2020 年全国高中数学联赛 A 卷（学生版）

books/national_high_school_league_students/papers/competition-national-league-2020-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，共 64 分. · 代数

在等比数列 $\htmlData{tutor-start=0,tutor-end=2}{\{}\htmlData{tutor-start=2,tutor-end=3}{a}_{\htmlData{tutor-start=5,tutor-end=6}{n}}\htmlData{tutor-start=7,tutor-end=9}{\}}$ 中， $\htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{9}}\htmlData{tutor-start=5,tutor-end=6}{=}\htmlData{tutor-start=6,tutor-end=7}{1}\htmlData{tutor-start=7,tutor-end=8}{3}$ ， $\htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{1}\htmlData{tutor-start=4,tutor-end=5}{3}}\htmlData{tutor-start=6,tutor-end=7}{=}\htmlData{tutor-start=7,tutor-end=8}{1}$ ，则 $\log_{\htmlData{tutor-start=6,tutor-end=7}{a}_{\htmlData{tutor-start=9,tutor-end=10}{1}}} \htmlData{tutor-start=13,tutor-end=14}{1}\htmlData{tutor-start=14,tutor-end=15}{3}$ 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，共 64 分. · 平面几何

在椭圆 $\htmlData{tutor-start=0,tutor-end=6}{\Gamma}$ 中， $\htmlData{tutor-start=0,tutor-end=1}{A}$ 为长轴的一个端点， $\htmlData{tutor-start=0,tutor-end=1}{B}$ 为短轴的一个端点， $\htmlData{tutor-start=0,tutor-end=1}{F}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{F}_{\htmlData{tutor-start=10,tutor-end=11}{2}}$ 为两个焦点. 若 $\overrightarrow{\htmlData{tutor-start=16,tutor-end=17}{A}\htmlData{tutor-start=17,tutor-end=18}{F}_{\htmlData{tutor-start=20,tutor-end=21}{1}}} \htmlData{tutor-start=24,tutor-end=30}{\cdot }\overrightarrow{\htmlData{tutor-start=46,tutor-end=47}{A}\htmlData{tutor-start=47,tutor-end=48}{F}_{\htmlData{tutor-start=50,tutor-end=51}{2}}} \htmlData{tutor-start=54,tutor-end=55}{+} \overrightarrow{\htmlData{tutor-start=72,tutor-end=73}{B}\htmlData{tutor-start=73,tutor-end=74}{F}_{\htmlData{tutor-start=76,tutor-end=77}{1}}} \htmlData{tutor-start=80,tutor-end=86}{\cdot }\overrightarrow{\htmlData{tutor-start=102,tutor-end=103}{B}\htmlData{tutor-start=103,tutor-end=104}{F}_{\htmlData{tutor-start=106,tutor-end=107}{2}}} \htmlData{tutor-start=110,tutor-end=111}{=} \htmlData{tutor-start=112,tutor-end=113}{0}$ ，则 $\frac{\htmlData{tutor-start=6,tutor-end=7}{|}\htmlData{tutor-start=7,tutor-end=8}{A}\htmlData{tutor-start=8,tutor-end=9}{B}\htmlData{tutor-start=9,tutor-end=10}{|}}{\htmlData{tutor-start=12,tutor-end=13}{|}\htmlData{tutor-start=13,tutor-end=14}{F}_{\htmlData{tutor-start=16,tutor-end=17}{1}}\htmlData{tutor-start=18,tutor-end=19}{F}_{\htmlData{tutor-start=21,tutor-end=22}{2}}\htmlData{tutor-start=23,tutor-end=24}{|}}$ 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，共 64 分. · 代数

