2022 年全国高中数学联赛 B 卷（学生版）

books/national_high_school_league_students/papers/competition-national-league-2022-b.pdf · HS-MATH-1024-v2.1-solution-aware

15 题0 个小问/题组

一试（B 卷） · 一、填空题：本大题共 8 小题，每小题 8 分，满分 64 分. · 代数

不等式 $\frac{\htmlData{tutor-start=6,tutor-end=7}{2}\htmlData{tutor-start=7,tutor-end=8}{0}}{\htmlData{tutor-start=10,tutor-end=11}{x}\htmlData{tutor-start=11,tutor-end=12}{-}\htmlData{tutor-start=12,tutor-end=13}{9}} \htmlData{tutor-start=15,tutor-end=16}{>} \frac{\htmlData{tutor-start=23,tutor-end=24}{2}\htmlData{tutor-start=24,tutor-end=25}{2}}{\htmlData{tutor-start=27,tutor-end=28}{x}\htmlData{tutor-start=28,tutor-end=29}{-}\htmlData{tutor-start=29,tutor-end=30}{1}\htmlData{tutor-start=30,tutor-end=31}{1}}$ 的解集为______.

题解状态：标准答案与规范题解待补充

题目标签：FIRST-B-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（B 卷） · 一、填空题：本大题共 8 小题，每小题 8 分，满分 64 分. · 平面几何

在平面直角坐标系中，以抛物线 $\htmlData{tutor-start=0,tutor-end=6}{\Gamma}\htmlData{tutor-start=6,tutor-end=7}{:} \htmlData{tutor-start=8,tutor-end=9}{y}^{\htmlData{tutor-start=11,tutor-end=12}{2}} \htmlData{tutor-start=14,tutor-end=15}{=} \htmlData{tutor-start=16,tutor-end=17}{6}\htmlData{tutor-start=17,tutor-end=18}{x}$ 的焦点为圆心作一个圆 $\htmlData{tutor-start=0,tutor-end=6}{\Omega}$ ，与 $\htmlData{tutor-start=0,tutor-end=6}{\Gamma}$ 的准线相切，则圆 $\htmlData{tutor-start=0,tutor-end=6}{\Omega}$ 的面积为______.

题解状态：标准答案与规范题解待补充

题目标签：FIRST-B-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（B 卷） · 一、填空题：本大题共 8 小题，每小题 8 分，满分 64 分. · 代数

函数 $\htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{x}\htmlData{tutor-start=3,tutor-end=4}{)} \htmlData{tutor-start=5,tutor-end=6}{=} \lg \htmlData{tutor-start=11,tutor-end=12}{2} \htmlData{tutor-start=13,tutor-end=19}{\cdot }\lg \htmlData{tutor-start=23,tutor-end=24}{5} \htmlData{tutor-start=25,tutor-end=26}{-} \lg \htmlData{tutor-start=31,tutor-end=32}{2}\htmlData{tutor-start=32,tutor-end=33}{x} \htmlData{tutor-start=34,tutor-end=40}{\cdot }\lg \htmlData{tutor-start=44,tutor-end=45}{5}\htmlData{tutor-start=45,tutor-end=46}{x}$ 的最大值为______.

题解状态：标准答案与规范题解待补充

题目标签：FIRST-B-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（B 卷） · 一、填空题：本大题共 8 小题，每小题 8 分，满分 64 分. · 数学竞赛/待细分

一枚不均匀的硬币，若随机抛掷它两次均得到正面的概率为 $\frac{\htmlData{tutor-start=6,tutor-end=7}{1}}{\htmlData{tutor-start=9,tutor-end=10}{2}}$ ，则随机抛掷它两次得到正面、反面各一次的概率为______.

