2021 年全国高中数学联赛 A1 卷（江苏、河南专用）（学生版）

books/national_high_school_league_students/papers/competition-national-league-2021-a1.pdf · HS-MATH-1024-v2.1-solution-aware

15 题0 个小问/题组

一试（A1 卷） · 一试试题（A1 卷） · 数学竞赛/待细分

设 $\htmlData{tutor-start=0,tutor-end=1}{A} \htmlData{tutor-start=2,tutor-end=3}{=} \htmlData{tutor-start=4,tutor-end=6}{\{}\htmlData{tutor-start=6,tutor-end=7}{1}\htmlData{tutor-start=7,tutor-end=8}{,} \htmlData{tutor-start=9,tutor-end=10}{2}\htmlData{tutor-start=10,tutor-end=11}{,} \htmlData{tutor-start=12,tutor-end=13}{3}\htmlData{tutor-start=13,tutor-end=15}{\}}$ ， $\htmlData{tutor-start=0,tutor-end=1}{B} \htmlData{tutor-start=2,tutor-end=3}{=} \htmlData{tutor-start=4,tutor-end=6}{\{}\htmlData{tutor-start=6,tutor-end=7}{2}\htmlData{tutor-start=7,tutor-end=8}{x} \htmlData{tutor-start=9,tutor-end=10}{+} \htmlData{tutor-start=11,tutor-end=12}{y} \htmlData{tutor-start=13,tutor-end=18}{\mid }\htmlData{tutor-start=18,tutor-end=19}{x}\htmlData{tutor-start=19,tutor-end=20}{,} \htmlData{tutor-start=21,tutor-end=22}{y} \htmlData{tutor-start=23,tutor-end=27}{\in }\htmlData{tutor-start=27,tutor-end=28}{A}\htmlData{tutor-start=28,tutor-end=29}{,} \htmlData{tutor-start=30,tutor-end=31}{x} \htmlData{tutor-start=32,tutor-end=33}{<} \htmlData{tutor-start=34,tutor-end=35}{y}\htmlData{tutor-start=35,tutor-end=37}{\}}$ ， $\htmlData{tutor-start=0,tutor-end=1}{C} \htmlData{tutor-start=2,tutor-end=3}{=} \htmlData{tutor-start=4,tutor-end=6}{\{}\htmlData{tutor-start=6,tutor-end=7}{2}\htmlData{tutor-start=7,tutor-end=8}{x} \htmlData{tutor-start=9,tutor-end=10}{+} \htmlData{tutor-start=11,tutor-end=12}{y} \htmlData{tutor-start=13,tutor-end=18}{\mid }\htmlData{tutor-start=18,tutor-end=19}{x}\htmlData{tutor-start=19,tutor-end=20}{,} \htmlData{tutor-start=21,tutor-end=22}{y} \htmlData{tutor-start=23,tutor-end=27}{\in }\htmlData{tutor-start=27,tutor-end=28}{A}\htmlData{tutor-start=28,tutor-end=29}{,} \htmlData{tutor-start=30,tutor-end=31}{x} \htmlData{tutor-start=32,tutor-end=33}{>} \htmlData{tutor-start=34,tutor-end=35}{y}\htmlData{tutor-start=35,tutor-end=37}{\}}$ ，则 $\htmlData{tutor-start=0,tutor-end=1}{B} \htmlData{tutor-start=2,tutor-end=7}{\cap }\htmlData{tutor-start=7,tutor-end=8}{C}$ 的所有元素之和为 \underline{\hspace{3cm}}.

题解状态：标准答案与规范题解待补充

题目标签：FIRST-A1-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（A1 卷） · 一试试题（A1 卷） · 数学竞赛/待细分

设 $\htmlData{tutor-start=0,tutor-end=1}{m}$ 为实数，向量 $\vec{\htmlData{tutor-start=5,tutor-end=6}{a}} \htmlData{tutor-start=8,tutor-end=9}{=} \htmlData{tutor-start=10,tutor-end=11}{(}\htmlData{tutor-start=11,tutor-end=12}{1} \htmlData{tutor-start=13,tutor-end=14}{+} \htmlData{tutor-start=15,tutor-end=16}{2}^{\htmlData{tutor-start=18,tutor-end=19}{m}}\htmlData{tutor-start=20,tutor-end=21}{,} \htmlData{tutor-start=22,tutor-end=23}{1} \htmlData{tutor-start=24,tutor-end=25}{-} \htmlData{tutor-start=26,tutor-end=27}{2}^{\htmlData{tutor-start=29,tutor-end=30}{m}}\htmlData{tutor-start=31,tutor-end=32}{)}\htmlData{tutor-start=32,tutor-end=33}{,} \vec{\htmlData{tutor-start=39,tutor-end=40}{b}} \htmlData{tutor-start=42,tutor-end=43}{=} \htmlData{tutor-start=44,tutor-end=45}{(}\htmlData{tutor-start=45,tutor-end=46}{4}^{\htmlData{tutor-start=48,tutor-end=49}{m}} \htmlData{tutor-start=51,tutor-end=52}{-} \htmlData{tutor-start=53,tutor-end=54}{3}\htmlData{tutor-start=54,tutor-end=55}{,} \htmlData{tutor-start=56,tutor-end=57}{4}^{\htmlData{tutor-start=59,tutor-end=60}{m}} \htmlData{tutor-start=62,tutor-end=63}{+} \htmlData{tutor-start=64,tutor-end=65}{5}\htmlData{tutor-start=65,tutor-end=66}{)}$ ，则数量积 $\vec{\htmlData{tutor-start=5,tutor-end=6}{a}} \htmlData{tutor-start=8,tutor-end=14}{\cdot }\vec{\htmlData{tutor-start=19,tutor-end=20}{b}}$ 的最小值为 \underline{\hspace{3cm}}.