设 $\htmlData{tutor-start=0,tutor-end=1}{a}\htmlData{tutor-start=1,tutor-end=2}{>}\htmlData{tutor-start=2,tutor-end=3}{0}$ ，函数 $\htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{x}\htmlData{tutor-start=3,tutor-end=4}{)}\htmlData{tutor-start=4,tutor-end=5}{=}\htmlData{tutor-start=5,tutor-end=6}{x}\htmlData{tutor-start=6,tutor-end=7}{+}\frac{\htmlData{tutor-start=13,tutor-end=14}{1}\htmlData{tutor-start=14,tutor-end=15}{0}\htmlData{tutor-start=15,tutor-end=16}{0}}{\htmlData{tutor-start=18,tutor-end=19}{x}}$ 在区间 $\htmlData{tutor-start=0,tutor-end=1}{(}\htmlData{tutor-start=1,tutor-end=2}{0}\htmlData{tutor-start=2,tutor-end=3}{,} \htmlData{tutor-start=4,tutor-end=5}{a}\htmlData{tutor-start=5,tutor-end=6}{]}$ 上的最小值为 $\htmlData{tutor-start=0,tutor-end=1}{m}_{\htmlData{tutor-start=3,tutor-end=4}{1}}$ ，在区间 $\htmlData{tutor-start=0,tutor-end=1}{[}\htmlData{tutor-start=1,tutor-end=2}{a}\htmlData{tutor-start=2,tutor-end=3}{,} \htmlData{tutor-start=4,tutor-end=5}{+}\htmlData{tutor-start=5,tutor-end=11}{\infty}\htmlData{tutor-start=11,tutor-end=12}{)}$ 上的最小值为 $\htmlData{tutor-start=0,tutor-end=1}{m}_{\htmlData{tutor-start=3,tutor-end=4}{2}}$ . 若 $\htmlData{tutor-start=0,tutor-end=1}{m}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{m}_{\htmlData{tutor-start=8,tutor-end=9}{2}}\htmlData{tutor-start=10,tutor-end=11}{=}\htmlData{tutor-start=11,tutor-end=12}{2}\htmlData{tutor-start=12,tutor-end=13}{0}\htmlData{tutor-start=13,tutor-end=14}{2}\htmlData{tutor-start=14,tutor-end=15}{0}$ ，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-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，共 64 分. · 数学竞赛/待细分

设 $\htmlData{tutor-start=0,tutor-end=1}{z}$ 为复数，若 $\frac{\htmlData{tutor-start=6,tutor-end=7}{z}\htmlData{tutor-start=7,tutor-end=8}{-}\htmlData{tutor-start=8,tutor-end=9}{2}}{\htmlData{tutor-start=11,tutor-end=12}{z}\htmlData{tutor-start=12,tutor-end=13}{-}\mathrm{\htmlData{tutor-start=21,tutor-end=22}{i}}}$ 为实数（ $\mathrm{\htmlData{tutor-start=8,tutor-end=9}{i}}$ 为虚数单位），则 $\htmlData{tutor-start=0,tutor-end=1}{|}\htmlData{tutor-start=1,tutor-end=2}{z}\htmlData{tutor-start=2,tutor-end=3}{+}\htmlData{tutor-start=3,tutor-end=4}{3}\htmlData{tutor-start=4,tutor-end=5}{|}$ 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-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，共 64 分. · 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， $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{B}\htmlData{tutor-start=2,tutor-end=3}{=}\htmlData{tutor-start=3,tutor-end=4}{6}$ ， $\htmlData{tutor-start=0,tutor-end=1}{B}\htmlData{tutor-start=1,tutor-end=2}{C}\htmlData{tutor-start=2,tutor-end=3}{=}\htmlData{tutor-start=3,tutor-end=4}{4}$ ，边 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{C}$ 上的中线长为 $\sqrt{\htmlData{tutor-start=6,tutor-end=7}{1}\htmlData{tutor-start=7,tutor-end=8}{0}}$ ，则 $\sin^{\htmlData{tutor-start=6,tutor-end=7}{6}} \frac{\htmlData{tutor-start=15,tutor-end=16}{A}}{\htmlData{tutor-start=18,tutor-end=19}{2}} \htmlData{tutor-start=21,tutor-end=22}{+} \cos^{\htmlData{tutor-start=29,tutor-end=30}{6}} \frac{\htmlData{tutor-start=38,tutor-end=39}{A}}{\htmlData{tutor-start=41,tutor-end=42}{2}}$ 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-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，共 64 分. · 数学竞赛/待细分

