2023 年全国高中数学联赛 A 卷（学生版）

books/national_high_school_league_students/papers/competition-national-league-2023-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. · 数论

设复数 $\htmlData{tutor-start=0,tutor-end=1}{z}\htmlData{tutor-start=1,tutor-end=2}{=}\htmlData{tutor-start=2,tutor-end=3}{9}\htmlData{tutor-start=3,tutor-end=4}{+}\htmlData{tutor-start=4,tutor-end=5}{1}\htmlData{tutor-start=5,tutor-end=6}{0}\text{\htmlData{tutor-start=12,tutor-end=13}{i}}$ （ $\text{\htmlData{tutor-start=6,tutor-end=7}{i}}$ 为虚数单位），若正整数 $\htmlData{tutor-start=0,tutor-end=1}{n}$ 满足 $\left|\htmlData{tutor-start=6,tutor-end=7}{z}^{\htmlData{tutor-start=9,tutor-end=10}{n}}\right| \htmlData{tutor-start=19,tutor-end=23}{\le }\htmlData{tutor-start=23,tutor-end=24}{2}\htmlData{tutor-start=24,tutor-end=25}{0}\htmlData{tutor-start=25,tutor-end=26}{2}\htmlData{tutor-start=26,tutor-end=27}{3}$ ，则 $\htmlData{tutor-start=0,tutor-end=1}{n}$ 的最大值为__________.

题解状态：标准答案与规范题解待补充

题目标签：FIRST-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. · 数学竞赛/待细分

若正实数 $\htmlData{tutor-start=0,tutor-end=1}{a}\htmlData{tutor-start=1,tutor-end=2}{,}\htmlData{tutor-start=2,tutor-end=3}{b}$ 满足 $\htmlData{tutor-start=0,tutor-end=1}{a}^{\lg \htmlData{tutor-start=7,tutor-end=8}{b}}\htmlData{tutor-start=9,tutor-end=10}{=}\htmlData{tutor-start=10,tutor-end=11}{2}$ ， $\htmlData{tutor-start=0,tutor-end=1}{a}^{\lg \htmlData{tutor-start=7,tutor-end=8}{a}} \htmlData{tutor-start=10,tutor-end=16}{\cdot }\htmlData{tutor-start=16,tutor-end=17}{b}^{\lg \htmlData{tutor-start=23,tutor-end=24}{b}}\htmlData{tutor-start=25,tutor-end=26}{=}\htmlData{tutor-start=26,tutor-end=27}{5}$ ，则 $\htmlData{tutor-start=0,tutor-end=1}{(}\htmlData{tutor-start=1,tutor-end=2}{a}\htmlData{tutor-start=2,tutor-end=3}{b}\htmlData{tutor-start=3,tutor-end=4}{)}^{\lg \htmlData{tutor-start=10,tutor-end=11}{a}\htmlData{tutor-start=11,tutor-end=12}{b}}$ 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-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

将一枚均匀的骰子独立投掷三次，所得的点数依次记为 $\htmlData{tutor-start=0,tutor-end=1}{x}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{y}\htmlData{tutor-start=4,tutor-end=5}{,} \htmlData{tutor-start=6,tutor-end=7}{z}$ ，则事件“ $\text{\htmlData{tutor-start=6,tutor-end=7}{C}}_{\htmlData{tutor-start=10,tutor-end=11}{7}}^{\htmlData{tutor-start=14,tutor-end=15}{x}} \htmlData{tutor-start=17,tutor-end=18}{<} \text{\htmlData{tutor-start=25,tutor-end=26}{C}}_{\htmlData{tutor-start=29,tutor-end=30}{7}}^{\htmlData{tutor-start=33,tutor-end=34}{y}} \htmlData{tutor-start=36,tutor-end=37}{<} \text{\htmlData{tutor-start=44,tutor-end=45}{C}}_{\htmlData{tutor-start=48,tutor-end=49}{7}}^{\htmlData{tutor-start=52,tutor-end=53}{z}}$ ”发生的概率为__________.

