2019 年全国高中数学联赛 A 卷（学生版）

books/national_high_school_league_students/papers/competition-national-league-2019-a.pdf · HS-MATH-1024-v2.1-solution-aware

15 题0 个小问/题组

一试（A 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

已知正实数 $\htmlData{tutor-start=0,tutor-end=1}{a}$ 满足 $\htmlData{tutor-start=0,tutor-end=1}{a}^{\htmlData{tutor-start=3,tutor-end=4}{a}}\htmlData{tutor-start=5,tutor-end=6}{=}\htmlData{tutor-start=6,tutor-end=7}{(}\htmlData{tutor-start=7,tutor-end=8}{9}\htmlData{tutor-start=8,tutor-end=9}{a}\htmlData{tutor-start=9,tutor-end=10}{)}^{\htmlData{tutor-start=12,tutor-end=13}{8}\htmlData{tutor-start=13,tutor-end=14}{a}}$ ，则 $\log_{\htmlData{tutor-start=6,tutor-end=7}{a}}\htmlData{tutor-start=8,tutor-end=9}{(}\htmlData{tutor-start=9,tutor-end=10}{3}\htmlData{tutor-start=10,tutor-end=11}{a}\htmlData{tutor-start=11,tutor-end=12}{)}$ 的值为__________.

题解状态：标准答案与规范题解待补充

题目标签：FIRST-A-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（A 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

若实数集合 $\htmlData{tutor-start=0,tutor-end=2}{\{}\htmlData{tutor-start=2,tutor-end=3}{1}\htmlData{tutor-start=3,tutor-end=4}{,}\htmlData{tutor-start=4,tutor-end=5}{2}\htmlData{tutor-start=5,tutor-end=6}{,}\htmlData{tutor-start=6,tutor-end=7}{3}\htmlData{tutor-start=7,tutor-end=8}{,}\htmlData{tutor-start=8,tutor-end=9}{x}\htmlData{tutor-start=9,tutor-end=11}{\}}$ 的最大元素与最小元素之差等于该集合的所有元素之和，则 $\htmlData{tutor-start=0,tutor-end=1}{x}$ 的值为__________.

题解状态：标准答案与规范题解待补充

题目标签：FIRST-A-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（A 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

平面直角坐标系中， $\vec{\htmlData{tutor-start=5,tutor-end=6}{e}}$ 是单位向量，向量 $\vec{\htmlData{tutor-start=5,tutor-end=6}{a}}$ 满足 $\vec{\htmlData{tutor-start=5,tutor-end=6}{a}} \htmlData{tutor-start=8,tutor-end=14}{\cdot }\vec{\htmlData{tutor-start=19,tutor-end=20}{e}}\htmlData{tutor-start=21,tutor-end=22}{=}\htmlData{tutor-start=22,tutor-end=23}{2}$ ，且 $\htmlData{tutor-start=0,tutor-end=1}{|}\vec{\htmlData{tutor-start=6,tutor-end=7}{a}}\htmlData{tutor-start=8,tutor-end=9}{|}^{\htmlData{tutor-start=11,tutor-end=12}{2}} \htmlData{tutor-start=14,tutor-end=18}{\le }\htmlData{tutor-start=18,tutor-end=19}{5}\htmlData{tutor-start=19,tutor-end=20}{|}\vec{\htmlData{tutor-start=25,tutor-end=26}{a}}\htmlData{tutor-start=27,tutor-end=28}{+}\htmlData{tutor-start=28,tutor-end=29}{t}\vec{\htmlData{tutor-start=34,tutor-end=35}{e}}\htmlData{tutor-start=36,tutor-end=37}{|}$ 对任意实数 $\htmlData{tutor-start=0,tutor-end=1}{t}$ 成立，则 $\htmlData{tutor-start=0,tutor-end=1}{|}\vec{\htmlData{tutor-start=6,tutor-end=7}{a}}\htmlData{tutor-start=8,tutor-end=9}{|}$ 的取值范围是__________.