正三棱锥 $\htmlData{tutor-start=0,tutor-end=1}{P}\htmlData{tutor-start=1,tutor-end=2}{-}\htmlData{tutor-start=2,tutor-end=3}{A}\htmlData{tutor-start=3,tutor-end=4}{B}\htmlData{tutor-start=4,tutor-end=5}{C}$ 的所有棱长均为 1， $\htmlData{tutor-start=0,tutor-end=1}{L}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{M}\htmlData{tutor-start=4,tutor-end=5}{,} \htmlData{tutor-start=6,tutor-end=7}{N}$ 分别为棱 $\htmlData{tutor-start=0,tutor-end=1}{P}\htmlData{tutor-start=1,tutor-end=2}{A}\htmlData{tutor-start=2,tutor-end=3}{,} \htmlData{tutor-start=4,tutor-end=5}{P}\htmlData{tutor-start=5,tutor-end=6}{B}\htmlData{tutor-start=6,tutor-end=7}{,} \htmlData{tutor-start=8,tutor-end=9}{P}\htmlData{tutor-start=9,tutor-end=10}{C}$ 的中点，则该正三棱锥的外接球被平面 $\htmlData{tutor-start=0,tutor-end=1}{L}\htmlData{tutor-start=1,tutor-end=2}{M}\htmlData{tutor-start=2,tutor-end=3}{N}$ 所截的截面面积为________.

题解状态：标准答案与规范题解待补充

题目标签：FIRST-A-FILL-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，共 64 分. · 数学竞赛/待细分

设 $\htmlData{tutor-start=0,tutor-end=1}{a}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{b}\htmlData{tutor-start=4,tutor-end=5}{>}\htmlData{tutor-start=5,tutor-end=6}{0}$ ，满足：关于 $\htmlData{tutor-start=0,tutor-end=1}{x}$ 的方程 $\sqrt{\htmlData{tutor-start=6,tutor-end=7}{|}\htmlData{tutor-start=7,tutor-end=8}{x}\htmlData{tutor-start=8,tutor-end=9}{|}} \htmlData{tutor-start=11,tutor-end=12}{+} \sqrt{\htmlData{tutor-start=19,tutor-end=20}{|}\htmlData{tutor-start=20,tutor-end=21}{x}\htmlData{tutor-start=21,tutor-end=22}{+}\htmlData{tutor-start=22,tutor-end=23}{a}\htmlData{tutor-start=23,tutor-end=24}{|}} \htmlData{tutor-start=26,tutor-end=27}{=} \htmlData{tutor-start=28,tutor-end=29}{b}$ 恰有三个不同的实数解 $\htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{x}_{\htmlData{tutor-start=10,tutor-end=11}{2}}\htmlData{tutor-start=12,tutor-end=13}{,} \htmlData{tutor-start=14,tutor-end=15}{x}_{\htmlData{tutor-start=17,tutor-end=18}{3}}$ ，且 $\htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{1}} \htmlData{tutor-start=6,tutor-end=7}{<} \htmlData{tutor-start=8,tutor-end=9}{x}_{\htmlData{tutor-start=11,tutor-end=12}{2}} \htmlData{tutor-start=14,tutor-end=15}{<} \htmlData{tutor-start=16,tutor-end=17}{x}_{\htmlData{tutor-start=19,tutor-end=20}{3}} \htmlData{tutor-start=22,tutor-end=23}{=} \htmlData{tutor-start=24,tutor-end=25}{b}$ ，则 $\htmlData{tutor-start=0,tutor-end=1}{a}\htmlData{tutor-start=1,tutor-end=2}{+}\htmlData{tutor-start=2,tutor-end=3}{b}$ 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-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，共 64 分. · 数学竞赛/待细分

现有 10 张卡片，每张卡片上写有 1, 2, 3, 4, 5 中两个不同的数，且任意两张卡片上的数不完全相同. 将这 10 张卡片放入标号为 1, 2, 3, 4, 5 的五个盒子中，规定写有 $\htmlData{tutor-start=0,tutor-end=1}{i}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{j}$ 的卡片只能放在 $\htmlData{tutor-start=0,tutor-end=1}{i}$ 号或 $\htmlData{tutor-start=0,tutor-end=1}{j}$ 号盒子中. 一种放法称为“好的”，如果 1 号盒子中的卡片数明显多于其他每个盒子中的卡片数，则“好的”方法共有________种.