题解状态：标准答案与规范题解待补充

题目标签：FIRST-A-FILL-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. · 数学竞赛/待细分

若平面上非零向量 $\vec{\htmlData{tutor-start=5,tutor-end=11}{\alpha}}\htmlData{tutor-start=12,tutor-end=13}{,} \vec{\htmlData{tutor-start=19,tutor-end=24}{\beta}}\htmlData{tutor-start=25,tutor-end=26}{,} \vec{\htmlData{tutor-start=32,tutor-end=38}{\gamma}}$ 满足 $\vec{\htmlData{tutor-start=5,tutor-end=11}{\alpha}} \htmlData{tutor-start=13,tutor-end=19}{\perp }\vec{\htmlData{tutor-start=24,tutor-end=29}{\beta}}$ ， $\vec{\htmlData{tutor-start=5,tutor-end=10}{\beta}} \htmlData{tutor-start=12,tutor-end=18}{\cdot }\vec{\htmlData{tutor-start=23,tutor-end=29}{\gamma}} \htmlData{tutor-start=31,tutor-end=32}{=} \htmlData{tutor-start=33,tutor-end=34}{2}\htmlData{tutor-start=34,tutor-end=35}{|}\vec{\htmlData{tutor-start=40,tutor-end=46}{\alpha}}\htmlData{tutor-start=47,tutor-end=48}{|}$ ， $\vec{\htmlData{tutor-start=5,tutor-end=11}{\gamma}} \htmlData{tutor-start=13,tutor-end=19}{\cdot }\vec{\htmlData{tutor-start=24,tutor-end=30}{\alpha}} \htmlData{tutor-start=32,tutor-end=33}{=} \htmlData{tutor-start=34,tutor-end=35}{3}\htmlData{tutor-start=35,tutor-end=36}{|}\vec{\htmlData{tutor-start=41,tutor-end=46}{\beta}}\htmlData{tutor-start=47,tutor-end=48}{|}$ ，则 $\htmlData{tutor-start=0,tutor-end=1}{|}\vec{\htmlData{tutor-start=6,tutor-end=12}{\gamma}}\htmlData{tutor-start=13,tutor-end=14}{|}$ 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） · 填空题 · 数学竞赛/待细分

方程 $\sin \htmlData{tutor-start=5,tutor-end=6}{x}\htmlData{tutor-start=6,tutor-end=7}{=}\cos \htmlData{tutor-start=12,tutor-end=13}{2}\htmlData{tutor-start=13,tutor-end=14}{x}$ 的最小的 20 个正实数解之和为________.

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） · 填空题 · 平面几何

设 $\htmlData{tutor-start=0,tutor-end=1}{a}\htmlData{tutor-start=1,tutor-end=2}{,}\htmlData{tutor-start=2,tutor-end=3}{b}\htmlData{tutor-start=3,tutor-end=4}{,}\htmlData{tutor-start=4,tutor-end=5}{c}$ 为正数， $\htmlData{tutor-start=0,tutor-end=1}{a}\htmlData{tutor-start=1,tutor-end=2}{<}\htmlData{tutor-start=2,tutor-end=3}{b}$ . 若 $\htmlData{tutor-start=0,tutor-end=1}{a}\htmlData{tutor-start=1,tutor-end=2}{,}\htmlData{tutor-start=2,tutor-end=3}{b}$ 为一元二次方程 $\htmlData{tutor-start=0,tutor-end=1}{a}\htmlData{tutor-start=1,tutor-end=2}{x}^{\htmlData{tutor-start=4,tutor-end=5}{2}}\htmlData{tutor-start=6,tutor-end=7}{-}\htmlData{tutor-start=7,tutor-end=8}{b}\htmlData{tutor-start=8,tutor-end=9}{x}\htmlData{tutor-start=9,tutor-end=10}{+}\htmlData{tutor-start=10,tutor-end=11}{c}\htmlData{tutor-start=11,tutor-end=12}{=}\htmlData{tutor-start=12,tutor-end=13}{0}$ 的两个根，且 $\htmlData{tutor-start=0,tutor-end=1}{a}\htmlData{tutor-start=1,tutor-end=2}{,}\htmlData{tutor-start=2,tutor-end=3}{b}\htmlData{tutor-start=3,tutor-end=4}{,}\htmlData{tutor-start=4,tutor-end=5}{c}$ 是一个三角形的三边长，则 $\htmlData{tutor-start=0,tutor-end=1}{a}\htmlData{tutor-start=1,tutor-end=2}{+}\htmlData{tutor-start=2,tutor-end=3}{b}\htmlData{tutor-start=3,tutor-end=4}{-}\htmlData{tutor-start=4,tutor-end=5}{c}$ 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-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） · 填空题 · 平面几何

