2016 年全国高中数学联赛 A 卷（学生版）

books/national_high_school_league_students/papers/competition-national-league-2016-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） · 数学竞赛/待细分

设实数 $\htmlData{tutor-start=0,tutor-end=1}{a}$ 满足 $\htmlData{tutor-start=0,tutor-end=1}{a} \htmlData{tutor-start=2,tutor-end=3}{<} \htmlData{tutor-start=4,tutor-end=5}{9}\htmlData{tutor-start=5,tutor-end=6}{a}^{\htmlData{tutor-start=8,tutor-end=9}{3}} \htmlData{tutor-start=11,tutor-end=12}{-} \htmlData{tutor-start=13,tutor-end=14}{1}\htmlData{tutor-start=14,tutor-end=15}{1}\htmlData{tutor-start=15,tutor-end=16}{a} \htmlData{tutor-start=17,tutor-end=18}{<} \htmlData{tutor-start=19,tutor-end=20}{|}\htmlData{tutor-start=20,tutor-end=21}{a}\htmlData{tutor-start=21,tutor-end=22}{|}$ ，则 $\htmlData{tutor-start=0,tutor-end=1}{a}$ 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-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

设复数 $\htmlData{tutor-start=0,tutor-end=1}{z}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{w}$ 满足 $\htmlData{tutor-start=0,tutor-end=1}{|}\htmlData{tutor-start=1,tutor-end=2}{z}\htmlData{tutor-start=2,tutor-end=3}{|}\htmlData{tutor-start=3,tutor-end=4}{=}\htmlData{tutor-start=4,tutor-end=5}{3}$ ， $\htmlData{tutor-start=0,tutor-end=1}{(}\htmlData{tutor-start=1,tutor-end=2}{z}\htmlData{tutor-start=2,tutor-end=3}{+}\overline{\htmlData{tutor-start=13,tutor-end=14}{w}}\htmlData{tutor-start=15,tutor-end=16}{)}\htmlData{tutor-start=16,tutor-end=17}{(}\overline{\htmlData{tutor-start=27,tutor-end=28}{z}}\htmlData{tutor-start=29,tutor-end=30}{-}\htmlData{tutor-start=30,tutor-end=31}{w}\htmlData{tutor-start=31,tutor-end=32}{)}\htmlData{tutor-start=32,tutor-end=33}{=}\htmlData{tutor-start=33,tutor-end=34}{7}\htmlData{tutor-start=34,tutor-end=35}{+}\htmlData{tutor-start=35,tutor-end=36}{4}\htmlData{tutor-start=36,tutor-end=37}{i}$ ，其中 $\htmlData{tutor-start=0,tutor-end=1}{i}$ 是虚数单位， $\overline{\htmlData{tutor-start=10,tutor-end=11}{z}}\htmlData{tutor-start=12,tutor-end=13}{,} \overline{\htmlData{tutor-start=24,tutor-end=25}{w}}$ 分别表示 $\htmlData{tutor-start=0,tutor-end=1}{z}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{w}$ 的共轭复数，则 $\htmlData{tutor-start=0,tutor-end=1}{(}\htmlData{tutor-start=1,tutor-end=2}{z}\htmlData{tutor-start=2,tutor-end=3}{+}\htmlData{tutor-start=3,tutor-end=4}{2}\overline{\htmlData{tutor-start=14,tutor-end=15}{w}}\htmlData{tutor-start=16,tutor-end=17}{)}\htmlData{tutor-start=17,tutor-end=18}{(}\overline{\htmlData{tutor-start=28,tutor-end=29}{z}}\htmlData{tutor-start=30,tutor-end=31}{-}\htmlData{tutor-start=31,tutor-end=32}{2}\htmlData{tutor-start=32,tutor-end=33}{w}\htmlData{tutor-start=33,tutor-end=34}{)}$ 的模为________.