题解状态：标准答案与规范题解待补充

题目标签：FIRST-B-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（B 卷） · 一、填空题：本大题共 8 小题，每小题 8 分，满分 64 分. · 数学竞赛/待细分

已知复数 $\htmlData{tutor-start=0,tutor-end=1}{z}$ 满足 $\htmlData{tutor-start=0,tutor-end=1}{|}\htmlData{tutor-start=1,tutor-end=2}{z}\htmlData{tutor-start=2,tutor-end=3}{|} \htmlData{tutor-start=4,tutor-end=5}{=} \htmlData{tutor-start=6,tutor-end=7}{1}$ ，且 $\text{\htmlData{tutor-start=6,tutor-end=7}{R}\htmlData{tutor-start=7,tutor-end=8}{e}}\frac{\htmlData{tutor-start=15,tutor-end=16}{z}\htmlData{tutor-start=16,tutor-end=17}{+}\htmlData{tutor-start=17,tutor-end=18}{1}}{\bar{\htmlData{tutor-start=25,tutor-end=26}{z}}\htmlData{tutor-start=27,tutor-end=28}{+}\htmlData{tutor-start=28,tutor-end=29}{1}} \htmlData{tutor-start=31,tutor-end=32}{=} \frac{\htmlData{tutor-start=39,tutor-end=40}{1}}{\htmlData{tutor-start=42,tutor-end=43}{3}}$ ，则 $\text{\htmlData{tutor-start=6,tutor-end=7}{R}\htmlData{tutor-start=7,tutor-end=8}{e}}\frac{\htmlData{tutor-start=15,tutor-end=16}{z}}{\bar{\htmlData{tutor-start=23,tutor-end=24}{z}}}$ 的值为______.

题解状态：标准答案与规范题解待补充

题目标签：FIRST-B-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（B 卷） · 一、填空题：本大题共 8 小题，每小题 8 分，满分 64 分. · 平面几何

若正四棱锥 $\htmlData{tutor-start=0,tutor-end=1}{P}\htmlData{tutor-start=1,tutor-end=2}{-}\htmlData{tutor-start=2,tutor-end=3}{A}\htmlData{tutor-start=3,tutor-end=4}{B}\htmlData{tutor-start=4,tutor-end=5}{C}\htmlData{tutor-start=5,tutor-end=6}{D}$ 的各条棱长均相等， $\htmlData{tutor-start=0,tutor-end=1}{M}$ 为棱 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{B}$ 的中点，则异面直线 $\htmlData{tutor-start=0,tutor-end=1}{B}\htmlData{tutor-start=1,tutor-end=2}{P}$ 与 $\htmlData{tutor-start=0,tutor-end=1}{C}\htmlData{tutor-start=1,tutor-end=2}{M}$ 所成的角的余弦值为______.

题解状态：标准答案与规范题解待补充

题目标签：FIRST-B-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（B 卷） · 一、填空题：本大题共 8 小题，每小题 8 分，满分 64 分. · 平面几何

若 $\htmlData{tutor-start=0,tutor-end=10}{\triangle }\htmlData{tutor-start=10,tutor-end=11}{A}\htmlData{tutor-start=11,tutor-end=12}{B}\htmlData{tutor-start=12,tutor-end=13}{C}$ 的三个内角 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{B}\htmlData{tutor-start=4,tutor-end=5}{,} \htmlData{tutor-start=6,tutor-end=7}{C}$ 满足 $\cos \htmlData{tutor-start=5,tutor-end=6}{A} \htmlData{tutor-start=7,tutor-end=8}{=} \sin \htmlData{tutor-start=14,tutor-end=15}{B} \htmlData{tutor-start=16,tutor-end=17}{=} \htmlData{tutor-start=18,tutor-end=19}{2}\tan\frac{\htmlData{tutor-start=29,tutor-end=30}{C}}{\htmlData{tutor-start=32,tutor-end=33}{2}}$ ，则 $\sin \htmlData{tutor-start=5,tutor-end=6}{A} \htmlData{tutor-start=7,tutor-end=8}{+} \cos \htmlData{tutor-start=14,tutor-end=15}{A} \htmlData{tutor-start=16,tutor-end=17}{+} \htmlData{tutor-start=18,tutor-end=19}{2}\tan \htmlData{tutor-start=24,tutor-end=25}{A}$ 的值为

题解状态：标准答案与规范题解待补充

题目标签：FIRST-B-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（B 卷） · 一、填空题：本大题共 8 小题，每小题 8 分，满分 64 分. · 组合数学

一个单位方格的四条边中，若存在三条边染了三种不同的颜色，则称该单位方格是“多彩”的. 如图，一个 $\htmlData{tutor-start=0,tutor-end=1}{1} \htmlData{tutor-start=2,tutor-end=9}{\times }\htmlData{tutor-start=9,tutor-end=10}{3}$ 方格表的表格线共含 10 条单位长线段，现要对这 10 条线段染色，每条线段染为红、黄、蓝三色之一，使得三个单位方格都是多采的. 这样的染色方式数为______.