题解状态：标准答案与规范题解待补充

题目标签：FIRST-A1-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（A1 卷） · 一试试题（A1 卷） · 平面几何

已知 $\htmlData{tutor-start=0,tutor-end=10}{\triangle }\htmlData{tutor-start=10,tutor-end=11}{A}\htmlData{tutor-start=11,tutor-end=12}{B}\htmlData{tutor-start=12,tutor-end=13}{C}$ 满足： $\tan \htmlData{tutor-start=5,tutor-end=6}{A}\htmlData{tutor-start=6,tutor-end=7}{,} \tan \htmlData{tutor-start=13,tutor-end=14}{B}$ 是方程 $\htmlData{tutor-start=0,tutor-end=1}{x}^{\htmlData{tutor-start=3,tutor-end=4}{2}} \htmlData{tutor-start=6,tutor-end=7}{-} \htmlData{tutor-start=8,tutor-end=9}{1}\htmlData{tutor-start=9,tutor-end=10}{0}\htmlData{tutor-start=10,tutor-end=11}{x} \htmlData{tutor-start=12,tutor-end=13}{+} \htmlData{tutor-start=14,tutor-end=15}{6} \htmlData{tutor-start=16,tutor-end=17}{=} \htmlData{tutor-start=18,tutor-end=19}{0}$ 的两个根，则 $\cos \htmlData{tutor-start=5,tutor-end=6}{C}$ 的值为 \underline{\hspace{3cm}}.

题解状态：标准答案与规范题解待补充

题目标签：FIRST-A1-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（A1 卷） · 一试试题（A1 卷） · 平面几何

在平面直角坐标系中，双曲线 $\htmlData{tutor-start=0,tutor-end=6}{\Gamma}\htmlData{tutor-start=6,tutor-end=7}{:} \frac{\htmlData{tutor-start=14,tutor-end=15}{x}^{\htmlData{tutor-start=17,tutor-end=18}{2}}}{\htmlData{tutor-start=21,tutor-end=22}{a}^{\htmlData{tutor-start=24,tutor-end=25}{2}}} \htmlData{tutor-start=28,tutor-end=29}{-} \frac{\htmlData{tutor-start=36,tutor-end=37}{y}^{\htmlData{tutor-start=39,tutor-end=40}{2}}}{\htmlData{tutor-start=43,tutor-end=44}{b}^{\htmlData{tutor-start=46,tutor-end=47}{2}}} \htmlData{tutor-start=50,tutor-end=51}{=} \htmlData{tutor-start=52,tutor-end=53}{1}\htmlData{tutor-start=53,tutor-end=54}{(}\htmlData{tutor-start=54,tutor-end=55}{a}\htmlData{tutor-start=55,tutor-end=56}{,} \htmlData{tutor-start=57,tutor-end=58}{b} \htmlData{tutor-start=59,tutor-end=60}{>} \htmlData{tutor-start=61,tutor-end=62}{0}\htmlData{tutor-start=62,tutor-end=63}{)}$ ，一条倾斜角为 $\frac{\htmlData{tutor-start=6,tutor-end=9}{\pi}}{\htmlData{tutor-start=11,tutor-end=12}{4}}$ 的直线经过 $\htmlData{tutor-start=0,tutor-end=6}{\Gamma}$ 的一个顶点及 $\htmlData{tutor-start=0,tutor-end=6}{\Gamma}$ 上另外一点 $\htmlData{tutor-start=0,tutor-end=1}{(}\htmlData{tutor-start=1,tutor-end=2}{2}\htmlData{tutor-start=2,tutor-end=3}{,} \htmlData{tutor-start=4,tutor-end=5}{3}\htmlData{tutor-start=5,tutor-end=6}{)}$ ，则 $\htmlData{tutor-start=0,tutor-end=6}{\Gamma}$ 的离心率为 \underline{\hspace{3cm}}.