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） · 数学竞赛/待细分

正实数 $\htmlData{tutor-start=0,tutor-end=1}{u}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{v}\htmlData{tutor-start=4,tutor-end=5}{,} \htmlData{tutor-start=6,tutor-end=7}{w}$ 均不等于 1，若 $\log_{\htmlData{tutor-start=6,tutor-end=7}{u}} \htmlData{tutor-start=9,tutor-end=10}{v}\htmlData{tutor-start=10,tutor-end=11}{w} \htmlData{tutor-start=12,tutor-end=13}{+} \log_{\htmlData{tutor-start=20,tutor-end=21}{v}} \htmlData{tutor-start=23,tutor-end=24}{w} \htmlData{tutor-start=25,tutor-end=26}{=} \htmlData{tutor-start=27,tutor-end=28}{5}$ ， $\log_{\htmlData{tutor-start=6,tutor-end=7}{v}} \htmlData{tutor-start=9,tutor-end=10}{u} \htmlData{tutor-start=11,tutor-end=12}{+} \log_{\htmlData{tutor-start=19,tutor-end=20}{w}} \htmlData{tutor-start=22,tutor-end=23}{v} \htmlData{tutor-start=24,tutor-end=25}{=} \htmlData{tutor-start=26,tutor-end=27}{3}$ ，则 $\log_{\htmlData{tutor-start=6,tutor-end=7}{w}} \htmlData{tutor-start=9,tutor-end=10}{u}$ 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

袋子 A 中装有 2 张 10 元纸币和 3 张 1 元纸币，袋子 B 中装有 4 张 5 元纸币和 3 张 1 元纸币.现随机从两个袋子中各取出两张纸币，则 A 中剩下的纸币面值之和大于 B 中剩下的纸币面值之和的概率为________.

题解状态：标准答案与规范题解待补充

题目标签：FIRST-A-FILL-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

设 $\htmlData{tutor-start=0,tutor-end=1}{P}$ 为一圆锥的顶点， $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{B}\htmlData{tutor-start=4,tutor-end=5}{,} \htmlData{tutor-start=6,tutor-end=7}{C}$ 是其底面圆周上的三点，满足 $\angle ABC=90^\circ$ ， $\htmlData{tutor-start=0,tutor-end=1}{M}$ 为 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{P}$ 的中点.若 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{B}\htmlData{tutor-start=2,tutor-end=3}{=}\htmlData{tutor-start=3,tutor-end=4}{1}$ ， $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{C}\htmlData{tutor-start=2,tutor-end=3}{=}\htmlData{tutor-start=3,tutor-end=4}{2}$ ， $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{P}\htmlData{tutor-start=2,tutor-end=3}{=}\sqrt{\htmlData{tutor-start=9,tutor-end=10}{2}}$ ，则二面角 $\htmlData{tutor-start=0,tutor-end=1}{M}\htmlData{tutor-start=1,tutor-end=2}{-}\htmlData{tutor-start=2,tutor-end=3}{B}\htmlData{tutor-start=3,tutor-end=4}{C}\htmlData{tutor-start=4,tutor-end=5}{-}\htmlData{tutor-start=5,tutor-end=6}{A}$ 的大小为________.

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） · 数论

设函数 $\htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{x}\htmlData{tutor-start=3,tutor-end=4}{)}\htmlData{tutor-start=4,tutor-end=5}{=}\sin^{\htmlData{tutor-start=11,tutor-end=12}{4}} \frac{\htmlData{tutor-start=20,tutor-end=21}{k}\htmlData{tutor-start=21,tutor-end=22}{x}}{\htmlData{tutor-start=24,tutor-end=25}{1}\htmlData{tutor-start=25,tutor-end=26}{0}} \htmlData{tutor-start=28,tutor-end=29}{+} \cos^{\htmlData{tutor-start=36,tutor-end=37}{4}} \frac{\htmlData{tutor-start=45,tutor-end=46}{k}\htmlData{tutor-start=46,tutor-end=47}{x}}{\htmlData{tutor-start=49,tutor-end=50}{1}\htmlData{tutor-start=50,tutor-end=51}{0}}$ ，其中 $\htmlData{tutor-start=0,tutor-end=1}{k}$ 是一个正整数.若对任意实数 $\htmlData{tutor-start=0,tutor-end=1}{a}$ ，均有 $\htmlData{tutor-start=0,tutor-end=2}{\{}\htmlData{tutor-start=2,tutor-end=3}{f}\htmlData{tutor-start=3,tutor-end=4}{(}\htmlData{tutor-start=4,tutor-end=5}{x}\htmlData{tutor-start=5,tutor-end=6}{)} \htmlData{tutor-start=7,tutor-end=12}{\mid }\htmlData{tutor-start=12,tutor-end=13}{a} \htmlData{tutor-start=14,tutor-end=15}{<} \htmlData{tutor-start=16,tutor-end=17}{x} \htmlData{tutor-start=18,tutor-end=19}{<} \htmlData{tutor-start=20,tutor-end=21}{a}\htmlData{tutor-start=21,tutor-end=22}{+}\htmlData{tutor-start=22,tutor-end=23}{1}\htmlData{tutor-start=23,tutor-end=25}{\}} \htmlData{tutor-start=26,tutor-end=27}{=} \htmlData{tutor-start=28,tutor-end=30}{\{}\htmlData{tutor-start=30,tutor-end=31}{f}\htmlData{tutor-start=31,tutor-end=32}{(}\htmlData{tutor-start=32,tutor-end=33}{x}\htmlData{tutor-start=33,tutor-end=34}{)} \htmlData{tutor-start=35,tutor-end=40}{\mid }\htmlData{tutor-start=40,tutor-end=41}{x} \htmlData{tutor-start=42,tutor-end=46}{\in }\htmlData{tutor-start=46,tutor-end=47}{R}\htmlData{tutor-start=47,tutor-end=49}{\}}$ ，则 $\htmlData{tutor-start=0,tutor-end=1}{k}$ 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-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） · 平面几何

