2015 年全国高中数学联赛 A 卷（学生版）

books/national_high_school_league_students/papers/competition-national-league-2015-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

设 $\htmlData{tutor-start=0,tutor-end=1}{a}\htmlData{tutor-start=1,tutor-end=2}{,}\htmlData{tutor-start=2,tutor-end=3}{b}$ 为不相等的实数,若二次函数 $\htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{x}\htmlData{tutor-start=3,tutor-end=4}{)}\htmlData{tutor-start=4,tutor-end=5}{=}\htmlData{tutor-start=5,tutor-end=6}{x}^{\htmlData{tutor-start=8,tutor-end=9}{2}}\htmlData{tutor-start=10,tutor-end=11}{+}\htmlData{tutor-start=11,tutor-end=12}{a}\htmlData{tutor-start=12,tutor-end=13}{x}\htmlData{tutor-start=13,tutor-end=14}{+}\htmlData{tutor-start=14,tutor-end=15}{b}$ 满足 $\htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{a}\htmlData{tutor-start=3,tutor-end=4}{)}\htmlData{tutor-start=4,tutor-end=5}{=}\htmlData{tutor-start=5,tutor-end=6}{f}\htmlData{tutor-start=6,tutor-end=7}{(}\htmlData{tutor-start=7,tutor-end=8}{b}\htmlData{tutor-start=8,tutor-end=9}{)}$ ,则 $\htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{2}\htmlData{tutor-start=3,tutor-end=4}{)}$ 的值是________.

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） · 数学竞赛/待细分

若实数 $\htmlData{tutor-start=0,tutor-end=6}{\theta}$ 满足 $\cos\htmlData{tutor-start=4,tutor-end=10}{\theta}\htmlData{tutor-start=10,tutor-end=11}{=}\tan\htmlData{tutor-start=15,tutor-end=21}{\theta}$ ,则 $\frac{\htmlData{tutor-start=6,tutor-end=7}{1}}{\sin\htmlData{tutor-start=13,tutor-end=19}{\theta}}\htmlData{tutor-start=20,tutor-end=21}{+}\cos^{\htmlData{tutor-start=27,tutor-end=28}{4}}\htmlData{tutor-start=29,tutor-end=35}{\theta}$ 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

已知复数数列 $\htmlData{tutor-start=0,tutor-end=2}{\{}\htmlData{tutor-start=2,tutor-end=3}{z}_{\htmlData{tutor-start=5,tutor-end=6}{n}}\htmlData{tutor-start=7,tutor-end=9}{\}}$ 满足 $\htmlData{tutor-start=0,tutor-end=1}{z}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{=}\htmlData{tutor-start=6,tutor-end=7}{1}\htmlData{tutor-start=7,tutor-end=8}{,} \htmlData{tutor-start=9,tutor-end=10}{z}_{\htmlData{tutor-start=12,tutor-end=13}{n}\htmlData{tutor-start=13,tutor-end=14}{+}\htmlData{tutor-start=14,tutor-end=15}{1}}\htmlData{tutor-start=16,tutor-end=17}{=}\overline{\htmlData{tutor-start=27,tutor-end=28}{z}}_{\htmlData{tutor-start=31,tutor-end=32}{n}}\htmlData{tutor-start=33,tutor-end=34}{+}\htmlData{tutor-start=34,tutor-end=35}{1}\htmlData{tutor-start=35,tutor-end=36}{+}\htmlData{tutor-start=36,tutor-end=37}{n}\htmlData{tutor-start=37,tutor-end=38}{i} \htmlData{tutor-start=39,tutor-end=40}{(}\htmlData{tutor-start=40,tutor-end=41}{n}\htmlData{tutor-start=41,tutor-end=42}{=}\htmlData{tutor-start=42,tutor-end=43}{1}\htmlData{tutor-start=43,tutor-end=44}{,}\htmlData{tutor-start=44,tutor-end=45}{2}\htmlData{tutor-start=45,tutor-end=46}{,}\cdots\htmlData{tutor-start=52,tutor-end=53}{)}$ ,其中 $\htmlData{tutor-start=0,tutor-end=1}{i}$ 为虚数单位, $\overline{\htmlData{tutor-start=10,tutor-end=11}{z}}_{\htmlData{tutor-start=14,tutor-end=15}{n}}$ 表示 $\htmlData{tutor-start=0,tutor-end=1}{z}_{\htmlData{tutor-start=3,tutor-end=4}{n}}$ 的共轭复数,则 $\htmlData{tutor-start=0,tutor-end=1}{z}_{\htmlData{tutor-start=3,tutor-end=4}{2}\htmlData{tutor-start=4,tutor-end=5}{0}\htmlData{tutor-start=5,tutor-end=6}{1}\htmlData{tutor-start=6,tutor-end=7}{5}}$ 的值是________.