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，已知圆 $\htmlData{tutor-start=0,tutor-end=6}{\Omega}$ 与 $\htmlData{tutor-start=0,tutor-end=1}{x}$ 轴、 $\htmlData{tutor-start=0,tutor-end=1}{y}$ 轴均相切，圆心在椭圆 $\htmlData{tutor-start=0,tutor-end=6}{\Gamma}\htmlData{tutor-start=6,tutor-end=7}{:}\frac{\htmlData{tutor-start=13,tutor-end=14}{x}^{\htmlData{tutor-start=16,tutor-end=17}{2}}}{\htmlData{tutor-start=20,tutor-end=21}{a}^{\htmlData{tutor-start=23,tutor-end=24}{2}}}\htmlData{tutor-start=26,tutor-end=27}{+}\frac{\htmlData{tutor-start=33,tutor-end=34}{y}^{\htmlData{tutor-start=36,tutor-end=37}{2}}}{\htmlData{tutor-start=40,tutor-end=41}{b}^{\htmlData{tutor-start=43,tutor-end=44}{2}}}\htmlData{tutor-start=46,tutor-end=47}{=}\htmlData{tutor-start=47,tutor-end=48}{1}\htmlData{tutor-start=48,tutor-end=49}{(}\htmlData{tutor-start=49,tutor-end=50}{a}\htmlData{tutor-start=50,tutor-end=51}{>}\htmlData{tutor-start=51,tutor-end=52}{b}\htmlData{tutor-start=52,tutor-end=53}{>}\htmlData{tutor-start=53,tutor-end=54}{0}\htmlData{tutor-start=54,tutor-end=55}{)}$ 内，且 $\htmlData{tutor-start=0,tutor-end=6}{\Omega}$ 与 $\htmlData{tutor-start=0,tutor-end=6}{\Gamma}$ 有唯一的公共点 $\htmlData{tutor-start=0,tutor-end=1}{(}\htmlData{tutor-start=1,tutor-end=2}{8}\htmlData{tutor-start=2,tutor-end=3}{,}\htmlData{tutor-start=3,tutor-end=4}{9}\htmlData{tutor-start=4,tutor-end=5}{)}$ . 则 $\htmlData{tutor-start=0,tutor-end=6}{\Gamma}$ 的焦距为________.

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） · 填空题 · 数学竞赛/待细分

八张标有 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{B}\htmlData{tutor-start=4,tutor-end=5}{,} \htmlData{tutor-start=6,tutor-end=7}{C}\htmlData{tutor-start=7,tutor-end=8}{,} \htmlData{tutor-start=9,tutor-end=10}{D}\htmlData{tutor-start=10,tutor-end=11}{,} \htmlData{tutor-start=12,tutor-end=13}{E}\htmlData{tutor-start=13,tutor-end=14}{,} \htmlData{tutor-start=15,tutor-end=16}{F}\htmlData{tutor-start=16,tutor-end=17}{,} \htmlData{tutor-start=18,tutor-end=19}{G}\htmlData{tutor-start=19,tutor-end=20}{,} \htmlData{tutor-start=21,tutor-end=22}{H}$ 的正方形卡片构成下图．现逐一取走这些卡片，要求每次取走一张卡片时，该卡片与剩下的卡片中至多一张有公共边（例如可按 $\htmlData{tutor-start=0,tutor-end=1}{D}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{A}\htmlData{tutor-start=4,tutor-end=5}{,} \htmlData{tutor-start=6,tutor-end=7}{B}\htmlData{tutor-start=7,tutor-end=8}{,} \htmlData{tutor-start=9,tutor-end=10}{E}\htmlData{tutor-start=10,tutor-end=11}{,} \htmlData{tutor-start=12,tutor-end=13}{C}\htmlData{tutor-start=13,tutor-end=14}{,} \htmlData{tutor-start=15,tutor-end=16}{F}\htmlData{tutor-start=16,tutor-end=17}{,} \htmlData{tutor-start=18,tutor-end=19}{G}\htmlData{tutor-start=19,tutor-end=20}{,} \htmlData{tutor-start=21,tutor-end=22}{H}$ 的次序取走卡片，但不可按 $\htmlData{tutor-start=0,tutor-end=1}{D}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{B}\htmlData{tutor-start=4,tutor-end=5}{,} \htmlData{tutor-start=6,tutor-end=7}{A}\htmlData{tutor-start=7,tutor-end=8}{,} \htmlData{tutor-start=9,tutor-end=10}{E}\htmlData{tutor-start=10,tutor-end=11}{,} \htmlData{tutor-start=12,tutor-end=13}{C}\htmlData{tutor-start=13,tutor-end=14}{,} \htmlData{tutor-start=15,tutor-end=16}{F}\htmlData{tutor-start=16,tutor-end=17}{,} \htmlData{tutor-start=18,tutor-end=19}{G}\htmlData{tutor-start=19,tutor-end=20}{,} \htmlData{tutor-start=21,tutor-end=22}{H}$ 的次序取走卡片），则取走这八张卡片的不同次序的数目为________．

