmlData{tutor-start=0,tutor-end=1}{C}$ 成等差数列，假设它们所对的边分别为 $\htmlData{tutor-start=0,tutor-end=1}{a}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{b}\htmlData{tutor-start=4,tutor-end=5}{,} \htmlData{tutor-start=6,tutor-end=7}{c}$ ，并且 $\htmlData{tutor-start=0,tutor-end=1}{c}\htmlData{tutor-start=1,tutor-end=2}{-}\htmlData{tutor-start=2,tutor-end=3}{a}$ 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 $\sin \frac{\htmlData{tutor-start=11,tutor-end=12}{C}\htmlData{tutor-start=12,tutor-end=13}{-}\htmlData{tutor-start=13,tutor-end=14}{A}}{\htmlData{tutor-start=16,tutor-end=17}{2}} \htmlData{tutor-start=19,tutor-end=20}{=}$ ________.

题解状态：标准答案与规范题解待补充

题目标签：FIRST-PROBLEM-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 · 一试题 · 组合数学

将正奇数集合 $\htmlData{tutor-start=0,tutor-end=2}{\{}\htmlData{tutor-start=2,tutor-end=3}{1}\htmlData{tutor-start=3,tutor-end=4}{,} \htmlData{tutor-start=5,tutor-end=6}{3}\htmlData{tutor-start=6,tutor-end=7}{,} \htmlData{tutor-start=8,tutor-end=9}{5}\htmlData{tutor-start=9,tutor-end=10}{,} \cdots\htmlData{tutor-start=17,tutor-end=19}{\}}$ 由小到大按第 $\htmlData{tutor-start=0,tutor-end=1}{n}$ 组有 $\htmlData{tutor-start=0,tutor-end=1}{(}\htmlData{tutor-start=1,tutor-end=2}{2}\htmlData{tutor-start=2,tutor-end=3}{n}\htmlData{tutor-start=3,tutor-end=4}{-}\htmlData{tutor-start=4,tutor-end=5}{1}\htmlData{tutor-start=5,tutor-end=6}{)}$ 个奇数进行分组： $\htmlData{tutor-start=0,tutor-end=2}{\{}\htmlData{tutor-start=2,tutor-end=3}{1}\htmlData{tutor-start=3,tutor-end=5}{\}}\htmlData{tutor-start=5,tutor-end=6}{,} \quad \htmlData{tutor-start=13,tutor-end=15}{\{}\htmlData{tutor-start=15,tutor-end=16}{3}\htmlData{tutor-start=16,tutor-end=17}{,} \htmlData{tutor-start=18,tutor-end=19}{5}\htmlData{tutor-start=19,tutor-end=20}{,} \htmlData{tutor-start=21,tutor-end=22}{7}\htmlData{tutor-start=22,tutor-end=24}{\}}\htmlData{tutor-start=24,tutor-end=25}{,} \quad \htmlData{tutor-start=32,tutor-end=34}{\{}\htmlData{tutor-start=34,tutor-end=35}{9}\htmlData{tutor-start=35,tutor-end=36}{,} \htmlData{tutor-start=37,tutor-end=38}{1}\htmlData{tutor-start=38,tutor-end=39}{1}\htmlData{tutor-start=39,tutor-end=40}{,} \htmlData{tutor-start=41,tutor-end=42}{1}\htmlData{tutor-start=42,tutor-end=43}{3}\htmlData{tutor-start=43,tutor-end=44}{,} \htmlData{tutor-start=45,tutor-end=46}{1}\htmlData{tutor-start=46,tutor-end=47}{5}\htmlData{tutor-start=47,tutor-end=48}{,} \htmlData{tutor-start=49,tutor-end=50}{1}\htmlData{tutor-start=50,tutor-end=51}{7}\htmlData{tutor-start=51,tutor-end=53}{\}}\htmlData{tutor-start=53,tutor-end=54}{,} \cdots\cdots$ (第一组) (第二组) (第三组) 则 1991 位于第________组.