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，共 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， $\sin \htmlData{tutor-start=5,tutor-end=6}{A} \htmlData{tutor-start=7,tutor-end=8}{=} \frac{\sqrt{\htmlData{tutor-start=21,tutor-end=22}{2}}}{\htmlData{tutor-start=25,tutor-end=26}{2}}$ . 求 $\cos \htmlData{tutor-start=5,tutor-end=6}{B} \htmlData{tutor-start=7,tutor-end=8}{+} \htmlData{tutor-start=9,tutor-end=10}{2}\cos \htmlData{tutor-start=15,tutor-end=16}{C}$ 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-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，共 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）对正整数 $\htmlData{tutor-start=0,tutor-end=1}{n}$ 及实数 $\htmlData{tutor-start=0,tutor-end=1}{x}$ ( $\htmlData{tutor-start=0,tutor-end=1}{0} \htmlData{tutor-start=2,tutor-end=12}{\leqslant }\htmlData{tutor-start=12,tutor-end=13}{x} \htmlData{tutor-start=14,tutor-end=15}{<} \htmlData{tutor-start=16,tutor-end=17}{n}$ )，定义 $\htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{n}\htmlData{tutor-start=3,tutor-end=4}{,} \htmlData{tutor-start=5,tutor-end=6}{x}\htmlData{tutor-start=6,tutor-end=7}{)} \htmlData{tutor-start=8,tutor-end=9}{=} \htmlData{tutor-start=10,tutor-end=11}{(}\htmlData{tutor-start=11,tutor-end=12}{1}\htmlData{tutor-start=12,tutor-end=13}{-}\htmlData{tutor-start=13,tutor-end=15}{\{}\htmlData{tutor-start=15,tutor-end=16}{x}\htmlData{tutor-start=16,tutor-end=18}{\}}\htmlData{tutor-start=18,tutor-end=19}{)} \htmlData{tutor-start=20,tutor-end=26}{\cdot }\mathrm{\htmlData{tutor-start=34,tutor-end=35}{C}}_{\htmlData{tutor-start=38,tutor-end=39}{n}}^{\htmlData{tutor-start=42,tutor-end=43}{[}\htmlData{tutor-start=43,tutor-end=44}{x}\htmlData{tutor-start=44,tutor-end=45}{]}} \htmlData{tutor-start=47,tutor-end=48}{+} \htmlData{tutor-start=49,tutor-end=51}{\{}\htmlData{tutor-start=51,tutor-end=52}{x}\htmlData{tutor-start=52,tutor-end=54}{\}} \htmlData{tutor-start=55,tutor-end=61}{\cdot }\mathrm{\htmlData{tutor-start=69,tutor-end=70}{C}}_{\htmlData{tutor-start=73,tutor-end=74}{n}}^{\htmlData{tutor-start=77,tutor-end=78}{[}\htmlData{tutor-start=78,tutor-end=79}{x}\htmlData{tutor-start=79,tutor-end=80}{]}\htmlData{tutor-start=80,tutor-end=81}{+}\htmlData{tutor-start=81,tutor-end=82}{1}}$ ，其中 $\htmlData{tutor-start=0,tutor-end=1}{[}\htmlData{tutor-start=1,tutor-end=2}{x}\htmlData{tutor-start=2,tutor-end=3}{]}$ 表示不超过实数 $\htmlData{tutor-start=0,tutor-end=1}{x}$ 的最大整数， $\htmlData{tutor-start=0,tutor-end=2}{\{}\htmlData{tutor-start=2,tutor-end=3}{x}\htmlData{tutor-start=3,tutor-end=5}{\}} \htmlData{tutor-start=6,tutor-end=7}{=} \htmlData{tutor-start=8,tutor-end=9}{x} \htmlData{tutor-start=10,tutor-end=11}{-} \htmlData{tutor-start=12,tutor-end=13}{[}\htmlData{tutor-start=13,tutor-end=14}{x}\htmlData{tutor-start=14,tutor-end=15}{]}$ . 