双曲线 $\htmlData{tutor-start=0,tutor-end=1}{C}$ 的方程为 $\htmlData{tutor-start=0,tutor-end=1}{x}^{\htmlData{tutor-start=3,tutor-end=4}{2}} \htmlData{tutor-start=6,tutor-end=7}{-} \frac{\htmlData{tutor-start=14,tutor-end=15}{y}^{\htmlData{tutor-start=17,tutor-end=18}{2}}}{\htmlData{tutor-start=21,tutor-end=22}{3}} \htmlData{tutor-start=24,tutor-end=25}{=} \htmlData{tutor-start=26,tutor-end=27}{1}$ ，左、右焦点分别为 $\htmlData{tutor-start=0,tutor-end=1}{F}_{\htmlData{tutor-start=3,tutor-end=4}{1}}$ 、 $\htmlData{tutor-start=0,tutor-end=1}{F}_{\htmlData{tutor-start=3,tutor-end=4}{2}}$ ，过点 $\htmlData{tutor-start=0,tutor-end=1}{F}_{\htmlData{tutor-start=3,tutor-end=4}{2}}$ 作直线与双曲线 $\htmlData{tutor-start=0,tutor-end=1}{C}$ 的右半支交于点 $\htmlData{tutor-start=0,tutor-end=1}{P}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{Q}$ ，使得 $\angle F_{1}PQ=90^\circ$ ，则 $\htmlData{tutor-start=0,tutor-end=7}{\Delta }\htmlData{tutor-start=7,tutor-end=8}{F}_{\htmlData{tutor-start=10,tutor-end=11}{1}}\htmlData{tutor-start=12,tutor-end=13}{P}\htmlData{tutor-start=13,tutor-end=14}{Q}$ 的内切圆半径是________.