题解状态：标准答案与规范题解待补充

题目标签：FIRST-PROBLEM-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 · 一试题 · 数学竞赛/待细分

$\htmlData{tutor-start=0,tutor-end=1}{1}\htmlData{tutor-start=1,tutor-end=2}{9}\htmlData{tutor-start=2,tutor-end=3}{9}\htmlData{tutor-start=3,tutor-end=4}{1}^{\htmlData{tutor-start=6,tutor-end=7}{2}\htmlData{tutor-start=7,tutor-end=8}{0}\htmlData{tutor-start=8,tutor-end=9}{0}\htmlData{tutor-start=9,tutor-end=10}{0}}$ 除以 $\htmlData{tutor-start=0,tutor-end=1}{1}\htmlData{tutor-start=1,tutor-end=2}{0}^{\htmlData{tutor-start=4,tutor-end=5}{6}}$ ，余数是________.

题解状态：标准答案与规范题解待补充

题目标签：FIRST-PROBLEM-P4-2

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 · 一试题 · 数学竞赛/待细分

设复数 $\htmlData{tutor-start=0,tutor-end=1}{z}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{z}_{\htmlData{tutor-start=10,tutor-end=11}{2}}$ 满足 $\htmlData{tutor-start=0,tutor-end=1}{|}\htmlData{tutor-start=1,tutor-end=2}{z}_{\htmlData{tutor-start=4,tutor-end=5}{1}}\htmlData{tutor-start=6,tutor-end=7}{|}\htmlData{tutor-start=7,tutor-end=8}{=}\htmlData{tutor-start=8,tutor-end=9}{|}\htmlData{tutor-start=9,tutor-end=10}{z}_{\htmlData{tutor-start=12,tutor-end=13}{1}}\htmlData{tutor-start=14,tutor-end=15}{+}\htmlData{tutor-start=15,tutor-end=16}{z}_{\htmlData{tutor-start=18,tutor-end=19}{2}}\htmlData{tutor-start=20,tutor-end=21}{|}\htmlData{tutor-start=21,tutor-end=22}{=}\htmlData{tutor-start=22,tutor-end=23}{3}$ ， $\htmlData{tutor-start=0,tutor-end=1}{|}\htmlData{tutor-start=1,tutor-end=2}{z}_{\htmlData{tutor-start=4,tutor-end=5}{1}}\htmlData{tutor-start=6,tutor-end=7}{-}\htmlData{tutor-start=7,tutor-end=8}{z}_{\htmlData{tutor-start=10,tutor-end=11}{2}}\htmlData{tutor-start=12,tutor-end=13}{|}\htmlData{tutor-start=13,tutor-end=14}{=}\htmlData{tutor-start=14,tutor-end=15}{3}\sqrt{\htmlData{tutor-start=21,tutor-end=22}{3}}$ ，则 $\log_{\htmlData{tutor-start=6,tutor-end=7}{3}} \htmlData{tutor-start=9,tutor-end=10}{|}\htmlData{tutor-start=10,tutor-end=11}{(}\htmlData{tutor-start=11,tutor-end=12}{z}_{\htmlData{tutor-start=14,tutor-end=15}{1}} \overline{\htmlData{tutor-start=27,tutor-end=28}{z}_{\htmlData{tutor-start=30,tutor-end=31}{2}}}\htmlData{tutor-start=33,tutor-end=34}{)}^{\htmlData{tutor-start=36,tutor-end=37}{2}\htmlData{tutor-start=37,tutor-end=38}{0}\htmlData{tutor-start=38,tutor-end=39}{0}\htmlData{tutor-start=39,tutor-end=40}{0}} \htmlData{tutor-start=42,tutor-end=43}{+} \htmlData{tutor-start=44,tutor-end=45}{(}\overline{\htmlData{tutor-start=55,tutor-end=56}{z}_{\htmlData{tutor-start=58,tutor-end=59}{1}}} \htmlData{tutor-start=62,tutor-end=63}{z}_{\htmlData{tutor-start=65,tutor-end=66}{2}}\htmlData{tutor-start=67,tutor-end=68}{)}^{\htmlData{tutor-start=70,tutor-end=71}{2}\htmlData{tutor-start=71,tutor-end=72}{0}\htmlData{tutor-start=72,tutor-end=73}{0}\htmlData{tutor-start=73,tutor-end=74}{0}}\htmlData{tutor-start=75,tutor-end=76}{|} \htmlData{tutor-start=77,tutor-end=78}{=}$ ________.