**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-B-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（B 卷） · 二、解答题：本大题共 3 小题，满分 56 分. 解答应写出文字说明、证明过程或演算步骤. · 数学竞赛/待细分

(本题满分 16 分) 在平面直角坐标系中， $\htmlData{tutor-start=0,tutor-end=1}{F}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{F}_{\htmlData{tutor-start=10,tutor-end=11}{2}}$ 是双曲线 $\htmlData{tutor-start=0,tutor-end=6}{\Gamma}\htmlData{tutor-start=6,tutor-end=7}{:} \frac{\htmlData{tutor-start=14,tutor-end=15}{x}^{\htmlData{tutor-start=17,tutor-end=18}{2}}}{\htmlData{tutor-start=21,tutor-end=22}{3}} \htmlData{tutor-start=24,tutor-end=25}{-} \htmlData{tutor-start=26,tutor-end=27}{y}^{\htmlData{tutor-start=29,tutor-end=30}{2}} \htmlData{tutor-start=32,tutor-end=33}{=} \htmlData{tutor-start=34,tutor-end=35}{1}$ 的两个焦点， $\htmlData{tutor-start=0,tutor-end=6}{\Gamma}$ 上一点 $\htmlData{tutor-start=0,tutor-end=1}{P}$ 满足 $\overrightarrow{\htmlData{tutor-start=16,tutor-end=17}{P}\htmlData{tutor-start=17,tutor-end=18}{F}_{\htmlData{tutor-start=20,tutor-end=21}{1}}} \htmlData{tutor-start=24,tutor-end=30}{\cdot }\overrightarrow{\htmlData{tutor-start=46,tutor-end=47}{P}\htmlData{tutor-start=47,tutor-end=48}{F}_{\htmlData{tutor-start=50,tutor-end=51}{2}}} \htmlData{tutor-start=54,tutor-end=55}{=} \htmlData{tutor-start=56,tutor-end=57}{1}$ . 求点 $\htmlData{tutor-start=0,tutor-end=1}{P}$ 到 $\htmlData{tutor-start=0,tutor-end=6}{\Gamma}$ 的两条渐近线的距离之和.

题解状态：标准答案与规范题解待补充

题目标签：FIRST-B-SOLUTION-P9

解题过程

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

一试（B 卷） · 二、解答题：本大题共 3 小题，满分 56 分. 解答应写出文字说明、证明过程或演算步骤. · 代数

(本题满分 20 分) 设正数 $\htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{a}_{\htmlData{tutor-start=10,tutor-end=11}{2}}\htmlData{tutor-start=12,tutor-end=13}{,} \htmlData{tutor-start=14,tutor-end=15}{a}_{\htmlData{tutor-start=17,tutor-end=18}{3}}\htmlData{tutor-start=19,tutor-end=20}{,} \htmlData{tutor-start=21,tutor-end=22}{b}_{\htmlData{tutor-start=24,tutor-end=25}{1}}\htmlData{tutor-start=26,tutor-end=27}{,} \htmlData{tutor-start=28,tutor-end=29}{b}_{\htmlData{tutor-start=31,tutor-end=32}{2}}\htmlData{tutor-start=33,tutor-end=34}{,} \htmlData{tutor-start=35,tutor-end=36}{b}_{\htmlData{tutor-start=38,tutor-end=39}{3}}$ 满足： $\htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{a}_{\htmlData{tutor-start=10,tutor-end=11}{2}}\htmlData{tutor-start=12,tutor-end=13}{,} \htmlData{tutor-start=14,tutor-end=15}{a}_{\htmlData{tutor-start=17,tutor-end=18}{3}}$ 成公差为 $\htmlData{tutor-start=0,tutor-end=1}{b}_{\htmlData{tutor-start=3,tutor-end=4}{1}}$ 的等差数列， $\htmlData{tutor-start=0,tutor-end=1}{b}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{b}_{\htmlData{tutor-start=10,tutor-end=11}{2}}\htmlData{tutor-start=12,tutor-end=13}{,} \htmlData{tutor-start=14,tutor-end=15}{b}_{\htmlData{tutor-start=17,tutor-end=18}{3}}$ 成公比为 $\htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{1}}$ 的等比数列，且 $\htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{3}} \htmlData{tutor-start=6,tutor-end=7}{=} \htmlData{tutor-start=8,tutor-end=9}{b}_{\htmlData{tutor-start=11,tutor-end=12}{3}}$ . 求 $\htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{3}}$ 的最小值，并确定当 $\htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{3}}$ 取到最小值时 $\htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{2}} \htmlData{tutor-start=6,tutor-end=7}{b}_{\htmlData{tutor-start=9,tutor-end=10}{2}}$ 的值.

