2024 年全国高中数学联赛 B 卷（学生版）

books/national_high_school_league_students/papers/competition-national-league-2024-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 · 数学竞赛/待细分

复数 $\htmlData{tutor-start=0,tutor-end=1}{z}\htmlData{tutor-start=1,tutor-end=2}{=}\htmlData{tutor-start=2,tutor-end=3}{1}\htmlData{tutor-start=3,tutor-end=4}{+}\sqrt{\htmlData{tutor-start=10,tutor-end=11}{3}}\text{\htmlData{tutor-start=18,tutor-end=19}{i}}$ （ $\text{\htmlData{tutor-start=6,tutor-end=7}{i}}$ 为虚数单位），则 $\htmlData{tutor-start=0,tutor-end=1}{z}\htmlData{tutor-start=1,tutor-end=2}{+}\htmlData{tutor-start=2,tutor-end=3}{2}\overline{\htmlData{tutor-start=13,tutor-end=14}{z}}\htmlData{tutor-start=15,tutor-end=16}{+}\htmlData{tutor-start=16,tutor-end=17}{3}\htmlData{tutor-start=17,tutor-end=18}{|}\htmlData{tutor-start=18,tutor-end=19}{z}\htmlData{tutor-start=19,tutor-end=20}{|}$ 的模为_______.

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 · 代数

设 $\htmlData{tutor-start=0,tutor-end=1}{a}\htmlData{tutor-start=1,tutor-end=2}{,}\htmlData{tutor-start=2,tutor-end=3}{b}$ 为实数，若一元二次不等式 $\htmlData{tutor-start=0,tutor-end=1}{x}^{\htmlData{tutor-start=3,tutor-end=4}{2}}\htmlData{tutor-start=5,tutor-end=6}{+}\htmlData{tutor-start=6,tutor-end=7}{b}\htmlData{tutor-start=7,tutor-end=8}{<}\htmlData{tutor-start=8,tutor-end=9}{a}\htmlData{tutor-start=9,tutor-end=10}{x}$ 的解集为 $\htmlData{tutor-start=0,tutor-end=1}{(}\htmlData{tutor-start=1,tutor-end=2}{1}\htmlData{tutor-start=2,tutor-end=3}{,}\htmlData{tutor-start=3,tutor-end=4}{5}\htmlData{tutor-start=4,tutor-end=5}{)}$ ，则一元二次不等式 $\htmlData{tutor-start=0,tutor-end=1}{x}^{\htmlData{tutor-start=3,tutor-end=4}{2}}\htmlData{tutor-start=5,tutor-end=6}{-}\htmlData{tutor-start=6,tutor-end=7}{a}\htmlData{tutor-start=7,tutor-end=8}{>}\htmlData{tutor-start=8,tutor-end=9}{b}\htmlData{tutor-start=9,tutor-end=10}{x}$ 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-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 $\htmlData{tutor-start=0,tutor-end=1}{y}\htmlData{tutor-start=1,tutor-end=2}{=}\frac{\htmlData{tutor-start=8,tutor-end=9}{4}^{\htmlData{tutor-start=11,tutor-end=12}{x}}\htmlData{tutor-start=13,tutor-end=14}{+}\htmlData{tutor-start=14,tutor-end=15}{1}}{\htmlData{tutor-start=17,tutor-end=18}{2}^{\htmlData{tutor-start=20,tutor-end=21}{x}}\htmlData{tutor-start=22,tutor-end=23}{+}\htmlData{tutor-start=23,tutor-end=24}{1}}$ 的最小值为_______.