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，共 64 分） · 数学竞赛/待细分

在矩形 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{B}\htmlData{tutor-start=2,tutor-end=3}{C}\htmlData{tutor-start=3,tutor-end=4}{D}$ 中, $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{B}\htmlData{tutor-start=2,tutor-end=3}{=}\htmlData{tutor-start=3,tutor-end=4}{2}\htmlData{tutor-start=4,tutor-end=5}{,} \htmlData{tutor-start=6,tutor-end=7}{A}\htmlData{tutor-start=7,tutor-end=8}{D}\htmlData{tutor-start=8,tutor-end=9}{=}\htmlData{tutor-start=9,tutor-end=10}{1}$ , 边 $\htmlData{tutor-start=0,tutor-end=1}{D}\htmlData{tutor-start=1,tutor-end=2}{C}$ 上(包括点 $\htmlData{tutor-start=0,tutor-end=1}{D}\htmlData{tutor-start=1,tutor-end=2}{,}\htmlData{tutor-start=2,tutor-end=3}{C}$ )的动点 $\htmlData{tutor-start=0,tutor-end=1}{P}$ 与 $\htmlData{tutor-start=0,tutor-end=1}{C}\htmlData{tutor-start=1,tutor-end=2}{B}$ 延长线上(包括点 $\htmlData{tutor-start=0,tutor-end=1}{B}$ )的动点 $\htmlData{tutor-start=0,tutor-end=1}{Q}$ 满足 $\htmlData{tutor-start=0,tutor-end=1}{|}\overrightarrow{\htmlData{tutor-start=17,tutor-end=18}{D}\htmlData{tutor-start=18,tutor-end=19}{P}}\htmlData{tutor-start=20,tutor-end=21}{|}\htmlData{tutor-start=21,tutor-end=22}{=}\htmlData{tutor-start=22,tutor-end=23}{|}\overrightarrow{\htmlData{tutor-start=39,tutor-end=40}{B}\htmlData{tutor-start=40,tutor-end=41}{Q}}\htmlData{tutor-start=42,tutor-end=43}{|}$ , 则向量 $\overrightarrow{\htmlData{tutor-start=16,tutor-end=17}{P}\htmlData{tutor-start=17,tutor-end=18}{A}}$ 与向量 $\overrightarrow{\htmlData{tutor-start=16,tutor-end=17}{P}\htmlData{tutor-start=17,tutor-end=18}{Q}}$ 的数量积 $\overrightarrow{\htmlData{tutor-start=16,tutor-end=17}{P}\htmlData{tutor-start=17,tutor-end=18}{A}}\htmlData{tutor-start=19,tutor-end=24}{\cdot}\overrightarrow{\htmlData{tutor-start=40,tutor-end=41}{P}\htmlData{tutor-start=41,tutor-end=42}{Q}}$ 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） · 数学竞赛/待细分

在正方体中随机取三条棱,它们两两异面的概率为________.

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，共 64 分） · 数学竞赛/待细分

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,点集 $\htmlData{tutor-start=0,tutor-end=1}{K}\htmlData{tutor-start=1,tutor-end=2}{=}\htmlData{tutor-start=2,tutor-end=4}{\{}\htmlData{tutor-start=4,tutor-end=5}{(}\htmlData{tutor-start=5,tutor-end=6}{x}\htmlData{tutor-start=6,tutor-end=7}{,}\htmlData{tutor-start=7,tutor-end=8}{y}\htmlData{tutor-start=8,tutor-end=9}{)}\htmlData{tutor-start=9,tutor-end=10}{|}\htmlData{tutor-start=10,tutor-end=11}{(}\htmlData{tutor-start=11,tutor-end=12}{|}\htmlData{tutor-start=12,tutor-end=13}{x}\htmlData{tutor-start=13,tutor-end=14}{|}\htmlData{tutor-start=14,tutor-end=15}{+}\htmlData{tutor-start=15,tutor-end=16}{|}\htmlData{tutor-start=16,tutor-end=17}{3}\htmlData{tutor-start=17,tutor-end=18}{y}\htmlData{tutor-start=18,tutor-end=19}{|}\htmlData{tutor-start=19,tutor-end=20}{-}\htmlData{tutor-start=20,tutor-end=21}{6}\htmlData{tutor-start=21,tutor-end=22}{)}\htmlData{tutor-start=22,tutor-end=23}{(}\htmlData{tutor-start=23,tutor-end=24}{|}\htmlData{tutor-start=24,tutor-end=25}{3}\htmlData{tutor-start=25,tutor-end=26}{x}\htmlData{tutor-start=26,tutor-end=27}{|}\htmlData{tutor-start=27,tutor-end=28}{+}\htmlData{tutor-start=28,tutor-end=29}{|}\htmlData{tutor-start=29,tutor-end=30}{y}\htmlData{tutor-start=30,tutor-end=31}{|}\htmlData{tutor-start=31,tutor-end=32}{-}\htmlData{tutor-start=32,tutor-end=33}{6}\htmlData{tutor-start=33,tutor-end=34}{)}\htmlData{tutor-start=34,tutor-end=38}{\le }\htmlData{tutor-start=38,tutor-end=39}{0}\htmlData{tutor-start=39,tutor-end=41}{\}}$ 所对应的平面区域的面积为________.