题解状态：标准答案与规范题解待补充

题目标签：FIRST-PROBLEM-P5-2

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 · 一试题 · 组合数学

设集合 $\htmlData{tutor-start=0,tutor-end=1}{M}\htmlData{tutor-start=1,tutor-end=2}{=}\htmlData{tutor-start=2,tutor-end=4}{\{}\htmlData{tutor-start=4,tutor-end=5}{1}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{2}\htmlData{tutor-start=8,tutor-end=9}{,} \cdots\htmlData{tutor-start=16,tutor-end=17}{,} \htmlData{tutor-start=18,tutor-end=19}{1}\htmlData{tutor-start=19,tutor-end=20}{0}\htmlData{tutor-start=20,tutor-end=21}{0}\htmlData{tutor-start=21,tutor-end=22}{0}\htmlData{tutor-start=22,tutor-end=24}{\}}$ ，现对 $\htmlData{tutor-start=0,tutor-end=1}{M}$ 中的任一非空子集 $\htmlData{tutor-start=0,tutor-end=1}{X}$ ，令 $\htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{X}}$ 表示 $\htmlData{tutor-start=0,tutor-end=1}{X}$ 中最大数与最小数的和. 那么，所有这样的 $\htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{X}}$ 的算术平均值为________.

题解状态：标准答案与规范题解待补充

题目标签：FIRST-PROBLEM-P6-2

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 · 一试题 · 数学竞赛/待细分

设正三棱锥 $\htmlData{tutor-start=0,tutor-end=1}{P}\htmlData{tutor-start=1,tutor-end=2}{-}\htmlData{tutor-start=2,tutor-end=3}{A}\htmlData{tutor-start=3,tutor-end=4}{B}\htmlData{tutor-start=4,tutor-end=5}{C}$ 的高为 $\htmlData{tutor-start=0,tutor-end=1}{P}\htmlData{tutor-start=1,tutor-end=2}{O}$ ， $\htmlData{tutor-start=0,tutor-end=1}{M}$ 为 $\htmlData{tutor-start=0,tutor-end=1}{P}\htmlData{tutor-start=1,tutor-end=2}{O}$ 的中点，过 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{M}$ 作与棱 $\htmlData{tutor-start=0,tutor-end=1}{B}\htmlData{tutor-start=1,tutor-end=2}{C}$ 平行的平面，将三棱锥截为上、下两部分，试求此两部分的体积比.

题解状态：标准答案与规范题解待补充

题目标签：FIRST-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。

一试 · 一试题 · 平面几何

设 $\htmlData{tutor-start=0,tutor-end=1}{O}$ 为抛物线的顶点， $\htmlData{tutor-start=0,tutor-end=1}{F}$ 为焦点，且 $\htmlData{tutor-start=0,tutor-end=1}{P}\htmlData{tutor-start=1,tutor-end=2}{Q}$ 为过 $\htmlData{tutor-start=0,tutor-end=1}{F}$ 的弦. 已知 $\htmlData{tutor-start=0,tutor-end=1}{|}\htmlData{tutor-start=1,tutor-end=2}{O}\htmlData{tutor-start=2,tutor-end=3}{F}\htmlData{tutor-start=3,tutor-end=4}{|}\htmlData{tutor-start=4,tutor-end=5}{=}\htmlData{tutor-start=5,tutor-end=6}{a}$ ， $\htmlData{tutor-start=0,tutor-end=1}{|}\htmlData{tutor-start=1,tutor-end=2}{P}\htmlData{tutor-start=2,tutor-end=3}{Q}\htmlData{tutor-start=3,tutor-end=4}{|}\htmlData{tutor-start=4,tutor-end=5}{=}\htmlData{tutor-start=5,tutor-end=6}{b}$ . 求 $\htmlData{tutor-start=0,tutor-end=10}{\triangle }\htmlData{tutor-start=10,tutor-end=11}{O}\htmlData{tutor-start=11,tutor-end=12}{P}\htmlData{tutor-start=12,tutor-end=13}{Q}$ 的面积.