题解状态：标准答案与规范题解待补充

题目标签：FIRST-A1-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（A1 卷） · 一试试题（A1 卷） · 代数

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 $\htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{1}} \htmlData{tutor-start=6,tutor-end=7}{=} \htmlData{tutor-start=8,tutor-end=9}{1}\htmlData{tutor-start=9,tutor-end=10}{,} \htmlData{tutor-start=11,tutor-end=12}{a}_{\htmlData{tutor-start=14,tutor-end=15}{n}} \htmlData{tutor-start=17,tutor-end=18}{=} \frac{\htmlData{tutor-start=25,tutor-end=26}{1}}{\htmlData{tutor-start=28,tutor-end=29}{4}\htmlData{tutor-start=29,tutor-end=30}{a}_{\htmlData{tutor-start=32,tutor-end=33}{n}\htmlData{tutor-start=33,tutor-end=34}{-}\htmlData{tutor-start=34,tutor-end=35}{1}}} \htmlData{tutor-start=38,tutor-end=39}{+} \frac{\htmlData{tutor-start=46,tutor-end=47}{1}}{\htmlData{tutor-start=49,tutor-end=50}{n}}\htmlData{tutor-start=51,tutor-end=52}{(}\htmlData{tutor-start=52,tutor-end=53}{n} \htmlData{tutor-start=54,tutor-end=58}{\ge }\htmlData{tutor-start=58,tutor-end=59}{2}\htmlData{tutor-start=59,tutor-end=60}{)}$ ，则 $\htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{1}\htmlData{tutor-start=4,tutor-end=5}{0}\htmlData{tutor-start=5,tutor-end=6}{0}}$ 的值为 \underline{\hspace{3cm}}.

题解状态：标准答案与规范题解待补充

题目标签：FIRST-A1-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（A1 卷） · 一试试题（A1 卷） · 平面几何

设正四棱锥 $\htmlData{tutor-start=0,tutor-end=1}{P}\htmlData{tutor-start=1,tutor-end=2}{-}\htmlData{tutor-start=2,tutor-end=3}{A}\htmlData{tutor-start=3,tutor-end=4}{B}\htmlData{tutor-start=4,tutor-end=5}{C}\htmlData{tutor-start=5,tutor-end=6}{D}$ 的底面边长与高相等，点 $\htmlData{tutor-start=0,tutor-end=1}{G}$ 为侧面 $\htmlData{tutor-start=0,tutor-end=10}{\triangle }\htmlData{tutor-start=10,tutor-end=11}{P}\htmlData{tutor-start=11,tutor-end=12}{B}\htmlData{tutor-start=12,tutor-end=13}{C}$ 的重心，则直线 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{G}$ 与底面 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{B}\htmlData{tutor-start=2,tutor-end=3}{C}\htmlData{tutor-start=3,tutor-end=4}{D}$ 所成角的正弦值为 \underline{\hspace{3cm}}.

题解状态：标准答案与规范题解待补充

题目标签：FIRST-A1-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（A1 卷） · 一试试题（A1 卷） · 数学竞赛/待细分

设 $\htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{a}_{\htmlData{tutor-start=10,tutor-end=11}{2}}\htmlData{tutor-start=12,tutor-end=13}{,} \cdots\htmlData{tutor-start=20,tutor-end=21}{,} \htmlData{tutor-start=22,tutor-end=23}{a}_{\htmlData{tutor-start=25,tutor-end=26}{2}\htmlData{tutor-start=26,tutor-end=27}{1}}$ 为 $\htmlData{tutor-start=0,tutor-end=1}{1}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{2}\htmlData{tutor-start=4,tutor-end=5}{,} \cdots\htmlData{tutor-start=12,tutor-end=13}{,} \htmlData{tutor-start=14,tutor-end=15}{2}\htmlData{tutor-start=15,tutor-end=16}{1}$ 的排列，满足 $\htmlData{tutor-start=0,tutor-end=1}{|}\htmlData{tutor-start=1,tutor-end=2}{a}_{\htmlData{tutor-start=4,tutor-end=5}{2}\htmlData{tutor-start=5,tutor-end=6}{0}} \htmlData{tutor-start=8,tutor-end=9}{-} \htmlData{tutor-start=10,tutor-end=11}{a}_{\htmlData{tutor-start=13,tutor-end=14}{2}\htmlData{tutor-start=14,tutor-end=15}{1}}\htmlData{tutor-start=16,tutor-end=17}{|} \htmlData{tutor-start=18,tutor-end=22}{\ge }\htmlData{tutor-start=22,tutor-end=23}{|}\htmlData{tutor-start=23,tutor-end=24}{a}_{\htmlData{tutor-start=26,tutor-end=27}{1}\htmlData{tutor-start=27,tutor-end=28}{9}} \htmlData{tutor-start=30,tutor-end=31}{-} \htmlData{tutor-start=32,tutor-end=33}{a}_{\htmlData{tutor-start=35,tutor-end=36}{2}\htmlData{tutor-start=36,tutor-end=37}{1}}\htmlData{tutor-start=38,tutor-end=39}{|} \htmlData{tutor-start=40,tutor-end=44}{\ge }\htmlData{tutor-start=44,tutor-end=45}{|}\htmlData{tutor-start=45,tutor-end=46}{a}_{\htmlData{tutor-start=48,tutor-end=49}{1}\htmlData{tutor-start=49,tutor-end=50}{8}} \htmlData{tutor-start=52,tutor-end=53}{-} \htmlData{tutor-start=54,tutor-end=55}{a}_{\htmlData{tutor-start=57,tutor-end=58}{2}\htmlData{tutor-start=58,tutor-end=59}{1}}\htmlData{tutor-start=60,tutor-end=61}{|} \htmlData{tutor-start=62,tutor-end=66}{\ge }\cdots \htmlData{tutor-start=73,tutor-end=77}{\ge }\htmlData{tutor-start=77,tutor-end=78}{|}\htmlData{tutor-start=78,tutor-end=79}{a}_{\htmlData{tutor-start=81,tutor-end=82}{1}} \htmlData{tutor-start=84,tutor-end=85}{-} \htmlData{tutor-start=86,tutor-end=87}{a}_{\htmlData{tutor-start=89,tutor-end=90}{2}\htmlData{tutor-start=90,tutor-end=91}{1}}\htmlData{tutor-start=92,tutor-end=93}{|}\htmlData{tutor-start=93,tutor-end=94}{.}$ 这样的排列的个数为 \underline{\hspace{3cm}}.