**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：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． · 数学竞赛/待细分

（本题满分 16 分）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，抛物线 $\htmlData{tutor-start=0,tutor-end=6}{\Gamma}\htmlData{tutor-start=6,tutor-end=7}{:} \htmlData{tutor-start=8,tutor-end=9}{y}^{\htmlData{tutor-start=11,tutor-end=12}{2}} \htmlData{tutor-start=14,tutor-end=15}{=} \htmlData{tutor-start=16,tutor-end=17}{4}\htmlData{tutor-start=17,tutor-end=18}{x}$ ， $\htmlData{tutor-start=0,tutor-end=1}{F}$ 为 $\htmlData{tutor-start=0,tutor-end=6}{\Gamma}$ 的焦点， $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{B}$ 为 $\htmlData{tutor-start=0,tutor-end=6}{\Gamma}$ 上的两个不重合的动点，使得线段 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{B}$ 的一个三等分点 $\htmlData{tutor-start=0,tutor-end=1}{P}$ 位于线段 $\htmlData{tutor-start=0,tutor-end=1}{O}\htmlData{tutor-start=1,tutor-end=2}{F}$ 上（含端点），记 $\htmlData{tutor-start=0,tutor-end=1}{Q}$ 为线段 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{B}$ 的另一个三等分点．求点 $\htmlData{tutor-start=0,tutor-end=1}{Q}$ 的轨迹方程．

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． · 数学竞赛/待细分

（本题满分 20 分）已知三棱柱 $\htmlData{tutor-start=0,tutor-end=6}{\Omega}\htmlData{tutor-start=6,tutor-end=7}{:} \htmlData{tutor-start=8,tutor-end=9}{A}\htmlData{tutor-start=9,tutor-end=10}{B}\htmlData{tutor-start=10,tutor-end=11}{C}\htmlData{tutor-start=11,tutor-end=12}{-}\htmlData{tutor-start=12,tutor-end=13}{A}_{\htmlData{tutor-start=15,tutor-end=16}{1}}\htmlData{tutor-start=17,tutor-end=18}{B}_{\htmlData{tutor-start=20,tutor-end=21}{1}}\htmlData{tutor-start=22,tutor-end=23}{C}_{\htmlData{tutor-start=25,tutor-end=26}{1}}$ 的 9 条棱长均相等. 记底面 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{B}\htmlData{tutor-start=2,tutor-end=3}{C}$ 所在平面为 $\htmlData{tutor-start=0,tutor-end=6}{\alpha}$ . 若 $\htmlData{tutor-start=0,tutor-end=6}{\Omega}$ 的另外四个面（即面 $\htmlData{tutor-start=0,tutor-end=1}{A}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{B}_{\htmlData{tutor-start=8,tutor-end=9}{1}}\htmlData{tutor-start=10,tutor-end=11}{C}_{\htmlData{tutor-start=13,tutor-end=14}{1}}$ , $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{B}\htmlData{tutor-start=2,tutor-end=3}{B}_{\htmlData{tutor-start=5,tutor-end=6}{1}}\htmlData{tutor-start=7,tutor-end=8}{A}_{\htmlData{tutor-start=10,tutor-end=11}{1}}$ , $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{C}\htmlData{tutor-start=2,tutor-end=3}{C}_{\htmlData{tutor-start=5,tutor-end=6}{1}}\htmlData{tutor-start=7,tutor-end=8}{A}_{\htmlData{tutor-start=10,tutor-end=11}{1}}$ , $\htmlData{tutor-start=0,tutor-end=1}{B}\htmlData{tutor-start=1,tutor-end=2}{C}\htmlData{tutor-start=2,tutor-end=3}{C}_{\htmlData{tutor-start=5,tutor-end=6}{1}}\htmlData{tutor-start=7,tutor-end=8}{B}_{\htmlData{tutor-start=10,tutor-end=11}{1}}$ ）在 $\htmlData{tutor-start=0,tutor-end=6}{\alpha}$ 上投影的面积从小到大重排后依次为 $\htmlData{tutor-start=0,tutor-end=1}{2}\sqrt{\htmlData{tutor-start=7,tutor-end=8}{3}}\htmlData{tutor-start=9,tutor-end=10}{,} \htmlData{tutor-start=11,tutor-end=12}{3}\sqrt{\htmlData{tutor-start=18,tutor-end=19}{3}}\htmlData{tutor-start=20,tutor-end=21}{,} \htmlData{tutor-start=22,tutor-end=23}{4}\sqrt{\htmlData{tutor-start=29,tutor-end=30}{3}}\htmlData{tutor-start=31,tutor-end=32}{,} \htmlData{tutor-start=33,tutor-end=34}{5}\sqrt{\htmlData{tutor-start=40,tutor-end=41}{3}}$ ，求 $\htmlData{tutor-start=0,tutor-end=6}{\Omega}$ 的体积.