题解状态：标准答案与规范题解待补充

题目标签：FIRST-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。

一试 · 一试题 · 数学竞赛/待细分

已知 $\htmlData{tutor-start=0,tutor-end=1}{0}\htmlData{tutor-start=1,tutor-end=2}{<}\htmlData{tutor-start=2,tutor-end=3}{a}\htmlData{tutor-start=3,tutor-end=4}{<}\htmlData{tutor-start=4,tutor-end=5}{1}$ ， $\htmlData{tutor-start=0,tutor-end=1}{x}^{\htmlData{tutor-start=3,tutor-end=4}{2}}\htmlData{tutor-start=5,tutor-end=6}{+}\htmlData{tutor-start=6,tutor-end=7}{y}\htmlData{tutor-start=7,tutor-end=8}{=}\htmlData{tutor-start=8,tutor-end=9}{0}$ ，求证： $\log_{\htmlData{tutor-start=6,tutor-end=7}{a}}\htmlData{tutor-start=8,tutor-end=9}{(}\htmlData{tutor-start=9,tutor-end=10}{a}^{\htmlData{tutor-start=12,tutor-end=13}{x}}\htmlData{tutor-start=14,tutor-end=15}{+}\htmlData{tutor-start=15,tutor-end=16}{a}^{\htmlData{tutor-start=18,tutor-end=19}{y}}\htmlData{tutor-start=20,tutor-end=21}{)} \htmlData{tutor-start=22,tutor-end=32}{\leqslant }\log_{\htmlData{tutor-start=38,tutor-end=39}{a}} \htmlData{tutor-start=41,tutor-end=42}{2} \htmlData{tutor-start=43,tutor-end=44}{+} \frac{\htmlData{tutor-start=51,tutor-end=52}{1}}{\htmlData{tutor-start=54,tutor-end=55}{8}}\htmlData{tutor-start=56,tutor-end=57}{.}$

题解状态：标准答案与规范题解待补充

题目标签：FIRST-PROBLEM-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 · 二试 · 代数

设 $\htmlData{tutor-start=0,tutor-end=1}{S}\htmlData{tutor-start=1,tutor-end=2}{=}\htmlData{tutor-start=2,tutor-end=4}{\{}\htmlData{tutor-start=4,tutor-end=5}{1}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{2}\htmlData{tutor-start=8,tutor-end=9}{,} \cdots\htmlData{tutor-start=16,tutor-end=17}{,} \htmlData{tutor-start=18,tutor-end=19}{n}\htmlData{tutor-start=19,tutor-end=21}{\}}$ ， $\htmlData{tutor-start=0,tutor-end=1}{A}$ 为至少含有两项的公差为正的等差数列，其项都在 $\htmlData{tutor-start=0,tutor-end=1}{S}$ 中，且添加 $\htmlData{tutor-start=0,tutor-end=1}{S}$ 的其他元素于 $\htmlData{tutor-start=0,tutor-end=1}{A}$ 后不能构成与 $\htmlData{tutor-start=0,tutor-end=1}{A}$ 有相同公差的等差数列。求这种 $\htmlData{tutor-start=0,tutor-end=1}{A}$ 的个数（这里只有两项的数列也看作等差数列）。

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 · 二试 · 平面几何

设凸四边形 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{B}\htmlData{tutor-start=2,tutor-end=3}{C}\htmlData{tutor-start=3,tutor-end=4}{D}$ 的面积为 1，求证：在它的边上（包括顶点）或内部可以找出四个点，使得以其中任意三点为顶点所构成的四个三角形的面积大于 $\frac{\htmlData{tutor-start=6,tutor-end=7}{1}}{\htmlData{tutor-start=9,tutor-end=10}{4}}$ 。

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 · 二试 · 数学竞赛/待细分

设 $\htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{n}}$ 是下述自然数 $\htmlData{tutor-start=0,tutor-end=1}{N}$ 的个数： $\htmlData{tutor-start=0,tutor-end=1}{N}$ 的各位数字之和为 $\htmlData{tutor-start=0,tutor-end=1}{n}$ 且每位数字只能取 1、3 或 4。求证： $\htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{2}\htmlData{tutor-start=4,tutor-end=5}{n}}$ 是完全平方数。这里， $\htmlData{tutor-start=0,tutor-end=1}{n}\htmlData{tutor-start=1,tutor-end=2}{=}\htmlData{tutor-start=2,tutor-end=3}{1}\htmlData{tutor-start=3,tutor-end=4}{,} \htmlData{tutor-start=5,tutor-end=6}{2}\htmlData{tutor-start=6,tutor-end=7}{,} \cdots$ 。

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。