题解状态：标准答案与规范题解待补充

题目标签：FIRST-B-SOLUTION-P10

解题过程

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

一试（B 卷） · 二、解答题：本大题共 3 小题，满分 56 分. 解答应写出文字说明、证明过程或演算步骤. · 代数

(本题满分 20 分) 若 $\htmlData{tutor-start=0,tutor-end=1}{a}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{b}$ 为实数， $\htmlData{tutor-start=0,tutor-end=1}{a} \htmlData{tutor-start=2,tutor-end=3}{<} \htmlData{tutor-start=4,tutor-end=5}{b}$ ，函数 $\htmlData{tutor-start=0,tutor-end=1}{y} \htmlData{tutor-start=2,tutor-end=3}{=} \sin \htmlData{tutor-start=9,tutor-end=10}{x}$ 在闭区间 $\htmlData{tutor-start=0,tutor-end=1}{[}\htmlData{tutor-start=1,tutor-end=2}{a}\htmlData{tutor-start=2,tutor-end=3}{,} \htmlData{tutor-start=4,tutor-end=5}{b}\htmlData{tutor-start=5,tutor-end=6}{]}$ 上的最大值与最小值之差为 1，求 $\htmlData{tutor-start=0,tutor-end=1}{b} \htmlData{tutor-start=2,tutor-end=3}{-} \htmlData{tutor-start=4,tutor-end=5}{a}$ 的取值范围.

题解状态：标准答案与规范题解待补充

题目标签：FIRST-B-SOLUTION-P11

解题过程

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

二试/加试（B 卷） · 加试 · 平面几何

(本题满分 40 分) 如图, 设 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{,}\htmlData{tutor-start=2,tutor-end=3}{B}\htmlData{tutor-start=3,tutor-end=4}{,}\htmlData{tutor-start=4,tutor-end=5}{C}\htmlData{tutor-start=5,tutor-end=6}{,}\htmlData{tutor-start=6,tutor-end=7}{D}$ 四点在圆 $\htmlData{tutor-start=0,tutor-end=6}{\omega}$ 上顺次排列, 其中 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{C}$ 经过 $\htmlData{tutor-start=0,tutor-end=6}{\omega}$ 的圆心 $\htmlData{tutor-start=0,tutor-end=1}{O}$ . 线段 $\htmlData{tutor-start=0,tutor-end=1}{B}\htmlData{tutor-start=1,tutor-end=2}{D}$ 上一点 $\htmlData{tutor-start=0,tutor-end=1}{P}$ 满足 $\htmlData{tutor-start=0,tutor-end=7}{\angle }\htmlData{tutor-start=7,tutor-end=8}{A}\htmlData{tutor-start=8,tutor-end=9}{P}\htmlData{tutor-start=9,tutor-end=10}{C} \htmlData{tutor-start=11,tutor-end=12}{=} \htmlData{tutor-start=13,tutor-end=20}{\angle }\htmlData{tutor-start=20,tutor-end=21}{B}\htmlData{tutor-start=21,tutor-end=22}{P}\htmlData{tutor-start=22,tutor-end=23}{C}$ , 线段 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{P}$ 上两点 $\htmlData{tutor-start=0,tutor-end=1}{X}\htmlData{tutor-start=1,tutor-end=2}{,}\htmlData{tutor-start=2,tutor-end=3}{Y}$ 满足 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{,}\htmlData{tutor-start=2,tutor-end=3}{O}\htmlData{tutor-start=3,tutor-end=4}{,}\htmlData{tutor-start=4,tutor-end=5}{X}\htmlData{tutor-start=5,tutor-end=6}{,}\htmlData{tutor-start=6,tutor-end=7}{B}$ 四点共圆, $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{,}\htmlData{tutor-start=2,tutor-end=3}{Y}\htmlData{tutor-start=3,tutor-end=4}{,}\htmlData{tutor-start=4,tutor-end=5}{O}\htmlData{tutor-start=5,tutor-end=6}{,}\htmlData{tutor-start=6,tutor-end=7}{D}$ 四点共圆. 证明: $\htmlData{tutor-start=0,tutor-end=1}{B}\htmlData{tutor-start=1,tutor-end=2}{D} \htmlData{tutor-start=3,tutor-end=4}{=} \htmlData{tutor-start=5,tutor-end=6}{2}\htmlData{tutor-start=6,tutor-end=7}{X}\htmlData{tutor-start=7,tutor-end=8}{Y}$ .