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 · 组合数学

一只青蛙在正方形 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{B}\htmlData{tutor-start=2,tutor-end=3}{C}\htmlData{tutor-start=3,tutor-end=4}{D}$ 的四个顶点间跳跃，每次跳跃总是等可能地跳至与当前所在顶点相邻的两个顶点之一，且各次跳跃是独立的. 若青蛙第一次跳跃前位于顶点 $\htmlData{tutor-start=0,tutor-end=1}{A}$ ，则它第 6 次跳跃后恰好仍位于顶点 $\htmlData{tutor-start=0,tutor-end=1}{A}$ 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 · 平面几何

在椭圆 $\htmlData{tutor-start=0,tutor-end=6}{\Omega}$ 中， $\htmlData{tutor-start=0,tutor-end=1}{F}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{,}\htmlData{tutor-start=6,tutor-end=7}{F}_{\htmlData{tutor-start=9,tutor-end=10}{2}}$ 为焦点， $\htmlData{tutor-start=0,tutor-end=1}{A}$ 为长轴的一个端点， $\htmlData{tutor-start=0,tutor-end=1}{B}$ 为短轴的一个端点，若 $\htmlData{tutor-start=0,tutor-end=7}{\angle }\htmlData{tutor-start=7,tutor-end=8}{F}_{\htmlData{tutor-start=10,tutor-end=11}{1}} \htmlData{tutor-start=13,tutor-end=14}{B} \htmlData{tutor-start=15,tutor-end=16}{F}_{\htmlData{tutor-start=18,tutor-end=19}{2}} \htmlData{tutor-start=21,tutor-end=22}{=} \htmlData{tutor-start=23,tutor-end=30}{\angle }\htmlData{tutor-start=30,tutor-end=31}{F}_{\htmlData{tutor-start=33,tutor-end=34}{1}} \htmlData{tutor-start=36,tutor-end=37}{A} \htmlData{tutor-start=38,tutor-end=39}{B}$ ，则 $\htmlData{tutor-start=0,tutor-end=6}{\Omega}$ 的离心率为_______.

题解状态：标准答案与规范题解待补充

题目标签：FIRST-B-FILL-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 · 数论

数列 $\htmlData{tutor-start=0,tutor-end=2}{\{}\htmlData{tutor-start=2,tutor-end=3}{a}_{\htmlData{tutor-start=5,tutor-end=6}{n}}\htmlData{tutor-start=7,tutor-end=9}{\}}$ 满足： $\htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{=}\htmlData{tutor-start=6,tutor-end=7}{1}$ ，且对任意正整数 $\htmlData{tutor-start=0,tutor-end=1}{n}$ ，均有 $\htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{n}\htmlData{tutor-start=4,tutor-end=5}{+}\htmlData{tutor-start=5,tutor-end=6}{1}}\htmlData{tutor-start=7,tutor-end=8}{=}\htmlData{tutor-start=8,tutor-end=9}{1}\htmlData{tutor-start=9,tutor-end=10}{0}^{\htmlData{tutor-start=12,tutor-end=13}{n}} \htmlData{tutor-start=15,tutor-end=16}{a}_{\htmlData{tutor-start=18,tutor-end=19}{n}}^{\htmlData{tutor-start=22,tutor-end=23}{2}}$ ，则 $\htmlData{tutor-start=0,tutor-end=2}{\{}\htmlData{tutor-start=2,tutor-end=3}{a}_{\htmlData{tutor-start=5,tutor-end=6}{n}}\htmlData{tutor-start=7,tutor-end=9}{\}}$ 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 · 数学竞赛/待细分