若整数 $\htmlData{tutor-start=0,tutor-end=1}{m}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{n} \htmlData{tutor-start=5,tutor-end=15}{\geqslant }\htmlData{tutor-start=15,tutor-end=16}{2}$ 满足 $\htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{m}\htmlData{tutor-start=3,tutor-end=4}{,} \frac{\htmlData{tutor-start=11,tutor-end=12}{1}}{\htmlData{tutor-start=14,tutor-end=15}{n}}\htmlData{tutor-start=16,tutor-end=17}{)} \htmlData{tutor-start=18,tutor-end=19}{+} \htmlData{tutor-start=20,tutor-end=21}{f}\htmlData{tutor-start=21,tutor-end=22}{(}\htmlData{tutor-start=22,tutor-end=23}{m}\htmlData{tutor-start=23,tutor-end=24}{,} \frac{\htmlData{tutor-start=31,tutor-end=32}{2}}{\htmlData{tutor-start=34,tutor-end=35}{n}}\htmlData{tutor-start=36,tutor-end=37}{)} \htmlData{tutor-start=38,tutor-end=39}{+} \cdots \htmlData{tutor-start=47,tutor-end=48}{+} \htmlData{tutor-start=49,tutor-end=50}{f}\htmlData{tutor-start=50,tutor-end=51}{(}\htmlData{tutor-start=51,tutor-end=52}{m}\htmlData{tutor-start=52,tutor-end=53}{,} \frac{\htmlData{tutor-start=60,tutor-end=61}{m}\htmlData{tutor-start=61,tutor-end=62}{n}\htmlData{tutor-start=62,tutor-end=63}{-}\htmlData{tutor-start=63,tutor-end=64}{1}}{\htmlData{tutor-start=66,tutor-end=67}{n}}\htmlData{tutor-start=68,tutor-end=69}{)} \htmlData{tutor-start=70,tutor-end=71}{=} \htmlData{tutor-start=72,tutor-end=73}{1}\htmlData{tutor-start=73,tutor-end=74}{2}\htmlData{tutor-start=74,tutor-end=75}{3}$ ，求 $\htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{n}\htmlData{tutor-start=3,tutor-end=4}{,} \frac{\htmlData{tutor-start=11,tutor-end=12}{1}}{\htmlData{tutor-start=14,tutor-end=15}{m}}\htmlData{tutor-start=16,tutor-end=17}{)} \htmlData{tutor-start=18,tutor-end=19}{+} \htmlData{tutor-start=20,tutor-end=21}{f}\htmlData{tutor-start=21,tutor-end=22}{(}\htmlData{tutor-start=22,tutor-end=23}{n}\htmlData{tutor-start=23,tutor-end=24}{,} \frac{\htmlData{tutor-start=31,tutor-end=32}{2}}{\htmlData{tutor-start=34,tutor-end=35}{m}}\htmlData{tutor-start=36,tutor-end=37}{)} \htmlData{tutor-start=38,tutor-end=39}{+} \cdots \htmlData{tutor-start=47,tutor-end=48}{+} \htmlData{tutor-start=49,tutor-end=50}{f}\htmlData{tutor-start=50,tutor-end=51}{(}\htmlData{tutor-start=51,tutor-end=52}{n}\htmlData{tutor-start=52,tutor-end=53}{,} \frac{\htmlData{tutor-start=60,tutor-end=61}{m}\htmlData{tutor-start=61,tutor-end=62}{n}\htmlData{tutor-start=62,tutor-end=63}{-}\htmlData{tutor-start=63,tutor-end=64}{1}}{\htmlData{tutor-start=66,tutor-end=67}{m}}\htmlData{tutor-start=68,tutor-end=69}{)}$ 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-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，共 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，点 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{B}\htmlData{tutor-start=4,tutor-end=5}{,} \htmlData{tutor-start=6,tutor-end=7}{C}$ 在双曲线 $\htmlData{tutor-start=0,tutor-end=1}{x}\htmlData{tutor-start=1,tutor-end=2}{y}\htmlData{tutor-start=2,tutor-end=3}{=}\htmlData{tutor-start=3,tutor-end=4}{1}$ 上，满足 $\htmlData{tutor-start=0,tutor-end=10}{\triangle }\htmlData{tutor-start=10,tutor-end=11}{A}\htmlData{tutor-start=11,tutor-end=12}{B}\htmlData{tutor-start=12,tutor-end=13}{C}$ 为等腰直角三角形. 求 $\htmlData{tutor-start=0,tutor-end=10}{\triangle }\htmlData{tutor-start=10,tutor-end=11}{A}\htmlData{tutor-start=11,tutor-end=12}{B}\htmlData{tutor-start=12,tutor-end=13}{C}$ 面积的最小值.