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． · 数学竞赛/待细分

（本题满分 20 分）求出所有满足下面要求的不小于 1 的实数 $\htmlData{tutor-start=0,tutor-end=1}{t}$ ：对任意 $\htmlData{tutor-start=0,tutor-end=1}{a}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{b} \htmlData{tutor-start=5,tutor-end=9}{\in }\htmlData{tutor-start=9,tutor-end=10}{[}\htmlData{tutor-start=10,tutor-end=11}{-}\htmlData{tutor-start=11,tutor-end=12}{1}\htmlData{tutor-start=12,tutor-end=13}{,} \htmlData{tutor-start=14,tutor-end=15}{t}\htmlData{tutor-start=15,tutor-end=16}{]}$ ，总存在 $\htmlData{tutor-start=0,tutor-end=1}{c}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{d} \htmlData{tutor-start=5,tutor-end=9}{\in }\htmlData{tutor-start=9,tutor-end=10}{[}\htmlData{tutor-start=10,tutor-end=11}{-}\htmlData{tutor-start=11,tutor-end=12}{1}\htmlData{tutor-start=12,tutor-end=13}{,} \htmlData{tutor-start=14,tutor-end=15}{t}\htmlData{tutor-start=15,tutor-end=16}{]}$ ，使得 $\htmlData{tutor-start=0,tutor-end=1}{(}\htmlData{tutor-start=1,tutor-end=2}{a}\htmlData{tutor-start=2,tutor-end=3}{+}\htmlData{tutor-start=3,tutor-end=4}{c}\htmlData{tutor-start=4,tutor-end=5}{)}\htmlData{tutor-start=5,tutor-end=6}{(}\htmlData{tutor-start=6,tutor-end=7}{b}\htmlData{tutor-start=7,tutor-end=8}{+}\htmlData{tutor-start=8,tutor-end=9}{d}\htmlData{tutor-start=9,tutor-end=10}{)}\htmlData{tutor-start=10,tutor-end=11}{=}\htmlData{tutor-start=11,tutor-end=12}{1}$ .