题解状态：标准答案与规范题解待补充

题目标签：SECOND-B-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（B 卷） · 加试 · 数学竞赛/待细分

(本题满分 40 分) 给定正实数 $\htmlData{tutor-start=0,tutor-end=1}{a}\htmlData{tutor-start=1,tutor-end=2}{,}\htmlData{tutor-start=2,tutor-end=3}{b}\htmlData{tutor-start=3,tutor-end=4}{,} \htmlData{tutor-start=5,tutor-end=6}{a} \htmlData{tutor-start=7,tutor-end=8}{<} \htmlData{tutor-start=9,tutor-end=10}{b}$ . 设 $\htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{x}_{\htmlData{tutor-start=10,tutor-end=11}{2}}\htmlData{tutor-start=12,tutor-end=13}{,} \cdots\htmlData{tutor-start=20,tutor-end=21}{,} \htmlData{tutor-start=22,tutor-end=23}{x}_{\htmlData{tutor-start=25,tutor-end=26}{2}\htmlData{tutor-start=26,tutor-end=27}{0}\htmlData{tutor-start=27,tutor-end=28}{2}\htmlData{tutor-start=28,tutor-end=29}{2}} \htmlData{tutor-start=31,tutor-end=35}{\in }\htmlData{tutor-start=35,tutor-end=36}{[}\htmlData{tutor-start=36,tutor-end=37}{a}\htmlData{tutor-start=37,tutor-end=38}{,} \htmlData{tutor-start=39,tutor-end=40}{b}\htmlData{tutor-start=40,tutor-end=41}{]}$ , 求 $\frac{|x_{1} - x_{2}| + |x_{2} - x_{3}| + \cdots + |x_{2021} - x_{2022}| + |x_{2022} - x_{1}|}{x_{1} + x_{2} + \cdots + x_{2022}}$ 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-B-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（B 卷） · 加试 · 数论