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，在等腰 $\htmlData{tutor-start=0,tutor-end=10}{\triangle }\htmlData{tutor-start=10,tutor-end=11}{A}\htmlData{tutor-start=11,tutor-end=12}{B}\htmlData{tutor-start=12,tutor-end=13}{C}$ 中， $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{B}\htmlData{tutor-start=2,tutor-end=3}{=}\htmlData{tutor-start=3,tutor-end=4}{B}\htmlData{tutor-start=4,tutor-end=5}{C}$ ， $\htmlData{tutor-start=0,tutor-end=1}{I}$ 为内心， $\htmlData{tutor-start=0,tutor-end=1}{M}$ 为 $\htmlData{tutor-start=0,tutor-end=1}{B}\htmlData{tutor-start=1,tutor-end=2}{I}$ 的中点， $\htmlData{tutor-start=0,tutor-end=1}{P}$ 为边 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{C}$ 上一点，满足 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{P}\htmlData{tutor-start=2,tutor-end=3}{=}\htmlData{tutor-start=3,tutor-end=4}{3}\htmlData{tutor-start=4,tutor-end=5}{P}\htmlData{tutor-start=5,tutor-end=6}{C}$ ， $\htmlData{tutor-start=0,tutor-end=1}{P}\htmlData{tutor-start=1,tutor-end=2}{I}$ 延长线上一点 $\htmlData{tutor-start=0,tutor-end=1}{H}$ 满足 $\htmlData{tutor-start=0,tutor-end=1}{M}\htmlData{tutor-start=1,tutor-end=2}{H} \htmlData{tutor-start=3,tutor-end=9}{\perp }\htmlData{tutor-start=9,tutor-end=10}{P}\htmlData{tutor-start=10,tutor-end=11}{H}$ ， $\htmlData{tutor-start=0,tutor-end=1}{Q}$ 为 $\htmlData{tutor-start=0,tutor-end=10}{\triangle }\htmlData{tutor-start=10,tutor-end=11}{A}\htmlData{tutor-start=11,tutor-end=12}{B}\htmlData{tutor-start=12,tutor-end=13}{C}$ 的外接圆上劣弧 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{B}$ 的中点。证明： $\htmlData{tutor-start=0,tutor-end=1}{B}\htmlData{tutor-start=1,tutor-end=2}{H} \htmlData{tutor-start=3,tutor-end=9}{\perp }\htmlData{tutor-start=9,tutor-end=10}{Q}\htmlData{tutor-start=10,tutor-end=11}{H}$ 。（答题时请将图画在答卷纸上）