题解状态：标准答案与规范题解待补充

题目标签：FIRST-A-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（A 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

设 A,B 为椭圆 $\htmlData{tutor-start=0,tutor-end=6}{\Gamma}$ 的长轴顶点，E,F 为 $\htmlData{tutor-start=0,tutor-end=6}{\Gamma}$ 的两个焦点， $\htmlData{tutor-start=0,tutor-end=1}{|}\htmlData{tutor-start=1,tutor-end=2}{A}\htmlData{tutor-start=2,tutor-end=3}{B}\htmlData{tutor-start=3,tutor-end=4}{|}\htmlData{tutor-start=4,tutor-end=5}{=}\htmlData{tutor-start=5,tutor-end=6}{4}\htmlData{tutor-start=6,tutor-end=7}{,} \htmlData{tutor-start=8,tutor-end=9}{|}\htmlData{tutor-start=9,tutor-end=10}{A}\htmlData{tutor-start=10,tutor-end=11}{F}\htmlData{tutor-start=11,tutor-end=12}{|}\htmlData{tutor-start=12,tutor-end=13}{=}\htmlData{tutor-start=13,tutor-end=14}{2}\htmlData{tutor-start=14,tutor-end=15}{+}\sqrt{\htmlData{tutor-start=21,tutor-end=22}{3}}$ ，P 为 $\htmlData{tutor-start=0,tutor-end=6}{\Gamma}$ 上一点，满足 $\htmlData{tutor-start=0,tutor-end=1}{|}\htmlData{tutor-start=1,tutor-end=2}{P}\htmlData{tutor-start=2,tutor-end=3}{E}\htmlData{tutor-start=3,tutor-end=4}{|} \htmlData{tutor-start=5,tutor-end=11}{\cdot }\htmlData{tutor-start=11,tutor-end=12}{|}\htmlData{tutor-start=12,tutor-end=13}{P}\htmlData{tutor-start=13,tutor-end=14}{F}\htmlData{tutor-start=14,tutor-end=15}{|}\htmlData{tutor-start=15,tutor-end=16}{=}\htmlData{tutor-start=16,tutor-end=17}{2}$ ，则 $\htmlData{tutor-start=0,tutor-end=7}{\Delta }\htmlData{tutor-start=7,tutor-end=8}{P}\htmlData{tutor-start=8,tutor-end=9}{E}\htmlData{tutor-start=9,tutor-end=10}{F}$ 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-A-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（A 卷） · 一、填空题：本大题共 8 小题，每小题 8 分，满分 64 分. · 数论

在 $\htmlData{tutor-start=0,tutor-end=1}{1}\htmlData{tutor-start=1,tutor-end=2}{,}\htmlData{tutor-start=2,tutor-end=3}{2}\htmlData{tutor-start=3,tutor-end=4}{,}\htmlData{tutor-start=4,tutor-end=5}{3}\htmlData{tutor-start=5,tutor-end=6}{,} \cdots\htmlData{tutor-start=13,tutor-end=14}{,} \htmlData{tutor-start=15,tutor-end=16}{1}\htmlData{tutor-start=16,tutor-end=17}{0}$ 中随机选出一个数 $\htmlData{tutor-start=0,tutor-end=1}{a}$ ，在 $\htmlData{tutor-start=0,tutor-end=1}{-}\htmlData{tutor-start=1,tutor-end=2}{1}\htmlData{tutor-start=2,tutor-end=3}{,}\htmlData{tutor-start=3,tutor-end=4}{-}\htmlData{tutor-start=4,tutor-end=5}{2}\htmlData{tutor-start=5,tutor-end=6}{,}\htmlData{tutor-start=6,tutor-end=7}{-}\htmlData{tutor-start=7,tutor-end=8}{3}\htmlData{tutor-start=8,tutor-end=9}{,} \cdots\htmlData{tutor-start=16,tutor-end=17}{,} \htmlData{tutor-start=18,tutor-end=19}{-}\htmlData{tutor-start=19,tutor-end=20}{1}\htmlData{tutor-start=20,tutor-end=21}{0}$ 中随机选出一个数 $\htmlData{tutor-start=0,tutor-end=1}{b}$ ，则 $\htmlData{tutor-start=0,tutor-end=1}{a}^{\htmlData{tutor-start=3,tutor-end=4}{2}}\htmlData{tutor-start=5,tutor-end=6}{+}\htmlData{tutor-start=6,tutor-end=7}{b}$ 被 3 整除的概率为__________.