题解状态：标准答案与规范题解待补充

题目标签：FIRST-A1-PROBLEM-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（A1 卷） · 一试试题（A1 卷） · 数学竞赛/待细分

正实数 $\htmlData{tutor-start=0,tutor-end=1}{x}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{y}$ 满足如下条件：存在 $\htmlData{tutor-start=0,tutor-end=1}{a} \htmlData{tutor-start=2,tutor-end=6}{\in }\htmlData{tutor-start=6,tutor-end=7}{[}\htmlData{tutor-start=7,tutor-end=8}{0}\htmlData{tutor-start=8,tutor-end=9}{,} \htmlData{tutor-start=10,tutor-end=11}{x}\htmlData{tutor-start=11,tutor-end=12}{]}\htmlData{tutor-start=12,tutor-end=13}{,} \htmlData{tutor-start=14,tutor-end=15}{b} \htmlData{tutor-start=16,tutor-end=20}{\in }\htmlData{tutor-start=20,tutor-end=21}{[}\htmlData{tutor-start=21,tutor-end=22}{0}\htmlData{tutor-start=22,tutor-end=23}{,} \htmlData{tutor-start=24,tutor-end=25}{y}\htmlData{tutor-start=25,tutor-end=26}{]}$ ，使得 $\htmlData{tutor-start=0,tutor-end=1}{a}^{\htmlData{tutor-start=3,tutor-end=4}{2}} \htmlData{tutor-start=6,tutor-end=7}{+} \htmlData{tutor-start=8,tutor-end=9}{y}^{\htmlData{tutor-start=11,tutor-end=12}{2}} \htmlData{tutor-start=14,tutor-end=15}{=} \htmlData{tutor-start=16,tutor-end=17}{2}\htmlData{tutor-start=17,tutor-end=18}{,} \quad \htmlData{tutor-start=25,tutor-end=26}{b}^{\htmlData{tutor-start=28,tutor-end=29}{2}} \htmlData{tutor-start=31,tutor-end=32}{+} \htmlData{tutor-start=33,tutor-end=34}{x}^{\htmlData{tutor-start=36,tutor-end=37}{2}} \htmlData{tutor-start=39,tutor-end=40}{=} \htmlData{tutor-start=41,tutor-end=42}{1}\htmlData{tutor-start=42,tutor-end=43}{,} \quad \htmlData{tutor-start=50,tutor-end=51}{a}\htmlData{tutor-start=51,tutor-end=52}{x} \htmlData{tutor-start=53,tutor-end=54}{+} \htmlData{tutor-start=55,tutor-end=56}{b}\htmlData{tutor-start=56,tutor-end=57}{y} \htmlData{tutor-start=58,tutor-end=59}{=} \htmlData{tutor-start=60,tutor-end=61}{1}\htmlData{tutor-start=61,tutor-end=62}{,}$ 则 $\htmlData{tutor-start=0,tutor-end=1}{x} \htmlData{tutor-start=2,tutor-end=3}{+} \htmlData{tutor-start=4,tutor-end=5}{y}$ 的最大值为 \underline{\hspace{3cm}}.