**原卷图示 1**原卷第 2 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-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） · 数学竞赛/待细分

给定正数 $\htmlData{tutor-start=0,tutor-end=1}{n} \htmlData{tutor-start=2,tutor-end=12}{\geqslant }\htmlData{tutor-start=12,tutor-end=13}{3}$ ，设 $\htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{a}_{\htmlData{tutor-start=10,tutor-end=11}{2}}\htmlData{tutor-start=12,tutor-end=13}{,} \cdots\htmlData{tutor-start=20,tutor-end=21}{,} \htmlData{tutor-start=22,tutor-end=23}{a}_{\htmlData{tutor-start=25,tutor-end=26}{2}\htmlData{tutor-start=26,tutor-end=27}{n}}$ ， $\htmlData{tutor-start=0,tutor-end=1}{b}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{b}_{\htmlData{tutor-start=10,tutor-end=11}{2}}\htmlData{tutor-start=12,tutor-end=13}{,} \cdots\htmlData{tutor-start=20,tutor-end=21}{,} \htmlData{tutor-start=22,tutor-end=23}{b}_{\htmlData{tutor-start=25,tutor-end=26}{2}\htmlData{tutor-start=26,tutor-end=27}{n}}$ 是 $\htmlData{tutor-start=0,tutor-end=1}{4}\htmlData{tutor-start=1,tutor-end=2}{n}$ 个非负实数，满足 $\htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{1}} \htmlData{tutor-start=6,tutor-end=7}{+} \htmlData{tutor-start=8,tutor-end=9}{a}_{\htmlData{tutor-start=11,tutor-end=12}{2}} \htmlData{tutor-start=14,tutor-end=15}{+} \cdots \htmlData{tutor-start=23,tutor-end=24}{+} \htmlData{tutor-start=25,tutor-end=26}{a}_{\htmlData{tutor-start=28,tutor-end=29}{2}\htmlData{tutor-start=29,tutor-end=30}{n}} \htmlData{tutor-start=32,tutor-end=33}{=} \htmlData{tutor-start=34,tutor-end=35}{b}_{\htmlData{tutor-start=37,tutor-end=38}{1}} \htmlData{tutor-start=40,tutor-end=41}{+} \htmlData{tutor-start=42,tutor-end=43}{b}_{\htmlData{tutor-start=45,tutor-end=46}{2}} \htmlData{tutor-start=48,tutor-end=49}{+} \cdots \htmlData{tutor-start=57,tutor-end=58}{+} \htmlData{tutor-start=59,tutor-end=60}{b}_{\htmlData{tutor-start=62,tutor-end=63}{2}\htmlData{tutor-start=63,tutor-end=64}{n}} \htmlData{tutor-start=66,tutor-end=67}{>} \htmlData{tutor-start=68,tutor-end=69}{0}\htmlData{tutor-start=69,tutor-end=70}{,}$ 且对任意 $\htmlData{tutor-start=0,tutor-end=1}{i}\htmlData{tutor-start=1,tutor-end=2}{=}\htmlData{tutor-start=2,tutor-end=3}{1}\htmlData{tutor-start=3,tutor-end=4}{,} \htmlData{tutor-start=5,tutor-end=6}{2}\htmlData{tutor-start=6,tutor-end=7}{,} \cdots\htmlData{tutor-start=14,tutor-end=15}{,} \htmlData{tutor-start=16,tutor-end=17}{2}\htmlData{tutor-start=17,tutor-end=18}{n}$ ，有 $\htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{i}} \htmlData{tutor-start=6,tutor-end=7}{a}_{\htmlData{tutor-start=9,tutor-end=10}{i}\htmlData{tutor-start=10,tutor-end=11}{+}\htmlData{tutor-start=11,tutor-end=12}{2}} \htmlData{tutor-start=14,tutor-end=24}{\geqslant }\htmlData{tutor-start=24,tutor-end=25}{b}_{\htmlData{tutor-start=27,tutor-end=28}{i}} \htmlData{tutor-start=30,tutor-end=31}{b}_{\htmlData{tutor-start=33,tutor-end=34}{i}\htmlData{tutor-start=34,tutor-end=35}{+}\htmlData{tutor-start=35,tutor-end=36}{1}}$ （这里 $\htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{2}\htmlData{tutor-start=4,tutor-end=5}{n}\htmlData{tutor-start=5,tutor-end=6}{+}\htmlData{tutor-start=6,tutor-end=7}{1}}\htmlData{tutor-start=8,tutor-end=9}{=}\htmlData{tutor-start=9,tutor-end=10}{a}_{\htmlData{tutor-start=12,tutor-end=13}{1}}$ ， $\htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{2}\htmlData{tutor-start=4,tutor-end=5}{n}\htmlData{tutor-start=5,tutor-end=6}{+}\htmlData{tutor-start=6,tutor-end=7}{2}}\htmlData{tutor-start=8,tutor-end=9}{=}\htmlData{tutor-start=9,tutor-end=10}{a}_{\htmlData{tutor-start=12,tutor-end=13}{2}}$ ， $\htmlData{tutor-start=0,tutor-end=1}{b}_{\htmlData{tutor-start=3,tutor-end=4}{2}\htmlData{tutor-start=4,tutor-end=5}{n}\htmlData{tutor-start=5,tutor-end=6}{+}\htmlData{tutor-start=6,tutor-end=7}{1}}\htmlData{tutor-start=8,tutor-end=9}{=}\htmlData{tutor-start=9,tutor-end=10}{b}_{\htmlData{tutor-start=12,tutor-end=13}{1}}$ ）。求 $\htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{1}} \htmlData{tutor-start=6,tutor-end=7}{+} \htmlData{tutor-start=8,tutor-end=9}{a}_{\htmlData{tutor-start=11,tutor-end=12}{2}} \htmlData{tutor-start=14,tutor-end=15}{+} \cdots \htmlData{tutor-start=23,tutor-end=24}{+} \htmlData{tutor-start=25,tutor-end=26}{a}_{\htmlData{tutor-start=28,tutor-end=29}{2}\htmlData{tutor-start=29,tutor-end=30}{n}}$ 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-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） · 三、（本题满分 50 分） · 数论