四面体 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{B}\htmlData{tutor-start=2,tutor-end=3}{C}\htmlData{tutor-start=3,tutor-end=4}{D}$ 中， $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{B} \htmlData{tutor-start=3,tutor-end=9}{\perp }\htmlData{tutor-start=9,tutor-end=10}{B}\htmlData{tutor-start=10,tutor-end=11}{C}$ ， $\htmlData{tutor-start=0,tutor-end=1}{B}\htmlData{tutor-start=1,tutor-end=2}{C} \htmlData{tutor-start=3,tutor-end=9}{\perp }\htmlData{tutor-start=9,tutor-end=10}{C}\htmlData{tutor-start=10,tutor-end=11}{D}$ ，且 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{B}\htmlData{tutor-start=2,tutor-end=3}{,}\htmlData{tutor-start=3,tutor-end=4}{B}\htmlData{tutor-start=4,tutor-end=5}{C}\htmlData{tutor-start=5,tutor-end=6}{,}\htmlData{tutor-start=6,tutor-end=7}{C}\htmlData{tutor-start=7,tutor-end=8}{D}\htmlData{tutor-start=8,tutor-end=9}{,}\htmlData{tutor-start=9,tutor-end=10}{D}\htmlData{tutor-start=10,tutor-end=11}{A}$ 的长分别为 1, 2, 3, 4，则四面体 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{B}\htmlData{tutor-start=2,tutor-end=3}{C}\htmlData{tutor-start=3,tutor-end=4}{D}$ 的体积为_______.

题解状态：标准答案与规范题解待补充

题目标签：FIRST-B-FILL-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 · 数论

若三个正整数 $\htmlData{tutor-start=0,tutor-end=1}{a}\htmlData{tutor-start=1,tutor-end=2}{,}\htmlData{tutor-start=2,tutor-end=3}{b}\htmlData{tutor-start=3,tutor-end=4}{,}\htmlData{tutor-start=4,tutor-end=5}{c}$ 的位数之和为 8，且组成 $\htmlData{tutor-start=0,tutor-end=1}{a}\htmlData{tutor-start=1,tutor-end=2}{,}\htmlData{tutor-start=2,tutor-end=3}{b}\htmlData{tutor-start=3,tutor-end=4}{,}\htmlData{tutor-start=4,tutor-end=5}{c}$ 的 8 个数码能排列为 2, 0, 2, 4, 0, 9, 0, 8，则称 $\htmlData{tutor-start=0,tutor-end=1}{(}\htmlData{tutor-start=1,tutor-end=2}{a}\htmlData{tutor-start=2,tutor-end=3}{,}\htmlData{tutor-start=3,tutor-end=4}{b}\htmlData{tutor-start=4,tutor-end=5}{,}\htmlData{tutor-start=5,tutor-end=6}{c}\htmlData{tutor-start=6,tutor-end=7}{)}$ 为“幸运数组”，例如 $\htmlData{tutor-start=0,tutor-end=1}{(}\htmlData{tutor-start=1,tutor-end=2}{9}\htmlData{tutor-start=2,tutor-end=3}{,}\htmlData{tutor-start=3,tutor-end=4}{8}\htmlData{tutor-start=4,tutor-end=5}{,}\htmlData{tutor-start=5,tutor-end=6}{2}\htmlData{tutor-start=6,tutor-end=7}{0}\htmlData{tutor-start=7,tutor-end=8}{2}\htmlData{tutor-start=8,tutor-end=9}{4}\htmlData{tutor-start=9,tutor-end=10}{0}\htmlData{tutor-start=10,tutor-end=11}{0}\htmlData{tutor-start=11,tutor-end=12}{)}$ 是一个幸运数组. 满足 $\htmlData{tutor-start=0,tutor-end=1}{1}\htmlData{tutor-start=1,tutor-end=2}{0}\htmlData{tutor-start=2,tutor-end=3}{<}\htmlData{tutor-start=3,tutor-end=4}{a}\htmlData{tutor-start=4,tutor-end=5}{<}\htmlData{tutor-start=5,tutor-end=6}{b}\htmlData{tutor-start=6,tutor-end=7}{<}\htmlData{tutor-start=7,tutor-end=8}{1}\htmlData{tutor-start=8,tutor-end=9}{0}\htmlData{tutor-start=9,tutor-end=10}{0}$ 的幸运数组 $\htmlData{tutor-start=0,tutor-end=1}{(}\htmlData{tutor-start=1,tutor-end=2}{a}\htmlData{tutor-start=2,tutor-end=3}{,}\htmlData{tutor-start=3,tutor-end=4}{b}\htmlData{tutor-start=4,tutor-end=5}{,}\htmlData{tutor-start=5,tutor-end=6}{c}\htmlData{tutor-start=6,tutor-end=7}{)}$ 的个数为_______.