题解状态：标准答案与规范题解待补充

题目标签：FIRST-A-FILL-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 $\htmlData{tutor-start=0,tutor-end=6}{\omega}$ 为正实数,若存在 $\htmlData{tutor-start=0,tutor-end=1}{a}\htmlData{tutor-start=1,tutor-end=2}{,}\htmlData{tutor-start=2,tutor-end=3}{b} \htmlData{tutor-start=4,tutor-end=5}{(}\htmlData{tutor-start=5,tutor-end=9}{\pi }\htmlData{tutor-start=9,tutor-end=13}{\le }\htmlData{tutor-start=13,tutor-end=14}{a} \htmlData{tutor-start=15,tutor-end=16}{<} \htmlData{tutor-start=17,tutor-end=18}{b} \htmlData{tutor-start=19,tutor-end=23}{\le }\htmlData{tutor-start=23,tutor-end=24}{2}\htmlData{tutor-start=24,tutor-end=27}{\pi}\htmlData{tutor-start=27,tutor-end=28}{)}$ ,使得 $\sin\htmlData{tutor-start=4,tutor-end=11}{\omega }\htmlData{tutor-start=11,tutor-end=12}{a}\htmlData{tutor-start=12,tutor-end=13}{+}\sin\htmlData{tutor-start=17,tutor-end=24}{\omega }\htmlData{tutor-start=24,tutor-end=25}{b}\htmlData{tutor-start=25,tutor-end=26}{=}\htmlData{tutor-start=26,tutor-end=27}{2}$ ,则实数 $\htmlData{tutor-start=0,tutor-end=6}{\omega}$ 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-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