题解状态：标准答案与规范题解待补充

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

对任意闭区间 $\htmlData{tutor-start=0,tutor-end=1}{I}$ ，用 $\htmlData{tutor-start=0,tutor-end=1}{M}_{\htmlData{tutor-start=3,tutor-end=4}{I}}$ 表示函数 $\htmlData{tutor-start=0,tutor-end=1}{y}\htmlData{tutor-start=1,tutor-end=2}{=}\sin \htmlData{tutor-start=7,tutor-end=8}{x}$ 在 $\htmlData{tutor-start=0,tutor-end=1}{I}$ 上的最大值. 若正数 $\htmlData{tutor-start=0,tutor-end=1}{a}$ 满足 $\htmlData{tutor-start=0,tutor-end=1}{M}_{\htmlData{tutor-start=3,tutor-end=4}{[}\htmlData{tutor-start=4,tutor-end=5}{0}\htmlData{tutor-start=5,tutor-end=6}{,}\htmlData{tutor-start=6,tutor-end=7}{a}\htmlData{tutor-start=7,tutor-end=8}{]}}\htmlData{tutor-start=9,tutor-end=10}{=}\htmlData{tutor-start=10,tutor-end=11}{2}\htmlData{tutor-start=11,tutor-end=12}{M}_{\htmlData{tutor-start=14,tutor-end=15}{[}\htmlData{tutor-start=15,tutor-end=16}{a}\htmlData{tutor-start=16,tutor-end=17}{,}\htmlData{tutor-start=17,tutor-end=18}{2}\htmlData{tutor-start=18,tutor-end=19}{a}\htmlData{tutor-start=19,tutor-end=20}{]}}$ ，则 $\htmlData{tutor-start=0,tutor-end=1}{a}$ 的值为__________.

题解状态：标准答案与规范题解待补充

题目标签：FIRST-A-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（A 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

如图，正方体 ABCD — EFGH 的一个截面经过顶点 A，C 及棱 EF 上一点 K，且将正方体分成体积比为 3:1 的两部分，则 $\frac{\htmlData{tutor-start=6,tutor-end=7}{E}\htmlData{tutor-start=7,tutor-end=8}{K}}{\htmlData{tutor-start=10,tutor-end=11}{K}\htmlData{tutor-start=11,tutor-end=12}{F}}$ 的值为__________.

**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-A-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（A 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

将 6 个数 2,0,1,9,20,19 按任意次序排成一行，拼成一个 8 位数（首位不为 0），则产生的不同的 8 位数的个数为__________.

题解状态：标准答案与规范题解待补充

题目标签：FIRST-A-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（A 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 16 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， $\htmlData{tutor-start=0,tutor-end=1}{B}\htmlData{tutor-start=1,tutor-end=2}{C}\htmlData{tutor-start=2,tutor-end=3}{=}\htmlData{tutor-start=3,tutor-end=4}{a}$ ， $\htmlData{tutor-start=0,tutor-end=1}{C}\htmlData{tutor-start=1,tutor-end=2}{A}\htmlData{tutor-start=2,tutor-end=3}{=}\htmlData{tutor-start=3,tutor-end=4}{b}$ ， $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{B}\htmlData{tutor-start=2,tutor-end=3}{=}\htmlData{tutor-start=3,tutor-end=4}{c}$ ．若 $\htmlData{tutor-start=0,tutor-end=1}{b}$ 是 $\htmlData{tutor-start=0,tutor-end=1}{a}$ 与 $\htmlData{tutor-start=0,tutor-end=1}{c}$ 的等比中项，且 $\sin \htmlData{tutor-start=5,tutor-end=6}{A}$ 是 $\sin\htmlData{tutor-start=4,tutor-end=5}{(}\htmlData{tutor-start=5,tutor-end=6}{B}\htmlData{tutor-start=6,tutor-end=7}{-}\htmlData{tutor-start=7,tutor-end=8}{A}\htmlData{tutor-start=8,tutor-end=9}{)}$ 与 $\sin \htmlData{tutor-start=5,tutor-end=6}{C}$ 的等差中项，求 $\cos \htmlData{tutor-start=5,tutor-end=6}{B}$ 的值．