题解状态：标准答案与规范题解待补充

题目标签：FIRST-A1-PROBLEM-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（A1 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 $\htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{x}\htmlData{tutor-start=3,tutor-end=4}{)} \htmlData{tutor-start=5,tutor-end=6}{=} \htmlData{tutor-start=7,tutor-end=8}{|}\htmlData{tutor-start=8,tutor-end=9}{2} \htmlData{tutor-start=10,tutor-end=11}{-} \log_{\htmlData{tutor-start=18,tutor-end=19}{3}} \htmlData{tutor-start=21,tutor-end=22}{x}\htmlData{tutor-start=22,tutor-end=23}{|}$ ，正实数 $\htmlData{tutor-start=0,tutor-end=1}{a}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{b}\htmlData{tutor-start=4,tutor-end=5}{,} \htmlData{tutor-start=6,tutor-end=7}{c}$ 满足 $\htmlData{tutor-start=0,tutor-end=1}{a} \htmlData{tutor-start=2,tutor-end=3}{<} \htmlData{tutor-start=4,tutor-end=5}{b} \htmlData{tutor-start=6,tutor-end=7}{<} \htmlData{tutor-start=8,tutor-end=9}{c}$ ，且 $\htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{a}\htmlData{tutor-start=3,tutor-end=4}{)} \htmlData{tutor-start=5,tutor-end=6}{=} \htmlData{tutor-start=7,tutor-end=8}{2}\htmlData{tutor-start=8,tutor-end=9}{f}\htmlData{tutor-start=9,tutor-end=10}{(}\htmlData{tutor-start=10,tutor-end=11}{b}\htmlData{tutor-start=11,tutor-end=12}{)} \htmlData{tutor-start=13,tutor-end=14}{=} \htmlData{tutor-start=15,tutor-end=16}{2}\htmlData{tutor-start=16,tutor-end=17}{f}\htmlData{tutor-start=17,tutor-end=18}{(}\htmlData{tutor-start=18,tutor-end=19}{c}\htmlData{tutor-start=19,tutor-end=20}{)}$ . 求 $\frac{\htmlData{tutor-start=6,tutor-end=7}{a}\htmlData{tutor-start=7,tutor-end=8}{c}}{\htmlData{tutor-start=10,tutor-end=11}{b}}$ 的值.

题解状态：标准答案与规范题解待补充

题目标签：FIRST-A1-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（A1 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. · 组合数学

（本题满分 20 分）设 $\htmlData{tutor-start=0,tutor-end=1}{a}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{b} \htmlData{tutor-start=5,tutor-end=9}{\in }\mathbb{\htmlData{tutor-start=17,tutor-end=18}{R}}$ . 若关于 $\htmlData{tutor-start=0,tutor-end=1}{z}$ 的方程 $\left(\htmlData{tutor-start=6,tutor-end=7}{z}^{\htmlData{tutor-start=9,tutor-end=10}{2}} \htmlData{tutor-start=12,tutor-end=13}{+} \htmlData{tutor-start=14,tutor-end=15}{a}\htmlData{tutor-start=15,tutor-end=16}{z} \htmlData{tutor-start=17,tutor-end=18}{+} \htmlData{tutor-start=19,tutor-end=20}{b}\right)\left(\htmlData{tutor-start=33,tutor-end=34}{z}^{\htmlData{tutor-start=36,tutor-end=37}{2}} \htmlData{tutor-start=39,tutor-end=40}{+} \htmlData{tutor-start=41,tutor-end=42}{a}\htmlData{tutor-start=42,tutor-end=43}{z} \htmlData{tutor-start=44,tutor-end=45}{+} \htmlData{tutor-start=46,tutor-end=47}{2}\htmlData{tutor-start=47,tutor-end=48}{b}\right) \htmlData{tutor-start=56,tutor-end=57}{=} \htmlData{tutor-start=58,tutor-end=59}{0}$ 有 4 个互不相等的复数根 $\htmlData{tutor-start=0,tutor-end=1}{z}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{z}_{\htmlData{tutor-start=10,tutor-end=11}{2}}\htmlData{tutor-start=12,tutor-end=13}{,} \htmlData{tutor-start=14,tutor-end=15}{z}_{\htmlData{tutor-start=17,tutor-end=18}{3}}\htmlData{tutor-start=19,tutor-end=20}{,} \htmlData{tutor-start=21,tutor-end=22}{z}_{\htmlData{tutor-start=24,tutor-end=25}{4}}$ ，且它们在复平面上对应的点恰是一个边长为 1 的正方形的四个顶点，求 $\htmlData{tutor-start=0,tutor-end=1}{|}\htmlData{tutor-start=1,tutor-end=2}{z}_{\htmlData{tutor-start=4,tutor-end=5}{1}}\htmlData{tutor-start=6,tutor-end=7}{|} \htmlData{tutor-start=8,tutor-end=9}{+} \htmlData{tutor-start=10,tutor-end=11}{|}\htmlData{tutor-start=11,tutor-end=12}{z}_{\htmlData{tutor-start=14,tutor-end=15}{2}}\htmlData{tutor-start=16,tutor-end=17}{|} \htmlData{tutor-start=18,tutor-end=19}{+} \htmlData{tutor-start=20,tutor-end=21}{|}\htmlData{tutor-start=21,tutor-end=22}{z}_{\htmlData{tutor-start=24,tutor-end=25}{3}}\htmlData{tutor-start=26,tutor-end=27}{|} \htmlData{tutor-start=28,tutor-end=29}{+} \htmlData{tutor-start=30,tutor-end=31}{|}\htmlData{tutor-start=31,tutor-end=32}{z}_{\htmlData{tutor-start=34,tutor-end=35}{4}}\htmlData{tutor-start=36,tutor-end=37}{|}$ 的值.