对四位数 $\overline{\htmlData{tutor-start=10,tutor-end=11}{a}\htmlData{tutor-start=11,tutor-end=12}{b}\htmlData{tutor-start=12,tutor-end=13}{c}\htmlData{tutor-start=13,tutor-end=14}{d}} \htmlData{tutor-start=16,tutor-end=17}{(}\htmlData{tutor-start=17,tutor-end=18}{1}\htmlData{tutor-start=18,tutor-end=22}{\le }\htmlData{tutor-start=22,tutor-end=23}{a}\htmlData{tutor-start=23,tutor-end=27}{\le }\htmlData{tutor-start=27,tutor-end=28}{9}\htmlData{tutor-start=28,tutor-end=29}{,} \htmlData{tutor-start=30,tutor-end=31}{0}\htmlData{tutor-start=31,tutor-end=35}{\le }\htmlData{tutor-start=35,tutor-end=36}{b}\htmlData{tutor-start=36,tutor-end=37}{,}\htmlData{tutor-start=37,tutor-end=38}{c}\htmlData{tutor-start=38,tutor-end=39}{,}\htmlData{tutor-start=39,tutor-end=40}{d}\htmlData{tutor-start=40,tutor-end=44}{\le }\htmlData{tutor-start=44,tutor-end=45}{9}\htmlData{tutor-start=45,tutor-end=46}{)}$ ,若 $\htmlData{tutor-start=0,tutor-end=1}{a}\htmlData{tutor-start=1,tutor-end=2}{>}\htmlData{tutor-start=2,tutor-end=3}{b}\htmlData{tutor-start=3,tutor-end=4}{,} \htmlData{tutor-start=5,tutor-end=6}{b}\htmlData{tutor-start=6,tutor-end=7}{<}\htmlData{tutor-start=7,tutor-end=8}{c}\htmlData{tutor-start=8,tutor-end=9}{,} \htmlData{tutor-start=10,tutor-end=11}{c}\htmlData{tutor-start=11,tutor-end=12}{>}\htmlData{tutor-start=12,tutor-end=13}{d}$ ,则称 $\overline{\htmlData{tutor-start=10,tutor-end=11}{a}\htmlData{tutor-start=11,tutor-end=12}{b}\htmlData{tutor-start=12,tutor-end=13}{c}\htmlData{tutor-start=13,tutor-end=14}{d}}$ 为 $\htmlData{tutor-start=0,tutor-end=1}{P}$ 类数,若 $\htmlData{tutor-start=0,tutor-end=1}{a}\htmlData{tutor-start=1,tutor-end=2}{<}\htmlData{tutor-start=2,tutor-end=3}{b}\htmlData{tutor-start=3,tutor-end=4}{,} \htmlData{tutor-start=5,tutor-end=6}{b}\htmlData{tutor-start=6,tutor-end=7}{>}\htmlData{tutor-start=7,tutor-end=8}{c}\htmlData{tutor-start=8,tutor-end=9}{,} \htmlData{tutor-start=10,tutor-end=11}{c}\htmlData{tutor-start=11,tutor-end=12}{<}\htmlData{tutor-start=12,tutor-end=13}{d}$ ,则称 $\overline{\htmlData{tutor-start=10,tutor-end=11}{a}\htmlData{tutor-start=11,tutor-end=12}{b}\htmlData{tutor-start=12,tutor-end=13}{c}\htmlData{tutor-start=13,tutor-end=14}{d}}$ 为 $\htmlData{tutor-start=0,tutor-end=1}{Q}$ 类数,用 $\htmlData{tutor-start=0,tutor-end=1}{N}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{P}\htmlData{tutor-start=3,tutor-end=4}{)}$ 与 $\htmlData{tutor-start=0,tutor-end=1}{N}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{Q}\htmlData{tutor-start=3,tutor-end=4}{)}$ 分别表示 $\htmlData{tutor-start=0,tutor-end=1}{P}$ 类数与 $\htmlData{tutor-start=0,tutor-end=1}{Q}$ 类数的个数,则 $\htmlData{tutor-start=0,tutor-end=1}{N}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{P}\htmlData{tutor-start=3,tutor-end=4}{)}\htmlData{tutor-start=4,tutor-end=5}{-}\htmlData{tutor-start=5,tutor-end=6}{N}\htmlData{tutor-start=6,tutor-end=7}{(}\htmlData{tutor-start=7,tutor-end=8}{Q}\htmlData{tutor-start=8,tutor-end=9}{)}$ 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-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 个小题，共 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）若实数 $\htmlData{tutor-start=0,tutor-end=1}{a}\htmlData{tutor-start=1,tutor-end=2}{,}\htmlData{tutor-start=2,tutor-end=3}{b}\htmlData{tutor-start=3,tutor-end=4}{,}\htmlData{tutor-start=4,tutor-end=5}{c}$ 满足 $\htmlData{tutor-start=0,tutor-end=1}{2}^{\htmlData{tutor-start=3,tutor-end=4}{a}}\htmlData{tutor-start=5,tutor-end=6}{+}\htmlData{tutor-start=6,tutor-end=7}{4}^{\htmlData{tutor-start=9,tutor-end=10}{b}}\htmlData{tutor-start=11,tutor-end=12}{=}\htmlData{tutor-start=12,tutor-end=13}{2}^{\htmlData{tutor-start=15,tutor-end=16}{c}}\htmlData{tutor-start=17,tutor-end=18}{,} \htmlData{tutor-start=19,tutor-end=20}{4}^{\htmlData{tutor-start=22,tutor-end=23}{a}}\htmlData{tutor-start=24,tutor-end=25}{+}\htmlData{tutor-start=25,tutor-end=26}{2}^{\htmlData{tutor-start=28,tutor-end=29}{b}}\htmlData{tutor-start=30,tutor-end=31}{=}\htmlData{tutor-start=31,tutor-end=32}{4}^{\htmlData{tutor-start=34,tutor-end=35}{c}}$ ,求 $\htmlData{tutor-start=0,tutor-end=1}{c}$ 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-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 个小题，共 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)设 $\htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{,}\htmlData{tutor-start=6,tutor-end=7}{a}_{\htmlData{tutor-start=9,tutor-end=10}{2}}\htmlData{tutor-start=11,tutor-end=12}{,}\htmlData{tutor-start=12,tutor-end=13}{a}_{\htmlData{tutor-start=15,tutor-end=16}{3}}\htmlData{tutor-start=17,tutor-end=18}{,}\htmlData{tutor-start=18,tutor-end=19}{a}_{\htmlData{tutor-start=21,tutor-end=22}{4}}$ 是四个有理数,使得 $\htmlData{tutor-start=0,tutor-end=2}{\{}\htmlData{tutor-start=2,tutor-end=3}{a}_{\htmlData{tutor-start=5,tutor-end=6}{i}} \htmlData{tutor-start=8,tutor-end=9}{a}_{\htmlData{tutor-start=11,tutor-end=12}{j}} \htmlData{tutor-start=14,tutor-end=19}{\mid }\htmlData{tutor-start=19,tutor-end=20}{1}\htmlData{tutor-start=20,tutor-end=24}{\le }\htmlData{tutor-start=24,tutor-end=25}{i} \htmlData{tutor-start=26,tutor-end=27}{<} \htmlData{tutor-start=28,tutor-end=29}{j} \htmlData{tutor-start=30,tutor-end=34}{\le }\htmlData{tutor-start=34,tutor-end=35}{4}\htmlData{tutor-start=35,tutor-end=37}{\}} \htmlData{tutor-start=38,tutor-end=39}{=} \htmlData{tutor-start=40,tutor-end=42}{\{}\htmlData{tutor-start=42,tutor-end=43}{-}\htmlData{tutor-start=43,tutor-end=44}{2}\htmlData{tutor-start=44,tutor-end=45}{4}\htmlData{tutor-start=45,tutor-end=46}{,} \htmlData{tutor-start=47,tutor-end=48}{-}\htmlData{tutor-start=48,tutor-end=49}{2}\htmlData{tutor-start=49,tutor-end=50}{,} \htmlData{tutor-start=51,tutor-end=52}{-}\frac{\htmlData{tutor-start=58,tutor-end=59}{3}}{\htmlData{tutor-start=61,tutor-end=62}{2}}\htmlData{tutor-start=63,tutor-end=64}{,} \htmlData{tutor-start=65,tutor-end=66}{-}\frac{\htmlData{tutor-start=72,tutor-end=73}{1}}{\htmlData{tutor-start=75,tutor-end=76}{8}}\htmlData{tutor-start=77,tutor-end=78}{,} \htmlData{tutor-start=79,tutor-end=80}{1}\htmlData{tutor-start=80,tutor-end=81}{,} \htmlData{tutor-start=82,tutor-end=83}{3}\htmlData{tutor-start=83,tutor-end=85}{\}}$ 求 $\htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{+}\htmlData{tutor-start=6,tutor-end=7}{a}_{\htmlData{tutor-start=9,tutor-end=10}{2}}\htmlData{tutor-start=11,tutor-end=12}{+}\htmlData{tutor-start=12,tutor-end=13}{a}_{\htmlData{tutor-start=15,tutor-end=16}{3}}\htmlData{tutor-start=17,tutor-end=18}{+}\htmlData{tutor-start=18,tutor-end=19}{a}_{\htmlData{tutor-start=21,tutor-end=22}{4}}$ 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 个小题，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 $\htmlData{tutor-start=0,tutor-end=1}{x}\htmlData{tutor-start=1,tutor-end=2}{O}\htmlData{tutor-start=2,tutor-end=3}{y}$ 中, $\htmlData{tutor-start=0,tutor-end=1}{F}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{F}_{\htmlData{tutor-start=10,tutor-end=11}{2}}$ 分别是椭圆 $\frac{\htmlData{tutor-start=6,tutor-end=7}{x}^{\htmlData{tutor-start=9,tutor-end=10}{2}}}{\htmlData{tutor-start=13,tutor-end=14}{2}}\htmlData{tutor-start=15,tutor-end=16}{+}\htmlData{tutor-start=16,tutor-end=17}{y}^{\htmlData{tutor-start=19,tutor-end=20}{2}}\htmlData{tutor-start=21,tutor-end=22}{=}\htmlData{tutor-start=22,tutor-end=23}{1}$ 的左,右焦点,设不经过焦点 $\htmlData{tutor-start=0,tutor-end=1}{F}_{\htmlData{tutor-start=3,tutor-end=4}{1}}$ 的直线 $\htmlData{tutor-start=0,tutor-end=1}{l}$ 与椭圆 $\htmlData{tutor-start=0,tutor-end=1}{C}$ 交于两个不同的点 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{,}\htmlData{tutor-start=2,tutor-end=3}{B}$ ,焦点 $\htmlData{tutor-start=0,tutor-end=1}{F}_{\htmlData{tutor-start=3,tutor-end=4}{2}}$ 到直线 $\htmlData{tutor-start=0,tutor-end=1}{l}$ 的距离为 $\htmlData{tutor-start=0,tutor-end=1}{d}$ . 如果直线 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{F}_{\htmlData{tutor-start=4,tutor-end=5}{1}}\htmlData{tutor-start=6,tutor-end=7}{,} \htmlData{tutor-start=8,tutor-end=9}{l}\htmlData{tutor-start=9,tutor-end=10}{,} \htmlData{tutor-start=11,tutor-end=12}{B}\htmlData{tutor-start=12,tutor-end=13}{F}_{\htmlData{tutor-start=15,tutor-end=16}{1}}$ 的斜率成等差数列,求 $\htmlData{tutor-start=0,tutor-end=1}{d}$ 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-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） · 二、解答题（本大题共 3 个小题，共 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 40 分) 设 $\htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{a}_{\htmlData{tutor-start=10,tutor-end=11}{2}}\htmlData{tutor-start=12,tutor-end=13}{,} \cdots\htmlData{tutor-start=20,tutor-end=21}{,} \htmlData{tutor-start=22,tutor-end=23}{a}_{\htmlData{tutor-start=25,tutor-end=26}{n}} \htmlData{tutor-start=28,tutor-end=29}{(}\htmlData{tutor-start=29,tutor-end=30}{n} \htmlData{tutor-start=31,tutor-end=35}{\ge }\htmlData{tutor-start=35,tutor-end=36}{2}\htmlData{tutor-start=36,tutor-end=37}{)}$ 是实数,证明:可以连取 $\varepsilon_{1}, \varepsilon_{2}, \cdots, \varepsilon_{n} \in \{1, -1\}$ 使得 $\htmlData{tutor-start=0,tutor-end=1}{(}\sum_{\htmlData{tutor-start=7,tutor-end=8}{i}\htmlData{tutor-start=8,tutor-end=9}{=}\htmlData{tutor-start=9,tutor-end=10}{1}}^{\htmlData{tutor-start=13,tutor-end=14}{n}} \htmlData{tutor-start=16,tutor-end=17}{a}_{\htmlData{tutor-start=19,tutor-end=20}{i}}\htmlData{tutor-start=21,tutor-end=22}{)}^{\htmlData{tutor-start=24,tutor-end=25}{2}} \htmlData{tutor-start=27,tutor-end=28}{+} \htmlData{tutor-start=29,tutor-end=30}{(}\sum_{\htmlData{tutor-start=36,tutor-end=37}{i}\htmlData{tutor-start=37,tutor-end=38}{=}\htmlData{tutor-start=38,tutor-end=39}{1}}^{\htmlData{tutor-start=42,tutor-end=43}{n}} \htmlData{tutor-start=45,tutor-end=56}{\varepsilon}_{\htmlData{tutor-start=58,tutor-end=59}{i}} \htmlData{tutor-start=61,tutor-end=62}{a}_{\htmlData{tutor-start=64,tutor-end=65}{i}}\htmlData{tutor-start=66,tutor-end=67}{)}^{\htmlData{tutor-start=69,tutor-end=70}{2}} \htmlData{tutor-start=72,tutor-end=76}{\le }\htmlData{tutor-start=76,tutor-end=77}{(}\htmlData{tutor-start=77,tutor-end=78}{n}\htmlData{tutor-start=78,tutor-end=79}{+}\htmlData{tutor-start=79,tutor-end=80}{1}\htmlData{tutor-start=80,tutor-end=81}{)}\htmlData{tutor-start=81,tutor-end=82}{(}\sum_{\htmlData{tutor-start=88,tutor-end=89}{i}\htmlData{tutor-start=89,tutor-end=90}{=}\htmlData{tutor-start=90,tutor-end=91}{1}}^{\htmlData{tutor-start=94,tutor-end=95}{n}} \htmlData{tutor-start=97,tutor-end=98}{a}_{\htmlData{tutor-start=100,tutor-end=101}{i}}^{\htmlData{tutor-start=104,tutor-end=105}{2}}\htmlData{tutor-start=106,tutor-end=107}{)}$