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

设 $\htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{a}_{\htmlData{tutor-start=10,tutor-end=11}{2}}\htmlData{tutor-start=12,tutor-end=13}{,} \htmlData{tutor-start=14,tutor-end=15}{a}_{\htmlData{tutor-start=17,tutor-end=18}{3}}\htmlData{tutor-start=19,tutor-end=20}{,} \htmlData{tutor-start=21,tutor-end=22}{a}_{\htmlData{tutor-start=24,tutor-end=25}{4}}$ 是 1, 2, $\cdots$ , 100 中的 4 个互不相同的数，满足 $\htmlData{tutor-start=0,tutor-end=1}{(}\htmlData{tutor-start=1,tutor-end=2}{a}_{\htmlData{tutor-start=4,tutor-end=5}{1}}^{\htmlData{tutor-start=8,tutor-end=9}{2}} \htmlData{tutor-start=11,tutor-end=12}{+} \htmlData{tutor-start=13,tutor-end=14}{a}_{\htmlData{tutor-start=16,tutor-end=17}{2}}^{\htmlData{tutor-start=20,tutor-end=21}{2}} \htmlData{tutor-start=23,tutor-end=24}{+} \htmlData{tutor-start=25,tutor-end=26}{a}_{\htmlData{tutor-start=28,tutor-end=29}{3}}^{\htmlData{tutor-start=32,tutor-end=33}{2}}\htmlData{tutor-start=34,tutor-end=35}{)}\htmlData{tutor-start=35,tutor-end=36}{(}\htmlData{tutor-start=36,tutor-end=37}{a}_{\htmlData{tutor-start=39,tutor-end=40}{2}}^{\htmlData{tutor-start=43,tutor-end=44}{2}} \htmlData{tutor-start=46,tutor-end=47}{+} \htmlData{tutor-start=48,tutor-end=49}{a}_{\htmlData{tutor-start=51,tutor-end=52}{3}}^{\htmlData{tutor-start=55,tutor-end=56}{2}} \htmlData{tutor-start=58,tutor-end=59}{+} \htmlData{tutor-start=60,tutor-end=61}{a}_{\htmlData{tutor-start=63,tutor-end=64}{4}}^{\htmlData{tutor-start=67,tutor-end=68}{2}}\htmlData{tutor-start=69,tutor-end=70}{)} \htmlData{tutor-start=71,tutor-end=72}{=} \htmlData{tutor-start=73,tutor-end=74}{(}\htmlData{tutor-start=74,tutor-end=75}{a}_{\htmlData{tutor-start=77,tutor-end=78}{1}}\htmlData{tutor-start=79,tutor-end=80}{a}_{\htmlData{tutor-start=82,tutor-end=83}{2}} \htmlData{tutor-start=85,tutor-end=86}{+} \htmlData{tutor-start=87,tutor-end=88}{a}_{\htmlData{tutor-start=90,tutor-end=91}{2}}\htmlData{tutor-start=92,tutor-end=93}{a}_{\htmlData{tutor-start=95,tutor-end=96}{3}} \htmlData{tutor-start=98,tutor-end=99}{+} \htmlData{tutor-start=100,tutor-end=101}{a}_{\htmlData{tutor-start=103,tutor-end=104}{3}}\htmlData{tutor-start=105,tutor-end=106}{a}_{\htmlData{tutor-start=108,tutor-end=109}{4}}\htmlData{tutor-start=110,tutor-end=111}{)}^{\htmlData{tutor-start=113,tutor-end=114}{2}}$ ，则这样的有序数组 $\htmlData{tutor-start=0,tutor-end=1}{(}\htmlData{tutor-start=1,tutor-end=2}{a}_{\htmlData{tutor-start=4,tutor-end=5}{1}}\htmlData{tutor-start=6,tutor-end=7}{,} \htmlData{tutor-start=8,tutor-end=9}{a}_{\htmlData{tutor-start=11,tutor-end=12}{2}}\htmlData{tutor-start=13,tutor-end=14}{,} \htmlData{tutor-start=15,tutor-end=16}{a}_{\htmlData{tutor-start=18,tutor-end=19}{3}}\htmlData{tutor-start=20,tutor-end=21}{,} \htmlData{tutor-start=22,tutor-end=23}{a}_{\htmlData{tutor-start=25,tutor-end=26}{4}}\htmlData{tutor-start=27,tutor-end=28}{)}$ 的个数为________.