题解状态：标准答案与规范题解待补充

题目标签：FIRST-A-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（A 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 $\htmlData{tutor-start=0,tutor-end=1}{x}\htmlData{tutor-start=1,tutor-end=2}{O}\htmlData{tutor-start=2,tutor-end=3}{y}$ 中，圆 W 与抛物线 G： $\htmlData{tutor-start=0,tutor-end=1}{y}^{\htmlData{tutor-start=3,tutor-end=4}{2}}\htmlData{tutor-start=5,tutor-end=6}{=}\htmlData{tutor-start=6,tutor-end=7}{4}\htmlData{tutor-start=7,tutor-end=8}{x}$ 恰有一个公共点，且圆 W 与 $\htmlData{tutor-start=0,tutor-end=1}{x}$ 轴相切于 G 的焦点 $\htmlData{tutor-start=0,tutor-end=1}{F}$ ．求圆 W 的半径．

题解状态：标准答案与规范题解待补充

题目标签：FIRST-A-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（A 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 20 分）称一个复数数列 $\htmlData{tutor-start=0,tutor-end=2}{\{}\htmlData{tutor-start=2,tutor-end=3}{z}_{\htmlData{tutor-start=5,tutor-end=6}{n}}\htmlData{tutor-start=7,tutor-end=9}{\}}$ 为“有趣的”，若 $\htmlData{tutor-start=0,tutor-end=1}{|}\htmlData{tutor-start=1,tutor-end=2}{z}_{\htmlData{tutor-start=4,tutor-end=5}{1}}\htmlData{tutor-start=6,tutor-end=7}{|}\htmlData{tutor-start=7,tutor-end=8}{=}\htmlData{tutor-start=8,tutor-end=9}{1}$ ，且对任意正整数 n，均有 $\htmlData{tutor-start=0,tutor-end=1}{4}\htmlData{tutor-start=1,tutor-end=2}{z}_{\htmlData{tutor-start=4,tutor-end=5}{n}\htmlData{tutor-start=5,tutor-end=6}{+}\htmlData{tutor-start=6,tutor-end=7}{1}}^{\htmlData{tutor-start=10,tutor-end=11}{2}}\htmlData{tutor-start=12,tutor-end=13}{+}\htmlData{tutor-start=13,tutor-end=14}{2}\htmlData{tutor-start=14,tutor-end=15}{z}_{\htmlData{tutor-start=17,tutor-end=18}{n}} \htmlData{tutor-start=20,tutor-end=21}{z}_{\htmlData{tutor-start=23,tutor-end=24}{n}\htmlData{tutor-start=24,tutor-end=25}{+}\htmlData{tutor-start=25,tutor-end=26}{1}}\htmlData{tutor-start=27,tutor-end=28}{+}\htmlData{tutor-start=28,tutor-end=29}{z}_{\htmlData{tutor-start=31,tutor-end=32}{n}}^{\htmlData{tutor-start=35,tutor-end=36}{2}}\htmlData{tutor-start=37,tutor-end=38}{=}\htmlData{tutor-start=38,tutor-end=39}{0}$ ．求最大的常数 C，使得对一切有趣的数列 $\htmlData{tutor-start=0,tutor-end=2}{\{}\htmlData{tutor-start=2,tutor-end=3}{z}_{\htmlData{tutor-start=5,tutor-end=6}{n}}\htmlData{tutor-start=7,tutor-end=9}{\}}$ 及任意正整数 m，均有 $\htmlData{tutor-start=0,tutor-end=1}{|}\htmlData{tutor-start=1,tutor-end=2}{z}_{\htmlData{tutor-start=4,tutor-end=5}{1}}\htmlData{tutor-start=6,tutor-end=7}{+}\htmlData{tutor-start=7,tutor-end=8}{z}_{\htmlData{tutor-start=10,tutor-end=11}{2}}\htmlData{tutor-start=12,tutor-end=13}{+}\cdots \htmlData{tutor-start=20,tutor-end=21}{z}_{\htmlData{tutor-start=23,tutor-end=24}{m}}\htmlData{tutor-start=25,tutor-end=26}{|}\htmlData{tutor-start=26,tutor-end=30}{\ge }\htmlData{tutor-start=30,tutor-end=31}{C}$ ．

题解状态：标准答案与规范题解待补充

题目标签：FIRST-A-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（A 卷） · 一、（本题满分 40 分） · 平面几何