题解状态：标准答案与规范题解待补充

题目标签：FIRST-A1-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（A1 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）如图所示，在平面直角坐标系中，椭圆 $\htmlData{tutor-start=0,tutor-end=6}{\Gamma}\htmlData{tutor-start=6,tutor-end=7}{:} \frac{\htmlData{tutor-start=14,tutor-end=15}{x}^{\htmlData{tutor-start=17,tutor-end=18}{2}}}{\htmlData{tutor-start=21,tutor-end=22}{2}} \htmlData{tutor-start=24,tutor-end=25}{+} \htmlData{tutor-start=26,tutor-end=27}{y}^{\htmlData{tutor-start=29,tutor-end=30}{2}} \htmlData{tutor-start=32,tutor-end=33}{=} \htmlData{tutor-start=34,tutor-end=35}{1}$ 的左、右焦点分别为 $\htmlData{tutor-start=0,tutor-end=1}{F}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{F}_{\htmlData{tutor-start=10,tutor-end=11}{2}}$ ，设 $\htmlData{tutor-start=0,tutor-end=1}{P}$ 是第一象限内 $\htmlData{tutor-start=0,tutor-end=6}{\Gamma}$ 上一点， $\htmlData{tutor-start=0,tutor-end=1}{P}\htmlData{tutor-start=1,tutor-end=2}{F}_{\htmlData{tutor-start=4,tutor-end=5}{1}}\htmlData{tutor-start=6,tutor-end=7}{,} \htmlData{tutor-start=8,tutor-end=9}{P}\htmlData{tutor-start=9,tutor-end=10}{F}_{\htmlData{tutor-start=12,tutor-end=13}{2}}$ 的延长线分别交 $\htmlData{tutor-start=0,tutor-end=6}{\Gamma}$ 于点 $\htmlData{tutor-start=0,tutor-end=1}{Q}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{Q}_{\htmlData{tutor-start=10,tutor-end=11}{2}}$ . 设 $\htmlData{tutor-start=0,tutor-end=1}{r}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{r}_{\htmlData{tutor-start=10,tutor-end=11}{2}}$ 分别为 $\htmlData{tutor-start=0,tutor-end=10}{\triangle }\htmlData{tutor-start=10,tutor-end=11}{P}\htmlData{tutor-start=11,tutor-end=12}{F}_{\htmlData{tutor-start=14,tutor-end=15}{1}}\htmlData{tutor-start=16,tutor-end=17}{Q}_{\htmlData{tutor-start=19,tutor-end=20}{2}}\htmlData{tutor-start=21,tutor-end=22}{,} \htmlData{tutor-start=23,tutor-end=33}{\triangle }\htmlData{tutor-start=33,tutor-end=34}{P}\htmlData{tutor-start=34,tutor-end=35}{F}_{\htmlData{tutor-start=37,tutor-end=38}{2}}\htmlData{tutor-start=39,tutor-end=40}{Q}_{\htmlData{tutor-start=42,tutor-end=43}{1}}$ 的内切圆半径. 求 $\htmlData{tutor-start=0,tutor-end=1}{r}_{\htmlData{tutor-start=3,tutor-end=4}{1}} \htmlData{tutor-start=6,tutor-end=7}{-} \htmlData{tutor-start=8,tutor-end=9}{r}_{\htmlData{tutor-start=11,tutor-end=12}{2}}$ 的最大值.

**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：FIRST-A1-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（A1 卷） · 加试试题（A1 卷） · 平面几何