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、解答题（本大题共 3 个小题，共 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 40 分)设 $\htmlData{tutor-start=0,tutor-end=1}{S}\htmlData{tutor-start=1,tutor-end=2}{=}\htmlData{tutor-start=2,tutor-end=4}{\{}\htmlData{tutor-start=4,tutor-end=5}{A}_{\htmlData{tutor-start=7,tutor-end=8}{1}}\htmlData{tutor-start=9,tutor-end=10}{,} \htmlData{tutor-start=11,tutor-end=12}{A}_{\htmlData{tutor-start=14,tutor-end=15}{2}}\htmlData{tutor-start=16,tutor-end=17}{,} \cdots\htmlData{tutor-start=24,tutor-end=25}{,} \htmlData{tutor-start=26,tutor-end=27}{A}_{\htmlData{tutor-start=29,tutor-end=30}{n}}\htmlData{tutor-start=31,tutor-end=33}{\}}$ ,其中 $\htmlData{tutor-start=0,tutor-end=1}{A}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{A}_{\htmlData{tutor-start=10,tutor-end=11}{2}}\htmlData{tutor-start=12,tutor-end=13}{,} \cdots\htmlData{tutor-start=20,tutor-end=21}{,} \htmlData{tutor-start=22,tutor-end=23}{A}_{\htmlData{tutor-start=25,tutor-end=26}{n}}$ 是 $\htmlData{tutor-start=0,tutor-end=1}{n}$ 个互不相同的有限集合 $\htmlData{tutor-start=0,tutor-end=1}{(}\htmlData{tutor-start=1,tutor-end=2}{n} \htmlData{tutor-start=3,tutor-end=7}{\ge }\htmlData{tutor-start=7,tutor-end=8}{2}\htmlData{tutor-start=8,tutor-end=9}{)}$ ,满足对任意 $\htmlData{tutor-start=0,tutor-end=1}{A}_{\htmlData{tutor-start=3,tutor-end=4}{i}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{A}_{\htmlData{tutor-start=10,tutor-end=11}{j}} \htmlData{tutor-start=13,tutor-end=17}{\in }\htmlData{tutor-start=17,tutor-end=18}{S}$ ,均有 $\htmlData{tutor-start=0,tutor-end=1}{A}_{\htmlData{tutor-start=3,tutor-end=4}{i}} \htmlData{tutor-start=6,tutor-end=11}{\cup }\htmlData{tutor-start=11,tutor-end=12}{A}_{\htmlData{tutor-start=14,tutor-end=15}{j}} \htmlData{tutor-start=17,tutor-end=21}{\in }\htmlData{tutor-start=21,tutor-end=22}{S}$ ,若 $\htmlData{tutor-start=0,tutor-end=1}{k}\htmlData{tutor-start=1,tutor-end=2}{=}\min_{\htmlData{tutor-start=8,tutor-end=9}{1} \htmlData{tutor-start=10,tutor-end=14}{\le }\htmlData{tutor-start=14,tutor-end=15}{i} \htmlData{tutor-start=16,tutor-end=20}{\le }\htmlData{tutor-start=20,tutor-end=21}{n}} \htmlData{tutor-start=23,tutor-end=24}{|}\htmlData{tutor-start=24,tutor-end=25}{A}_{\htmlData{tutor-start=27,tutor-end=28}{i}}\htmlData{tutor-start=29,tutor-end=30}{|} \htmlData{tutor-start=31,tutor-end=35}{\ge }\htmlData{tutor-start=35,tutor-end=36}{2}$ ,证明:存在 $\htmlData{tutor-start=0,tutor-end=1}{x} \htmlData{tutor-start=2,tutor-end=6}{\in }\bigcup_{\htmlData{tutor-start=15,tutor-end=16}{i}\htmlData{tutor-start=16,tutor-end=17}{=}\htmlData{tutor-start=17,tutor-end=18}{1}}^{\htmlData{tutor-start=21,tutor-end=22}{n}} \htmlData{tutor-start=24,tutor-end=25}{A}_{\htmlData{tutor-start=27,tutor-end=28}{i}}$ ,使得 $\htmlData{tutor-start=0,tutor-end=1}{x}$ 属于 $\htmlData{tutor-start=0,tutor-end=1}{A}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{A}_{\htmlData{tutor-start=10,tutor-end=11}{2}}\htmlData{tutor-start=12,tutor-end=13}{,} \cdots\htmlData{tutor-start=20,tutor-end=21}{,} \htmlData{tutor-start=22,tutor-end=23}{A}_{\htmlData{tutor-start=25,tutor-end=26}{n}}$ 中的至少 $\frac{\htmlData{tutor-start=6,tutor-end=7}{n}}{\htmlData{tutor-start=9,tutor-end=10}{k}}$ 个集合(这里 $\htmlData{tutor-start=0,tutor-end=1}{|}\htmlData{tutor-start=1,tutor-end=2}{X}\htmlData{tutor-start=2,tutor-end=3}{|}$ 表示有限集合 $\htmlData{tutor-start=0,tutor-end=1}{X}$ 的元素个数)