如图，在锐角 $\htmlData{tutor-start=0,tutor-end=10}{\triangle }\htmlData{tutor-start=10,tutor-end=11}{A}\htmlData{tutor-start=11,tutor-end=12}{B}\htmlData{tutor-start=12,tutor-end=13}{C}$ 中， $\htmlData{tutor-start=0,tutor-end=1}{M}$ 是 $\htmlData{tutor-start=0,tutor-end=1}{B}\htmlData{tutor-start=1,tutor-end=2}{C}$ 边的中点，点 $\htmlData{tutor-start=0,tutor-end=1}{P}$ 在 $\htmlData{tutor-start=0,tutor-end=10}{\triangle }\htmlData{tutor-start=10,tutor-end=11}{A}\htmlData{tutor-start=11,tutor-end=12}{B}\htmlData{tutor-start=12,tutor-end=13}{C}$ 内，使得 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{P}$ 平分 $\htmlData{tutor-start=0,tutor-end=7}{\angle }\htmlData{tutor-start=7,tutor-end=8}{B}\htmlData{tutor-start=8,tutor-end=9}{A}\htmlData{tutor-start=9,tutor-end=10}{C}$ 。直线 $\htmlData{tutor-start=0,tutor-end=1}{M}\htmlData{tutor-start=1,tutor-end=2}{P}$ 与 $\htmlData{tutor-start=0,tutor-end=10}{\triangle }\htmlData{tutor-start=10,tutor-end=11}{A}\htmlData{tutor-start=11,tutor-end=12}{B}\htmlData{tutor-start=12,tutor-end=13}{P}\htmlData{tutor-start=13,tutor-end=14}{,} \htmlData{tutor-start=15,tutor-end=25}{\triangle }\htmlData{tutor-start=25,tutor-end=26}{A}\htmlData{tutor-start=26,tutor-end=27}{C}\htmlData{tutor-start=27,tutor-end=28}{P}$ 的外接圆分别相交于不同于点 $\htmlData{tutor-start=0,tutor-end=1}{P}$ 的两点 $\htmlData{tutor-start=0,tutor-end=1}{D}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{E}$ 。证明：若 $\htmlData{tutor-start=0,tutor-end=1}{D}\htmlData{tutor-start=1,tutor-end=2}{E}\htmlData{tutor-start=2,tutor-end=3}{=}\htmlData{tutor-start=3,tutor-end=4}{M}\htmlData{tutor-start=4,tutor-end=5}{P}$ ，则 $\htmlData{tutor-start=0,tutor-end=1}{B}\htmlData{tutor-start=1,tutor-end=2}{C}\htmlData{tutor-start=2,tutor-end=3}{=}\htmlData{tutor-start=3,tutor-end=4}{2}\htmlData{tutor-start=4,tutor-end=5}{B}\htmlData{tutor-start=5,tutor-end=6}{P}$ 。

题解状态：标准答案与规范题解待补充

题目标签：SECOND-A-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（A 卷） · 二、（本题满分 40 分） · 数论