（本题满分 40 分）如图，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， $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{B} \htmlData{tutor-start=3,tutor-end=4}{>} \htmlData{tutor-start=5,tutor-end=6}{A}\htmlData{tutor-start=6,tutor-end=7}{C}$ ， $\htmlData{tutor-start=0,tutor-end=10}{\triangle }\htmlData{tutor-start=10,tutor-end=11}{A}\htmlData{tutor-start=11,tutor-end=12}{B}\htmlData{tutor-start=12,tutor-end=13}{C}$ 内两点 $\htmlData{tutor-start=0,tutor-end=1}{X}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{Y}$ 均在 $\htmlData{tutor-start=0,tutor-end=7}{\angle }\htmlData{tutor-start=7,tutor-end=8}{B}\htmlData{tutor-start=8,tutor-end=9}{A}\htmlData{tutor-start=9,tutor-end=10}{C}$ 的平分线上，且满足 $\htmlData{tutor-start=0,tutor-end=7}{\angle }\htmlData{tutor-start=7,tutor-end=8}{A}\htmlData{tutor-start=8,tutor-end=9}{B}\htmlData{tutor-start=9,tutor-end=10}{X} \htmlData{tutor-start=11,tutor-end=12}{=} \htmlData{tutor-start=13,tutor-end=20}{\angle }\htmlData{tutor-start=20,tutor-end=21}{A}\htmlData{tutor-start=21,tutor-end=22}{C}\htmlData{tutor-start=22,tutor-end=23}{Y}$ . 设 $\htmlData{tutor-start=0,tutor-end=1}{B}\htmlData{tutor-start=1,tutor-end=2}{X}$ 的延长线与线段 $\htmlData{tutor-start=0,tutor-end=1}{C}\htmlData{tutor-start=1,tutor-end=2}{Y}$ 交于点 $\htmlData{tutor-start=0,tutor-end=1}{P}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=13}{\triangle }\htmlData{tutor-start=13,tutor-end=14}{B}\htmlData{tutor-start=14,tutor-end=15}{P}\htmlData{tutor-start=15,tutor-end=16}{Y}$ 的外接圆 $\htmlData{tutor-start=0,tutor-end=6}{\omega}_{\htmlData{tutor-start=8,tutor-end=9}{1}}$ 与 $\htmlData{tutor-start=0,tutor-end=10}{\triangle }\htmlData{tutor-start=10,tutor-end=11}{C}\htmlData{tutor-start=11,tutor-end=12}{P}\htmlData{tutor-start=12,tutor-end=13}{X}$ 的外接圆 $\htmlData{tutor-start=0,tutor-end=6}{\omega}_{\htmlData{tutor-start=8,tutor-end=9}{2}}$ 交于 $\htmlData{tutor-start=0,tutor-end=1}{P}$ 及另一点 $\htmlData{tutor-start=0,tutor-end=1}{Q}$ . 证明： $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{P}\htmlData{tutor-start=4,tutor-end=5}{,} \htmlData{tutor-start=6,tutor-end=7}{Q}$ 三点共线. （答题时将图画在答题卷上）

题解状态：标准答案与规范题解待补充

题目标签：SECOND-A1-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（A1 卷） · 加试试题（A1 卷） · 数论

（本题满分 40 分）求正整数 $\htmlData{tutor-start=0,tutor-end=1}{a}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{b}\htmlData{tutor-start=4,tutor-end=5}{,} \htmlData{tutor-start=6,tutor-end=7}{n}$ （其中 $\htmlData{tutor-start=0,tutor-end=1}{n} \htmlData{tutor-start=2,tutor-end=6}{\ge }\htmlData{tutor-start=6,tutor-end=7}{2}$ ）满足的充分必要条件，使得存在一个从集合 $\htmlData{tutor-start=0,tutor-end=1}{S} \htmlData{tutor-start=2,tutor-end=3}{=} \htmlData{tutor-start=4,tutor-end=6}{\{}\htmlData{tutor-start=6,tutor-end=7}{a} \htmlData{tutor-start=8,tutor-end=9}{+} \htmlData{tutor-start=10,tutor-end=11}{b}\htmlData{tutor-start=11,tutor-end=12}{t} \htmlData{tutor-start=13,tutor-end=18}{\mid }\htmlData{tutor-start=18,tutor-end=19}{t} \htmlData{tutor-start=20,tutor-end=21}{=} \htmlData{tutor-start=22,tutor-end=23}{0}\htmlData{tutor-start=23,tutor-end=24}{,} \htmlData{tutor-start=25,tutor-end=26}{1}\htmlData{tutor-start=26,tutor-end=27}{,} \cdots\htmlData{tutor-start=34,tutor-end=35}{,} \htmlData{tutor-start=36,tutor-end=37}{n} \htmlData{tutor-start=38,tutor-end=39}{-} \htmlData{tutor-start=40,tutor-end=41}{1}\htmlData{tutor-start=41,tutor-end=43}{\}}$ 到自身的一一映射 $\htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{:} \htmlData{tutor-start=3,tutor-end=4}{S} \htmlData{tutor-start=5,tutor-end=9}{\to }\htmlData{tutor-start=9,tutor-end=10}{S}$ ，满足：对任意 $\htmlData{tutor-start=0,tutor-end=1}{x} \htmlData{tutor-start=2,tutor-end=6}{\in }\htmlData{tutor-start=6,tutor-end=7}{S}$ ，均有 $\htmlData{tutor-start=0,tutor-end=1}{x}$ 与 $\htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{x}\htmlData{tutor-start=3,tutor-end=4}{)}$ 互素.