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. · 代数

（本题满分16分）设无穷等比数列 $\htmlData{tutor-start=0,tutor-end=2}{\{}\htmlData{tutor-start=2,tutor-end=3}{a}_{\htmlData{tutor-start=5,tutor-end=6}{n}}\htmlData{tutor-start=7,tutor-end=9}{\}}$ 的公比 $\htmlData{tutor-start=0,tutor-end=1}{q}$ 满足 $\htmlData{tutor-start=0,tutor-end=1}{0}\htmlData{tutor-start=1,tutor-end=2}{<}\htmlData{tutor-start=2,tutor-end=3}{|}\htmlData{tutor-start=3,tutor-end=4}{q}\htmlData{tutor-start=4,tutor-end=5}{|}\htmlData{tutor-start=5,tutor-end=6}{<}\htmlData{tutor-start=6,tutor-end=7}{1}$ . 若 $\htmlData{tutor-start=0,tutor-end=2}{\{}\htmlData{tutor-start=2,tutor-end=3}{a}_{\htmlData{tutor-start=5,tutor-end=6}{n}}\htmlData{tutor-start=7,tutor-end=9}{\}}$ 的各项和等于 $\htmlData{tutor-start=0,tutor-end=2}{\{}\htmlData{tutor-start=2,tutor-end=3}{a}_{\htmlData{tutor-start=5,tutor-end=6}{n}}\htmlData{tutor-start=7,tutor-end=9}{\}}$ 各项的平方和，求 $\htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{2}}$ 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-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. · 平面几何

（本题满分 20 分）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 $\cos \htmlData{tutor-start=5,tutor-end=6}{C} \htmlData{tutor-start=7,tutor-end=8}{=} \frac{\sin \htmlData{tutor-start=20,tutor-end=21}{A} \htmlData{tutor-start=22,tutor-end=23}{+} \cos \htmlData{tutor-start=29,tutor-end=30}{A}}{\htmlData{tutor-start=32,tutor-end=33}{2}} \htmlData{tutor-start=35,tutor-end=36}{=} \frac{\sin \htmlData{tutor-start=48,tutor-end=49}{B} \htmlData{tutor-start=50,tutor-end=51}{+} \cos \htmlData{tutor-start=57,tutor-end=58}{B}}{\htmlData{tutor-start=60,tutor-end=61}{2}}$ ，求 $\sin \htmlData{tutor-start=5,tutor-end=6}{C}$ 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. · 平面几何

（本题满分 20 分）在平面直角坐标系中，将圆心在 $\htmlData{tutor-start=0,tutor-end=1}{y}$ 轴上，且与双曲线 $\htmlData{tutor-start=0,tutor-end=6}{\Gamma}\htmlData{tutor-start=6,tutor-end=7}{:} \htmlData{tutor-start=8,tutor-end=9}{x}^{\htmlData{tutor-start=11,tutor-end=12}{2}} \htmlData{tutor-start=14,tutor-end=15}{-} \htmlData{tutor-start=16,tutor-end=17}{y}^{\htmlData{tutor-start=19,tutor-end=20}{2}} \htmlData{tutor-start=22,tutor-end=23}{=} \htmlData{tutor-start=24,tutor-end=25}{1}$ 的两支各恰有一个公共点的圆称为“好圆”。是否存在常数 $\htmlData{tutor-start=0,tutor-end=7}{\lambda}$ 及 $\htmlData{tutor-start=0,tutor-end=1}{x}$ 轴上的一个定点 $\htmlData{tutor-start=0,tutor-end=1}{A}$ ，满足：若两个好圆外切于点 $\htmlData{tutor-start=0,tutor-end=1}{P}$ ，则它们的圆心距 $\htmlData{tutor-start=0,tutor-end=1}{d} \htmlData{tutor-start=2,tutor-end=3}{=} \htmlData{tutor-start=4,tutor-end=12}{\lambda }\htmlData{tutor-start=12,tutor-end=13}{|}\htmlData{tutor-start=13,tutor-end=14}{P}\htmlData{tutor-start=14,tutor-end=15}{A}\htmlData{tutor-start=15,tutor-end=16}{|}$ ？