设整数 $\htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{a}_{\htmlData{tutor-start=10,tutor-end=11}{2}}\htmlData{tutor-start=12,tutor-end=13}{,} \cdots\htmlData{tutor-start=20,tutor-end=21}{,} \htmlData{tutor-start=22,tutor-end=23}{a}_{\htmlData{tutor-start=25,tutor-end=26}{2}\htmlData{tutor-start=26,tutor-end=27}{0}\htmlData{tutor-start=27,tutor-end=28}{1}\htmlData{tutor-start=28,tutor-end=29}{9}}$ 满足 $\htmlData{tutor-start=0,tutor-end=1}{1}\htmlData{tutor-start=1,tutor-end=2}{=}\htmlData{tutor-start=2,tutor-end=3}{a}_{\htmlData{tutor-start=5,tutor-end=6}{1}} \htmlData{tutor-start=8,tutor-end=12}{\le }\htmlData{tutor-start=12,tutor-end=13}{a}_{\htmlData{tutor-start=15,tutor-end=16}{2}} \htmlData{tutor-start=18,tutor-end=22}{\le }\cdots \htmlData{tutor-start=29,tutor-end=33}{\le }\htmlData{tutor-start=33,tutor-end=34}{a}_{\htmlData{tutor-start=36,tutor-end=37}{2}\htmlData{tutor-start=37,tutor-end=38}{0}\htmlData{tutor-start=38,tutor-end=39}{1}\htmlData{tutor-start=39,tutor-end=40}{9}}\htmlData{tutor-start=41,tutor-end=42}{=}\htmlData{tutor-start=42,tutor-end=43}{9}\htmlData{tutor-start=43,tutor-end=44}{9}$ 。记 $\htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{=}\htmlData{tutor-start=2,tutor-end=3}{(}\htmlData{tutor-start=3,tutor-end=4}{a}_{\htmlData{tutor-start=6,tutor-end=7}{1}}^{\htmlData{tutor-start=10,tutor-end=11}{2}}\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=20,tutor-end=21}{2}}\htmlData{tutor-start=22,tutor-end=23}{+}\cdots\htmlData{tutor-start=29,tutor-end=30}{+}\htmlData{tutor-start=30,tutor-end=31}{a}_{\htmlData{tutor-start=33,tutor-end=34}{2}\htmlData{tutor-start=34,tutor-end=35}{0}\htmlData{tutor-start=35,tutor-end=36}{1}\htmlData{tutor-start=36,tutor-end=37}{9}}^{\htmlData{tutor-start=40,tutor-end=41}{2}}\htmlData{tutor-start=42,tutor-end=43}{)}\htmlData{tutor-start=43,tutor-end=44}{-}\htmlData{tutor-start=44,tutor-end=45}{(}\htmlData{tutor-start=45,tutor-end=46}{a}_{\htmlData{tutor-start=48,tutor-end=49}{1}}\htmlData{tutor-start=50,tutor-end=51}{a}_{\htmlData{tutor-start=53,tutor-end=54}{3}}\htmlData{tutor-start=55,tutor-end=56}{+}\htmlData{tutor-start=56,tutor-end=57}{a}_{\htmlData{tutor-start=59,tutor-end=60}{2}}\htmlData{tutor-start=61,tutor-end=62}{a}_{\htmlData{tutor-start=64,tutor-end=65}{4}}\htmlData{tutor-start=66,tutor-end=67}{+}\htmlData{tutor-start=67,tutor-end=68}{a}_{\htmlData{tutor-start=70,tutor-end=71}{3}}\htmlData{tutor-start=72,tutor-end=73}{a}_{\htmlData{tutor-start=75,tutor-end=76}{5}}\htmlData{tutor-start=77,tutor-end=78}{+}\cdots\htmlData{tutor-start=84,tutor-end=85}{+}\htmlData{tutor-start=85,tutor-end=86}{a}_{\htmlData{tutor-start=88,tutor-end=89}{2}\htmlData{tutor-start=89,tutor-end=90}{0}\htmlData{tutor-start=90,tutor-end=91}{1}\htmlData{tutor-start=91,tutor-end=92}{7}}\htmlData{tutor-start=93,tutor-end=94}{a}_{\htmlData{tutor-start=96,tutor-end=97}{2}\htmlData{tutor-start=97,tutor-end=98}{0}\htmlData{tutor-start=98,tutor-end=99}{1}\htmlData{tutor-start=99,tutor-end=100}{9}}\htmlData{tutor-start=101,tutor-end=102}{)}$ 。求 $\htmlData{tutor-start=0,tutor-end=1}{f}$ 的最小值 $\htmlData{tutor-start=0,tutor-end=1}{f}_{\htmlData{tutor-start=3,tutor-end=4}{0}}$ ，并确定使 $\htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{=}\htmlData{tutor-start=2,tutor-end=3}{f}_{\htmlData{tutor-start=5,tutor-end=6}{0}}$ 成立的数组 $\htmlData{tutor-start=0,tutor-end=1}{(}\htmlData{tutor-start=1,tutor-end=2}{a}_{\htmlData{tutor-start=4,tutor-end=5}{1}}\htmlData{tutor-start=6,tutor-end=7}{,} \htmlData{tutor-start=8,tutor-end=9}{a}_{\htmlData{tutor-start=11,tutor-end=12}{2}}\htmlData{tutor-start=13,tutor-end=14}{,} \cdots\htmlData{tutor-start=21,tutor-end=22}{,} \htmlData{tutor-start=23,tutor-end=24}{a}_{\htmlData{tutor-start=26,tutor-end=27}{2}\htmlData{tutor-start=27,tutor-end=28}{0}\htmlData{tutor-start=28,tutor-end=29}{1}\htmlData{tutor-start=29,tutor-end=30}{9}}\htmlData{tutor-start=31,tutor-end=32}{)}$ 的个数。