设 $\htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{=}\htmlData{tutor-start=6,tutor-end=7}{1}$ ， $\htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{2}}\htmlData{tutor-start=5,tutor-end=6}{=}\htmlData{tutor-start=6,tutor-end=7}{2}$ ， $\htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{n}} \htmlData{tutor-start=6,tutor-end=7}{=} \htmlData{tutor-start=8,tutor-end=9}{2}\htmlData{tutor-start=9,tutor-end=10}{a}_{\htmlData{tutor-start=12,tutor-end=13}{n}\htmlData{tutor-start=13,tutor-end=14}{-}\htmlData{tutor-start=14,tutor-end=15}{1}} \htmlData{tutor-start=17,tutor-end=18}{+} \htmlData{tutor-start=19,tutor-end=20}{a}_{\htmlData{tutor-start=22,tutor-end=23}{n}\htmlData{tutor-start=23,tutor-end=24}{-}\htmlData{tutor-start=24,tutor-end=25}{2}}$ ， $\htmlData{tutor-start=0,tutor-end=1}{n}\htmlData{tutor-start=1,tutor-end=2}{=}\htmlData{tutor-start=2,tutor-end=3}{3}\htmlData{tutor-start=3,tutor-end=4}{,} \htmlData{tutor-start=5,tutor-end=6}{4}\htmlData{tutor-start=6,tutor-end=7}{,} \cdots$ 。证明：对整数 $\htmlData{tutor-start=0,tutor-end=1}{n} \htmlData{tutor-start=2,tutor-end=12}{\geqslant }\htmlData{tutor-start=12,tutor-end=13}{5}$ ， $\htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{n}}$ 必有一个模 4 余 1 的素因子。

题解状态：标准答案与规范题解待补充

题目标签：SECOND-A-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（A 卷） · 四、（本题满分 50 分） · 平面几何

给定凸 20 边形 $\htmlData{tutor-start=0,tutor-end=1}{P}$ 。用 $\htmlData{tutor-start=0,tutor-end=1}{P}$ 的 17 条在内部不相交的对角线将 $\htmlData{tutor-start=0,tutor-end=1}{P}$ 分割成 18 个三角形，所得图形称为 $\htmlData{tutor-start=0,tutor-end=1}{P}$ 的一个三角剖分图。对 $\htmlData{tutor-start=0,tutor-end=1}{P}$ 的任意一个三角剖分图 $\htmlData{tutor-start=0,tutor-end=1}{T}$ ， $\htmlData{tutor-start=0,tutor-end=1}{P}$ 的 20 条边以及添加的 17 条对角线均成为 $\htmlData{tutor-start=0,tutor-end=1}{T}$ 的边。 $\htmlData{tutor-start=0,tutor-end=1}{T}$ 的任意 10 条两两无公共端点的边的集合成为 $\htmlData{tutor-start=0,tutor-end=1}{T}$ 的一个完美匹配。当 $\htmlData{tutor-start=0,tutor-end=1}{T}$ 取遍 $\htmlData{tutor-start=0,tutor-end=1}{P}$ 的所有三角剖分图时，求 $\htmlData{tutor-start=0,tutor-end=1}{T}$ 的完美匹配个数的最大值。

题解状态：标准答案与规范题解待补充

题目标签：SECOND-A-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。