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） · 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. · 数论

（本题满分40分）给定正整数 $\htmlData{tutor-start=0,tutor-end=1}{a}$ . 求函数 $\htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{x}\htmlData{tutor-start=3,tutor-end=4}{)}\htmlData{tutor-start=4,tutor-end=5}{=}\min\htmlData{tutor-start=9,tutor-end=11}{\{}\htmlData{tutor-start=11,tutor-end=13}{\|}\htmlData{tutor-start=13,tutor-end=14}{x}\htmlData{tutor-start=14,tutor-end=16}{\|}\htmlData{tutor-start=16,tutor-end=17}{,} \htmlData{tutor-start=18,tutor-end=20}{\|}\htmlData{tutor-start=20,tutor-end=21}{a}\htmlData{tutor-start=21,tutor-end=22}{x}\htmlData{tutor-start=22,tutor-end=24}{\|}\htmlData{tutor-start=24,tutor-end=26}{\}}$ 的最大值.（ $\htmlData{tutor-start=0,tutor-end=2}{\|}\htmlData{tutor-start=2,tutor-end=3}{x}\htmlData{tutor-start=3,tutor-end=5}{\|}$ 表示实数 $\htmlData{tutor-start=0,tutor-end=1}{x}$ 到与它最近整数的距离； $\min\htmlData{tutor-start=4,tutor-end=6}{\{}\htmlData{tutor-start=6,tutor-end=7}{s}\htmlData{tutor-start=7,tutor-end=8}{,} \htmlData{tutor-start=9,tutor-end=10}{t}\htmlData{tutor-start=10,tutor-end=12}{\}}$ 表示实数 $\htmlData{tutor-start=0,tutor-end=1}{s}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{t}$ 中较小的数.）

题解状态：标准答案与规范题解待补充

题目标签：SECOND-B-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（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. · 平面几何

（本题满分 40 分）如图，在凸四边形 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{B}\htmlData{tutor-start=2,tutor-end=3}{C}\htmlData{tutor-start=3,tutor-end=4}{D}$ 中， $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{C}$ 平分 $\htmlData{tutor-start=0,tutor-end=7}{\angle }\htmlData{tutor-start=7,tutor-end=8}{B}\htmlData{tutor-start=8,tutor-end=9}{A}\htmlData{tutor-start=9,tutor-end=10}{D}$ ，且 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{C}^{\htmlData{tutor-start=4,tutor-end=5}{2}}\htmlData{tutor-start=6,tutor-end=7}{=}\htmlData{tutor-start=7,tutor-end=8}{A}\htmlData{tutor-start=8,tutor-end=9}{B} \htmlData{tutor-start=10,tutor-end=16}{\cdot }\htmlData{tutor-start=16,tutor-end=17}{A}\htmlData{tutor-start=17,tutor-end=18}{D}$ ，点 $\htmlData{tutor-start=0,tutor-end=1}{E}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{F}$ 分别在边 $\htmlData{tutor-start=0,tutor-end=1}{B}\htmlData{tutor-start=1,tutor-end=2}{C}\htmlData{tutor-start=2,tutor-end=3}{,} \htmlData{tutor-start=4,tutor-end=5}{C}\htmlData{tutor-start=5,tutor-end=6}{D}$ 上，满足 $\frac{\htmlData{tutor-start=6,tutor-end=7}{B}\htmlData{tutor-start=7,tutor-end=8}{E}}{\htmlData{tutor-start=10,tutor-end=11}{E}\htmlData{tutor-start=11,tutor-end=12}{C}}\htmlData{tutor-start=13,tutor-end=14}{=}\frac{\htmlData{tutor-start=20,tutor-end=21}{D}\htmlData{tutor-start=21,tutor-end=22}{F}}{\htmlData{tutor-start=24,tutor-end=25}{F}\htmlData{tutor-start=25,tutor-end=26}{C}}$ . 过点 $\htmlData{tutor-start=0,tutor-end=1}{C}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{B}\htmlData{tutor-start=4,tutor-end=5}{,} \htmlData{tutor-start=6,tutor-end=7}{F}$ 的圆 $\htmlData{tutor-start=0,tutor-end=6}{\omega}_{\htmlData{tutor-start=8,tutor-end=9}{1}}$ 与过点 $\htmlData{tutor-start=0,tutor-end=1}{C}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{D}\htmlData{tutor-start=4,tutor-end=5}{,} \htmlData{tutor-start=6,tutor-end=7}{E}$ 的圆 $\htmlData{tutor-start=0,tutor-end=6}{\omega}_{\htmlData{tutor-start=8,tutor-end=9}{2}}$ 交于 $\htmlData{tutor-start=0,tutor-end=1}{C}$ 及另一点 $\htmlData{tutor-start=0,tutor-end=1}{T}$ . 证明：点 $\htmlData{tutor-start=0,tutor-end=1}{T}$ 在直线 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{C}$ 上.