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） · 加试 · 平面几何

（本题满分 40 分）如图， $\htmlData{tutor-start=0,tutor-end=6}{\Omega}$ 是以 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{B}$ 为直径的固定的半圆弧， $\htmlData{tutor-start=0,tutor-end=6}{\omega}$ 是经过点 $\htmlData{tutor-start=0,tutor-end=1}{A}$ 及 $\htmlData{tutor-start=0,tutor-end=6}{\Omega}$ 上另一个定点 $\htmlData{tutor-start=0,tutor-end=1}{T}$ 的定圆，且 $\htmlData{tutor-start=0,tutor-end=6}{\omega}$ 的圆心位于 $\htmlData{tutor-start=0,tutor-end=10}{\triangle }\htmlData{tutor-start=10,tutor-end=11}{A}\htmlData{tutor-start=11,tutor-end=12}{B}\htmlData{tutor-start=12,tutor-end=13}{T}$ 内。设 $\htmlData{tutor-start=0,tutor-end=1}{P}$ 是 $\htmlData{tutor-start=0,tutor-end=6}{\Omega}$ 的弧 $\widehat{\htmlData{tutor-start=9,tutor-end=10}{T}\htmlData{tutor-start=10,tutor-end=11}{B}}$ （不含端点）上的动点， $\htmlData{tutor-start=0,tutor-end=1}{C}\htmlData{tutor-start=1,tutor-end=2}{,}\htmlData{tutor-start=2,tutor-end=3}{D}$ 是 $\htmlData{tutor-start=0,tutor-end=6}{\omega}$ 上的两个动点，满足： $\htmlData{tutor-start=0,tutor-end=1}{C}$ 在线段 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{P}$ 上， $\htmlData{tutor-start=0,tutor-end=1}{C}\htmlData{tutor-start=1,tutor-end=2}{,}\htmlData{tutor-start=2,tutor-end=3}{D}$ 位于直线 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{B}$ 的异侧，且 $\htmlData{tutor-start=0,tutor-end=1}{C}\htmlData{tutor-start=1,tutor-end=2}{D} \htmlData{tutor-start=3,tutor-end=9}{\perp }\htmlData{tutor-start=9,tutor-end=10}{A}\htmlData{tutor-start=10,tutor-end=11}{B}$ 。记 $\htmlData{tutor-start=0,tutor-end=10}{\triangle }\htmlData{tutor-start=10,tutor-end=11}{C}\htmlData{tutor-start=11,tutor-end=12}{D}\htmlData{tutor-start=12,tutor-end=13}{P}$ 的外心为 $\htmlData{tutor-start=0,tutor-end=1}{K}$ 。证明： (1) 点 $\htmlData{tutor-start=0,tutor-end=1}{K}$ 在 $\htmlData{tutor-start=0,tutor-end=10}{\triangle }\htmlData{tutor-start=10,tutor-end=11}{T}\htmlData{tutor-start=11,tutor-end=12}{D}\htmlData{tutor-start=12,tutor-end=13}{P}$ 的外接圆上； (2) $\htmlData{tutor-start=0,tutor-end=1}{K}$ 为定点。

题解状态：标准答案与规范题解待补充

题目标签：SECOND-A-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（A 卷） · 二试 · 数论

（本题满分 40 分）正整数 $\htmlData{tutor-start=0,tutor-end=1}{n}$ 称为“好数”，如果对任意不同于 $\htmlData{tutor-start=0,tutor-end=1}{n}$ 的正整数 $\htmlData{tutor-start=0,tutor-end=1}{m}$ ，均有 $\left\{\frac{2^{n}}{n^{2}}\right\} \neq \left\{\frac{2^{m}}{m^{2}}\right\}$ ，这里， $\htmlData{tutor-start=0,tutor-end=2}{\{}\htmlData{tutor-start=2,tutor-end=3}{x}\htmlData{tutor-start=3,tutor-end=5}{\}}$ 表示实数 $\htmlData{tutor-start=0,tutor-end=1}{x}$ 的小数部分. 证明：存在无穷多个两两互素的合数均为好数.

题解状态：标准答案与规范题解待补充

题目标签：SECOND-A-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（A 卷） · 二试 · 数论