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、解答题（本大题共 3 个小题，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 50 分)如图, $\htmlData{tutor-start=0,tutor-end=10}{\triangle }\htmlData{tutor-start=10,tutor-end=11}{A}\htmlData{tutor-start=11,tutor-end=12}{B}\htmlData{tutor-start=12,tutor-end=13}{C}$ 内接于圆 $\htmlData{tutor-start=0,tutor-end=1}{O}\htmlData{tutor-start=1,tutor-end=2}{,}\htmlData{tutor-start=2,tutor-end=3}{P}$ 为 $\htmlData{tutor-start=0,tutor-end=1}{B}\htmlData{tutor-start=1,tutor-end=2}{C}$ 上一点,点 $\htmlData{tutor-start=0,tutor-end=1}{K}$ 在线段 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{P}$ 上,使得 $\htmlData{tutor-start=0,tutor-end=1}{B}\htmlData{tutor-start=1,tutor-end=2}{K}$ 平分 $\htmlData{tutor-start=0,tutor-end=7}{\angle }\htmlData{tutor-start=7,tutor-end=8}{A}\htmlData{tutor-start=8,tutor-end=9}{B}\htmlData{tutor-start=9,tutor-end=10}{C}$ ,过 $\htmlData{tutor-start=0,tutor-end=1}{K}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{P}\htmlData{tutor-start=4,tutor-end=5}{,} \htmlData{tutor-start=6,tutor-end=7}{C}$ 三点的圆 $\htmlData{tutor-start=0,tutor-end=6}{\Omega}$ 与边 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{C}$ 交于点 $\htmlData{tutor-start=0,tutor-end=1}{D}$ ,连结 $\htmlData{tutor-start=0,tutor-end=1}{B}\htmlData{tutor-start=1,tutor-end=2}{D}$ 交圆 $\htmlData{tutor-start=0,tutor-end=6}{\Omega}$ 于点 $\htmlData{tutor-start=0,tutor-end=1}{E}$ ,连结 $\htmlData{tutor-start=0,tutor-end=1}{P}\htmlData{tutor-start=1,tutor-end=2}{E}$ 并延长与边 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{B}$ 交于点 $\htmlData{tutor-start=0,tutor-end=1}{F}$ ,证明: $\htmlData{tutor-start=0,tutor-end=7}{\angle }\htmlData{tutor-start=7,tutor-end=8}{A}\htmlData{tutor-start=8,tutor-end=9}{B}\htmlData{tutor-start=9,tutor-end=10}{C}\htmlData{tutor-start=10,tutor-end=11}{=}\htmlData{tutor-start=11,tutor-end=12}{2}\htmlData{tutor-start=12,tutor-end=19}{\angle }\htmlData{tutor-start=19,tutor-end=20}{F}\htmlData{tutor-start=20,tutor-end=21}{C}\htmlData{tutor-start=21,tutor-end=22}{B}$

**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、解答题（本大题共 3 个小题，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 50 分)求具有下述性质的所有正整数 $\htmlData{tutor-start=0,tutor-end=1}{k}$ :对任意正整数 $\htmlData{tutor-start=0,tutor-end=1}{n}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{2}^{\htmlData{tutor-start=6,tutor-end=7}{(}\htmlData{tutor-start=7,tutor-end=8}{k}\htmlData{tutor-start=8,tutor-end=9}{-}\htmlData{tutor-start=9,tutor-end=10}{1}\htmlData{tutor-start=10,tutor-end=11}{)}\htmlData{tutor-start=11,tutor-end=12}{n}\htmlData{tutor-start=12,tutor-end=13}{+}\htmlData{tutor-start=13,tutor-end=14}{1}}$ 不整除 $\frac{\htmlData{tutor-start=6,tutor-end=7}{(}\htmlData{tutor-start=7,tutor-end=8}{k}\htmlData{tutor-start=8,tutor-end=9}{n}\htmlData{tutor-start=9,tutor-end=10}{)}\htmlData{tutor-start=10,tutor-end=11}{!}}{\htmlData{tutor-start=13,tutor-end=14}{n}\htmlData{tutor-start=14,tutor-end=15}{!}}$ .

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。