**原卷图示 1**原卷第 3 页 · qwen3.7-plus_layout_detection · 需复核

题解状态：标准答案与规范题解待补充

题目标签：SECOND-B-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（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. · 数论

（本题满分 50 分）给定素数 $\htmlData{tutor-start=0,tutor-end=1}{p}$ ， $p \equiv 2 \pmod 3$ 。求同时满足下面两个条件的三元有序数组 $\htmlData{tutor-start=0,tutor-end=1}{(}\htmlData{tutor-start=1,tutor-end=2}{a}\htmlData{tutor-start=2,tutor-end=3}{,} \htmlData{tutor-start=4,tutor-end=5}{b}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{c}\htmlData{tutor-start=8,tutor-end=9}{)}$ 的数目：（1） $\htmlData{tutor-start=0,tutor-end=1}{a}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{b}\htmlData{tutor-start=4,tutor-end=5}{,} \htmlData{tutor-start=6,tutor-end=7}{c} \htmlData{tutor-start=8,tutor-end=12}{\in }\htmlData{tutor-start=12,tutor-end=14}{\{}\htmlData{tutor-start=14,tutor-end=15}{0}\htmlData{tutor-start=15,tutor-end=16}{,} \htmlData{tutor-start=17,tutor-end=18}{1}\htmlData{tutor-start=18,tutor-end=19}{,} \htmlData{tutor-start=20,tutor-end=21}{2}\htmlData{tutor-start=21,tutor-end=22}{,} \cdots\htmlData{tutor-start=29,tutor-end=30}{,} \htmlData{tutor-start=31,tutor-end=32}{p}\htmlData{tutor-start=32,tutor-end=33}{-}\htmlData{tutor-start=33,tutor-end=34}{1}\htmlData{tutor-start=34,tutor-end=36}{\}}$ ；（2）三个数 $\htmlData{tutor-start=0,tutor-end=1}{a}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{b}\htmlData{tutor-start=3,tutor-end=4}{+}\htmlData{tutor-start=4,tutor-end=5}{1}\htmlData{tutor-start=5,tutor-end=6}{)}\htmlData{tutor-start=6,tutor-end=7}{,} \htmlData{tutor-start=8,tutor-end=9}{b}\htmlData{tutor-start=9,tutor-end=10}{(}\htmlData{tutor-start=10,tutor-end=11}{c}\htmlData{tutor-start=11,tutor-end=12}{+}\htmlData{tutor-start=12,tutor-end=13}{1}\htmlData{tutor-start=13,tutor-end=14}{)}\htmlData{tutor-start=14,tutor-end=15}{,} \htmlData{tutor-start=16,tutor-end=17}{c}\htmlData{tutor-start=17,tutor-end=18}{(}\htmlData{tutor-start=18,tutor-end=19}{a}\htmlData{tutor-start=19,tutor-end=20}{+}\htmlData{tutor-start=20,tutor-end=21}{1}\htmlData{tutor-start=21,tutor-end=22}{)}$ 除以 $\htmlData{tutor-start=0,tutor-end=1}{p}$ 所得的余数相同。