（本题满分 50 分）求具有下述性质的最小正整数 $\htmlData{tutor-start=0,tutor-end=1}{k}$ ：若将 $\htmlData{tutor-start=0,tutor-end=1}{1}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{2}\htmlData{tutor-start=4,tutor-end=5}{,} \cdots\htmlData{tutor-start=12,tutor-end=13}{,} \htmlData{tutor-start=14,tutor-end=15}{k}$ 中的每个数任意染为红色或者蓝色，则或者存在 9 个互不相同的红色的数 $\htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{x}_{\htmlData{tutor-start=10,tutor-end=11}{2}}\htmlData{tutor-start=12,tutor-end=13}{,} \cdots\htmlData{tutor-start=20,tutor-end=21}{,} \htmlData{tutor-start=22,tutor-end=23}{x}_{\htmlData{tutor-start=25,tutor-end=26}{9}}$ 满足 $\htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{1}} \htmlData{tutor-start=6,tutor-end=7}{+} \htmlData{tutor-start=8,tutor-end=9}{x}_{\htmlData{tutor-start=11,tutor-end=12}{2}} \htmlData{tutor-start=14,tutor-end=15}{+} \cdots \htmlData{tutor-start=23,tutor-end=24}{+} \htmlData{tutor-start=25,tutor-end=26}{x}_{\htmlData{tutor-start=28,tutor-end=29}{8}} \htmlData{tutor-start=31,tutor-end=32}{<} \htmlData{tutor-start=33,tutor-end=34}{x}_{\htmlData{tutor-start=36,tutor-end=37}{9}}$ ，或者存在 10 个互不相同的蓝色的数 $\htmlData{tutor-start=0,tutor-end=1}{y}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{y}_{\htmlData{tutor-start=10,tutor-end=11}{2}}\htmlData{tutor-start=12,tutor-end=13}{,} \cdots\htmlData{tutor-start=20,tutor-end=21}{,} \htmlData{tutor-start=22,tutor-end=23}{y}_{\htmlData{tutor-start=25,tutor-end=26}{1}\htmlData{tutor-start=26,tutor-end=27}{0}}$ 满足 $\htmlData{tutor-start=0,tutor-end=1}{y}_{\htmlData{tutor-start=3,tutor-end=4}{1}} \htmlData{tutor-start=6,tutor-end=7}{+} \htmlData{tutor-start=8,tutor-end=9}{y}_{\htmlData{tutor-start=11,tutor-end=12}{2}} \htmlData{tutor-start=14,tutor-end=15}{+} \cdots \htmlData{tutor-start=23,tutor-end=24}{+} \htmlData{tutor-start=25,tutor-end=26}{y}_{\htmlData{tutor-start=28,tutor-end=29}{9}} \htmlData{tutor-start=31,tutor-end=32}{<} \htmlData{tutor-start=33,tutor-end=34}{y}_{\htmlData{tutor-start=36,tutor-end=37}{1}\htmlData{tutor-start=37,tutor-end=38}{0}}$ .

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 · 数学竞赛/待细分

（本题满分 50 分）设 $\htmlData{tutor-start=0,tutor-end=1}{a}\htmlData{tutor-start=1,tutor-end=2}{=}\htmlData{tutor-start=2,tutor-end=3}{1}\htmlData{tutor-start=3,tutor-end=4}{+}\htmlData{tutor-start=4,tutor-end=5}{1}\htmlData{tutor-start=5,tutor-end=6}{0}^{\htmlData{tutor-start=8,tutor-end=9}{-}\htmlData{tutor-start=9,tutor-end=10}{4}}$ ．在 $\htmlData{tutor-start=0,tutor-end=1}{2}\htmlData{tutor-start=1,tutor-end=2}{0}\htmlData{tutor-start=2,tutor-end=3}{2}\htmlData{tutor-start=3,tutor-end=4}{3}\htmlData{tutor-start=4,tutor-end=11}{\times }\htmlData{tutor-start=11,tutor-end=12}{2}\htmlData{tutor-start=12,tutor-end=13}{0}\htmlData{tutor-start=13,tutor-end=14}{2}\htmlData{tutor-start=14,tutor-end=15}{3}$ 的方格表的每个小方格中填入区间 $\htmlData{tutor-start=0,tutor-end=1}{[}\htmlData{tutor-start=1,tutor-end=2}{1}\htmlData{tutor-start=2,tutor-end=3}{,}\htmlData{tutor-start=3,tutor-end=4}{a}\htmlData{tutor-start=4,tutor-end=5}{]}$ 中的一个实数．设第 $\htmlData{tutor-start=0,tutor-end=1}{i}$ 行的总和为 $\htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{i}}$ ，第 $\htmlData{tutor-start=0,tutor-end=1}{i}$ 列的总和为 $\htmlData{tutor-start=0,tutor-end=1}{y}_{\htmlData{tutor-start=3,tutor-end=4}{i}}$ ， $\htmlData{tutor-start=0,tutor-end=1}{1}\htmlData{tutor-start=1,tutor-end=5}{\le }\htmlData{tutor-start=5,tutor-end=6}{i}\htmlData{tutor-start=6,tutor-end=10}{\le }\htmlData{tutor-start=10,tutor-end=11}{2}\htmlData{tutor-start=11,tutor-end=12}{0}\htmlData{tutor-start=12,tutor-end=13}{2}\htmlData{tutor-start=13,tutor-end=14}{3}$ ．求 $\frac{y_{1} y_{2} \cdots y_{2023}}{x_{1} x_{2} \cdots x_{2023}}$ 的最大值（答案用含 $\htmlData{tutor-start=0,tutor-end=1}{a}$ 的式子表示）．

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。