题解状态：标准答案与规范题解待补充

题目标签：SECOND-A1-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（A1 卷） · 加试试题（A1 卷） · 数论

（本题满分 50 分）设正实数数列 $\htmlData{tutor-start=0,tutor-end=2}{\{}\htmlData{tutor-start=2,tutor-end=3}{a}_{\htmlData{tutor-start=5,tutor-end=6}{n}}\htmlData{tutor-start=7,tutor-end=9}{\}}\htmlData{tutor-start=9,tutor-end=10}{,} \htmlData{tutor-start=11,tutor-end=13}{\{}\htmlData{tutor-start=13,tutor-end=14}{b}_{\htmlData{tutor-start=16,tutor-end=17}{n}}\htmlData{tutor-start=18,tutor-end=20}{\}}$ 满足：对任意整数 $\htmlData{tutor-start=0,tutor-end=1}{n} \htmlData{tutor-start=2,tutor-end=6}{\ge }\htmlData{tutor-start=6,tutor-end=7}{1}\htmlData{tutor-start=7,tutor-end=8}{0}\htmlData{tutor-start=8,tutor-end=9}{1}$ ，有 $\htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{n}} \htmlData{tutor-start=6,tutor-end=7}{=} \sqrt{\frac{\htmlData{tutor-start=20,tutor-end=21}{1}}{\htmlData{tutor-start=23,tutor-end=24}{1}\htmlData{tutor-start=24,tutor-end=25}{0}\htmlData{tutor-start=25,tutor-end=26}{0}} \sum_{\htmlData{tutor-start=34,tutor-end=35}{j}\htmlData{tutor-start=35,tutor-end=36}{=}\htmlData{tutor-start=36,tutor-end=37}{1}}^{\htmlData{tutor-start=40,tutor-end=41}{1}\htmlData{tutor-start=41,tutor-end=42}{0}\htmlData{tutor-start=42,tutor-end=43}{0}} \htmlData{tutor-start=45,tutor-end=46}{b}_{\htmlData{tutor-start=48,tutor-end=49}{n}\htmlData{tutor-start=49,tutor-end=50}{-}\htmlData{tutor-start=50,tutor-end=51}{j}}^{\htmlData{tutor-start=54,tutor-end=55}{2}}}\htmlData{tutor-start=57,tutor-end=58}{,} \quad \htmlData{tutor-start=65,tutor-end=66}{b}_{\htmlData{tutor-start=68,tutor-end=69}{n}} \htmlData{tutor-start=71,tutor-end=72}{=} \sqrt{\frac{\htmlData{tutor-start=85,tutor-end=86}{1}}{\htmlData{tutor-start=88,tutor-end=89}{1}\htmlData{tutor-start=89,tutor-end=90}{0}\htmlData{tutor-start=90,tutor-end=91}{0}} \sum_{\htmlData{tutor-start=99,tutor-end=100}{j}\htmlData{tutor-start=100,tutor-end=101}{=}\htmlData{tutor-start=101,tutor-end=102}{1}}^{\htmlData{tutor-start=105,tutor-end=106}{1}\htmlData{tutor-start=106,tutor-end=107}{0}\htmlData{tutor-start=107,tutor-end=108}{0}} \htmlData{tutor-start=110,tutor-end=111}{a}_{\htmlData{tutor-start=113,tutor-end=114}{n}\htmlData{tutor-start=114,tutor-end=115}{-}\htmlData{tutor-start=115,tutor-end=116}{j}}^{\htmlData{tutor-start=119,tutor-end=120}{2}}}\htmlData{tutor-start=122,tutor-end=123}{.}$ 证明：存在正整数 $\htmlData{tutor-start=0,tutor-end=1}{m}$ ，使得 $\htmlData{tutor-start=0,tutor-end=1}{|}\htmlData{tutor-start=1,tutor-end=2}{a}_{\htmlData{tutor-start=4,tutor-end=5}{m}} \htmlData{tutor-start=7,tutor-end=8}{-} \htmlData{tutor-start=9,tutor-end=10}{b}_{\htmlData{tutor-start=12,tutor-end=13}{m}}\htmlData{tutor-start=14,tutor-end=15}{|} \htmlData{tutor-start=16,tutor-end=17}{<} \htmlData{tutor-start=18,tutor-end=19}{0}\htmlData{tutor-start=19,tutor-end=20}{.}\htmlData{tutor-start=20,tutor-end=21}{0}\htmlData{tutor-start=21,tutor-end=22}{0}\htmlData{tutor-start=22,tutor-end=23}{1}$ .

题解状态：标准答案与规范题解待补充

题目标签：SECOND-A1-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（A1 卷） · 加试试题（A1 卷） · 平面几何

（本题满分 50 分）圆周上给定 100 个不同的点. 试确定最大的正整数 $\htmlData{tutor-start=0,tutor-end=1}{k}$ ：将这 100 个点中任意 $\htmlData{tutor-start=0,tutor-end=1}{k}$ 个点任意染为红、蓝两色之一，均可将其余的点适当地染为红色或蓝色，使得可用这 100 个点为端点作 50 条线段，任意两条线段没有公共点，且每条线段的端点同色.

题解状态：标准答案与规范题解待补充

题目标签：SECOND-A1-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。