(本题满分 50 分) 设正整数 $\htmlData{tutor-start=0,tutor-end=1}{a}\htmlData{tutor-start=1,tutor-end=2}{,}\htmlData{tutor-start=2,tutor-end=3}{b}$ 都恰有 $\htmlData{tutor-start=0,tutor-end=1}{m}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{m} \htmlData{tutor-start=4,tutor-end=8}{\ge }\htmlData{tutor-start=8,tutor-end=9}{3}\htmlData{tutor-start=9,tutor-end=10}{)}$ 个正约数, 其中 $\htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{a}_{\htmlData{tutor-start=10,tutor-end=11}{2}}\htmlData{tutor-start=12,tutor-end=13}{,} \htmlData{tutor-start=14,tutor-end=15}{a}_{\htmlData{tutor-start=17,tutor-end=18}{3}}\htmlData{tutor-start=19,tutor-end=20}{,} \cdots\htmlData{tutor-start=27,tutor-end=28}{,} \htmlData{tutor-start=29,tutor-end=30}{a}_{\htmlData{tutor-start=32,tutor-end=33}{m}}$ 是 $\htmlData{tutor-start=0,tutor-end=1}{a}$ 的所有正约数的一个排列. 问 $\htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{a}_{\htmlData{tutor-start=10,tutor-end=11}{1}} \htmlData{tutor-start=13,tutor-end=14}{+} \htmlData{tutor-start=15,tutor-end=16}{a}_{\htmlData{tutor-start=18,tutor-end=19}{2}}\htmlData{tutor-start=20,tutor-end=21}{,} \htmlData{tutor-start=22,tutor-end=23}{a}_{\htmlData{tutor-start=25,tutor-end=26}{2}} \htmlData{tutor-start=28,tutor-end=29}{+} \htmlData{tutor-start=30,tutor-end=31}{a}_{\htmlData{tutor-start=33,tutor-end=34}{3}}\htmlData{tutor-start=35,tutor-end=36}{,} \cdots\htmlData{tutor-start=43,tutor-end=44}{,} \htmlData{tutor-start=45,tutor-end=46}{a}_{\htmlData{tutor-start=48,tutor-end=49}{m}\htmlData{tutor-start=49,tutor-end=50}{-}\htmlData{tutor-start=50,tutor-end=51}{1}} \htmlData{tutor-start=53,tutor-end=54}{+} \htmlData{tutor-start=55,tutor-end=56}{a}_{\htmlData{tutor-start=58,tutor-end=59}{m}}$ 是否可能恰好是 $\htmlData{tutor-start=0,tutor-end=1}{b}$ 的所有正约数的一个排列? 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：SECOND-B-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（B 卷） · 加试 · 平面几何

(本题满分 50 分) 圆周上依次有 100 个点 $\htmlData{tutor-start=0,tutor-end=1}{P}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{P}_{\htmlData{tutor-start=10,tutor-end=11}{2}}\htmlData{tutor-start=12,tutor-end=13}{,} \cdots\htmlData{tutor-start=20,tutor-end=21}{,} \htmlData{tutor-start=22,tutor-end=23}{P}_{\htmlData{tutor-start=25,tutor-end=26}{1}\htmlData{tutor-start=26,tutor-end=27}{0}\htmlData{tutor-start=27,tutor-end=28}{0}}$ (其中 $\htmlData{tutor-start=0,tutor-end=1}{P}_{\htmlData{tutor-start=3,tutor-end=4}{1}\htmlData{tutor-start=4,tutor-end=5}{0}\htmlData{tutor-start=5,tutor-end=6}{0}}$ 与 $\htmlData{tutor-start=0,tutor-end=1}{P}_{\htmlData{tutor-start=3,tutor-end=4}{1}}$ 相邻). 现有 5 种颜色, 要求 $\htmlData{tutor-start=0,tutor-end=1}{P}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{P}_{\htmlData{tutor-start=10,tutor-end=11}{2}}\htmlData{tutor-start=12,tutor-end=13}{,} \cdots\htmlData{tutor-start=20,tutor-end=21}{,} \htmlData{tutor-start=22,tutor-end=23}{P}_{\htmlData{tutor-start=25,tutor-end=26}{1}\htmlData{tutor-start=26,tutor-end=27}{0}\htmlData{tutor-start=27,tutor-end=28}{0}}$ 中每个点染 5 种颜色之一, 每种颜色至少染一个点. 若对任意这样的染色方式, $\htmlData{tutor-start=0,tutor-end=1}{P}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{P}_{\htmlData{tutor-start=10,tutor-end=11}{2}}\htmlData{tutor-start=12,tutor-end=13}{,} \cdots\htmlData{tutor-start=20,tutor-end=21}{,} \htmlData{tutor-start=22,tutor-end=23}{P}_{\htmlData{tutor-start=25,tutor-end=26}{1}\htmlData{tutor-start=26,tutor-end=27}{0}\htmlData{tutor-start=27,tutor-end=28}{0}}$ 中总存在 $\htmlData{tutor-start=0,tutor-end=1}{t}$ 个连续的点含有至少 3 种颜色, 求 $\htmlData{tutor-start=0,tutor-end=1}{t}$ 的最小值.

题解状态：标准答案与规范题解待补充

题目标签：SECOND-B-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。