题解状态：标准答案与规范题解待补充

题目标签：SECOND-B-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。

二试/加试（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. · 数论

（本题满分 50 分）给定正整数 $\htmlData{tutor-start=0,tutor-end=1}{n}$ 。在一个 $\htmlData{tutor-start=0,tutor-end=1}{2} \htmlData{tutor-start=2,tutor-end=9}{\times }\htmlData{tutor-start=9,tutor-end=10}{n}$ 的方格表上，由一些方格构成的集合 $\htmlData{tutor-start=0,tutor-end=1}{S}$ 称为“连通的”，如果对 $\htmlData{tutor-start=0,tutor-end=1}{S}$ 中任意两个不同的小方格 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{B}$ ，存在整数 $\htmlData{tutor-start=0,tutor-end=1}{l} \htmlData{tutor-start=2,tutor-end=12}{\geqslant }\htmlData{tutor-start=12,tutor-end=13}{2}$ 及 $\htmlData{tutor-start=0,tutor-end=1}{S}$ 中 $\htmlData{tutor-start=0,tutor-end=1}{l}$ 个方格 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{=}\htmlData{tutor-start=2,tutor-end=3}{C}_{\htmlData{tutor-start=5,tutor-end=6}{1}}\htmlData{tutor-start=7,tutor-end=8}{,} \htmlData{tutor-start=9,tutor-end=10}{C}_{\htmlData{tutor-start=12,tutor-end=13}{2}}\htmlData{tutor-start=14,tutor-end=15}{,} \cdots\htmlData{tutor-start=22,tutor-end=23}{,} \htmlData{tutor-start=24,tutor-end=25}{C}_{\htmlData{tutor-start=27,tutor-end=28}{l}}\htmlData{tutor-start=29,tutor-end=30}{=}\htmlData{tutor-start=30,tutor-end=31}{B}$ ，满足 $\htmlData{tutor-start=0,tutor-end=1}{C}_{\htmlData{tutor-start=3,tutor-end=4}{i}}$ 与 $\htmlData{tutor-start=0,tutor-end=1}{C}_{\htmlData{tutor-start=3,tutor-end=4}{i}\htmlData{tutor-start=4,tutor-end=5}{+}\htmlData{tutor-start=5,tutor-end=6}{1}}$ 有公共边（ $\htmlData{tutor-start=0,tutor-end=1}{i}\htmlData{tutor-start=1,tutor-end=2}{=}\htmlData{tutor-start=2,tutor-end=3}{1}\htmlData{tutor-start=3,tutor-end=4}{,} \htmlData{tutor-start=5,tutor-end=6}{2}\htmlData{tutor-start=6,tutor-end=7}{,} \cdots\htmlData{tutor-start=14,tutor-end=15}{,} \htmlData{tutor-start=16,tutor-end=17}{l}\htmlData{tutor-start=17,tutor-end=18}{-}\htmlData{tutor-start=18,tutor-end=19}{1}$ ）。求具有下述性质的最大整数 $\htmlData{tutor-start=0,tutor-end=1}{K}$ ：若将该方格表中的每个小方格任意染为黑色或白色，总存在一个连通的集合 $\htmlData{tutor-start=0,tutor-end=1}{S}$ ，使得 $\htmlData{tutor-start=0,tutor-end=1}{S}$ 中的黑格个数与白格个数之差的绝对值不小于 $\htmlData{tutor-start=0,tutor-end=1}{K}$ 。

题解状态：标准答案与规范题解待补充

题目标签：SECOND-B-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。