题解状态：标准答案与规范题解待补充

题目标签：SECOND-A-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（A 卷） · 二试 · 数论

（本题满分 50 分）设 $\htmlData{tutor-start=0,tutor-end=1}{m}$ 为整数， $\htmlData{tutor-start=0,tutor-end=1}{|}\htmlData{tutor-start=1,tutor-end=2}{m}\htmlData{tutor-start=2,tutor-end=3}{|} \htmlData{tutor-start=4,tutor-end=8}{\ge }\htmlData{tutor-start=8,tutor-end=9}{2}$ ．整数数列 $\htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{a}_{\htmlData{tutor-start=10,tutor-end=11}{2}}\htmlData{tutor-start=12,tutor-end=13}{,} \cdots$ 满足： $\htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{a}_{\htmlData{tutor-start=10,tutor-end=11}{2}}$ 不全为零，且对任意正整数 $\htmlData{tutor-start=0,tutor-end=1}{n}$ ，均有 $\htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{n}\htmlData{tutor-start=4,tutor-end=5}{+}\htmlData{tutor-start=5,tutor-end=6}{2}} \htmlData{tutor-start=8,tutor-end=9}{=} \htmlData{tutor-start=10,tutor-end=11}{a}_{\htmlData{tutor-start=13,tutor-end=14}{n}\htmlData{tutor-start=14,tutor-end=15}{+}\htmlData{tutor-start=15,tutor-end=16}{1}} \htmlData{tutor-start=18,tutor-end=19}{-} \htmlData{tutor-start=20,tutor-end=21}{m} \htmlData{tutor-start=22,tutor-end=23}{a}_{\htmlData{tutor-start=25,tutor-end=26}{n}}$ ．证明：若存在整数 $\htmlData{tutor-start=0,tutor-end=1}{r}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{s}$ ( $\htmlData{tutor-start=0,tutor-end=1}{r} \htmlData{tutor-start=2,tutor-end=3}{>} \htmlData{tutor-start=4,tutor-end=5}{s} \htmlData{tutor-start=6,tutor-end=10}{\ge }\htmlData{tutor-start=10,tutor-end=11}{2}$ ) 使得 $\htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{r}} \htmlData{tutor-start=6,tutor-end=7}{=} \htmlData{tutor-start=8,tutor-end=9}{a}_{\htmlData{tutor-start=11,tutor-end=12}{s}} \htmlData{tutor-start=14,tutor-end=15}{=} \htmlData{tutor-start=16,tutor-end=17}{a}_{\htmlData{tutor-start=19,tutor-end=20}{1}}$ ，则 $\htmlData{tutor-start=0,tutor-end=1}{r} \htmlData{tutor-start=2,tutor-end=3}{-} \htmlData{tutor-start=4,tutor-end=5}{s} \htmlData{tutor-start=6,tutor-end=10}{\ge }\htmlData{tutor-start=10,tutor-end=11}{|}\htmlData{tutor-start=11,tutor-end=12}{m}\htmlData{tutor-start=12,tutor-end=13}{|}$ ．

题解状态：标准答案与规范题解待补充

题目标签：SECOND-A-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（A 卷） · 二试 · 组合数学

（本题满分 50 分）设 $\htmlData{tutor-start=0,tutor-end=1}{V}$ 是空间中 2019 个点构成的集合，其中任意四点不共面．某些点之间连有线段，记 $\htmlData{tutor-start=0,tutor-end=1}{E}$ 为这些线段构成的集合．试求最小的正整数 $\htmlData{tutor-start=0,tutor-end=1}{n}$ ，满足条件：若 $\htmlData{tutor-start=0,tutor-end=1}{E}$ 至少有 $\htmlData{tutor-start=0,tutor-end=1}{n}$ 个元素，则 $\htmlData{tutor-start=0,tutor-end=1}{E}$ 一定含有 908 个二元子集，其中每个二元子集中的两条线段有公共端点，且任意两个二元子集的交为空集．

题解状态：标准答案与规范题解待补充

题目标签：SECOND-A-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。