题解状态：标准答案与规范题解待补充

题目标签：FIRST-A-FILL-P8

解题过程

该题已完成题面、原题号、PDF 来源锚定和题目级特征标注；答案、证明步骤与教材关联尚未录入。

一试（A 卷） · 二、解答题（本大题共 3 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）在 $\htmlData{tutor-start=0,tutor-end=7}{\Delta }\htmlData{tutor-start=7,tutor-end=8}{A}\htmlData{tutor-start=8,tutor-end=9}{B}\htmlData{tutor-start=9,tutor-end=10}{C}$ 中，已知 $\overrightarrow{\htmlData{tutor-start=16,tutor-end=17}{A}\htmlData{tutor-start=17,tutor-end=18}{B}} \htmlData{tutor-start=20,tutor-end=26}{\cdot }\overrightarrow{\htmlData{tutor-start=42,tutor-end=43}{A}\htmlData{tutor-start=43,tutor-end=44}{C}} \htmlData{tutor-start=46,tutor-end=47}{+} \htmlData{tutor-start=48,tutor-end=49}{2}\overrightarrow{\htmlData{tutor-start=65,tutor-end=66}{B}\htmlData{tutor-start=66,tutor-end=67}{A}} \htmlData{tutor-start=69,tutor-end=75}{\cdot }\overrightarrow{\htmlData{tutor-start=91,tutor-end=92}{B}\htmlData{tutor-start=92,tutor-end=93}{C}} \htmlData{tutor-start=95,tutor-end=96}{=} \htmlData{tutor-start=97,tutor-end=98}{3}\overrightarrow{\htmlData{tutor-start=114,tutor-end=115}{C}\htmlData{tutor-start=115,tutor-end=116}{A}} \htmlData{tutor-start=118,tutor-end=124}{\cdot }\overrightarrow{\htmlData{tutor-start=140,tutor-end=141}{C}\htmlData{tutor-start=141,tutor-end=142}{B}}$ .求 $\sin \htmlData{tutor-start=5,tutor-end=6}{C}$ 的最大值.

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） · 一试 · 代数

（本题满分 20 分）已知 $\htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{x}\htmlData{tutor-start=3,tutor-end=4}{)}$ 是 $\htmlData{tutor-start=0,tutor-end=1}{R}$ 上的奇函数， $\htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{1}\htmlData{tutor-start=3,tutor-end=4}{)}\htmlData{tutor-start=4,tutor-end=5}{=}\htmlData{tutor-start=5,tutor-end=6}{1}$ ，且对任意 $\htmlData{tutor-start=0,tutor-end=1}{x}\htmlData{tutor-start=1,tutor-end=2}{<}\htmlData{tutor-start=2,tutor-end=3}{0}$ ，均有 $\htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\frac{\htmlData{tutor-start=8,tutor-end=9}{x}}{\htmlData{tutor-start=11,tutor-end=12}{x}\htmlData{tutor-start=12,tutor-end=13}{-}\htmlData{tutor-start=13,tutor-end=14}{1}}\htmlData{tutor-start=15,tutor-end=16}{)}\htmlData{tutor-start=16,tutor-end=17}{=}\htmlData{tutor-start=17,tutor-end=18}{x}\htmlData{tutor-start=18,tutor-end=19}{f}\htmlData{tutor-start=19,tutor-end=20}{(}\htmlData{tutor-start=20,tutor-end=21}{x}\htmlData{tutor-start=21,tutor-end=22}{)}$ 。求 $\htmlData{tutor-start=0,tutor-end=1}{f}\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}{f}\htmlData{tutor-start=5,tutor-end=6}{(}\frac{\htmlData{tutor-start=12,tutor-end=13}{1}}{\htmlData{tutor-start=15,tutor-end=16}{1}\htmlData{tutor-start=16,tutor-end=17}{0}\htmlData{tutor-start=17,tutor-end=18}{0}}\htmlData{tutor-start=19,tutor-end=20}{)}\htmlData{tutor-start=20,tutor-end=21}{+}\htmlData{tutor-start=21,tutor-end=22}{f}\htmlData{tutor-start=22,tutor-end=23}{(}\frac{\htmlData{tutor-start=29,tutor-end=30}{1}}{\htmlData{tutor-start=32,tutor-end=33}{2}}\htmlData{tutor-start=34,tutor-end=35}{)}\htmlData{tutor-start=35,tutor-end=36}{f}\htmlData{tutor-start=36,tutor-end=37}{(}\frac{\htmlData{tutor-start=43,tutor-end=44}{1}}{\htmlData{tutor-start=46,tutor-end=47}{9}\htmlData{tutor-start=47,tutor-end=48}{9}}\htmlData{tutor-start=49,tutor-end=50}{)}\htmlData{tutor-start=50,tutor-end=51}{+}\htmlData{tutor-start=51,tutor-end=52}{f}\htmlData{tutor-start=52,tutor-end=53}{(}\frac{\htmlData{tutor-start=59,tutor-end=60}{1}}{\htmlData{tutor-start=62,tutor-end=63}{3}}\htmlData{tutor-start=64,tutor-end=65}{)}\htmlData{tutor-start=65,tutor-end=66}{f}\htmlData{tutor-start=66,tutor-end=67}{(}\frac{\htmlData{tutor-start=73,tutor-end=74}{1}}{\htmlData{tutor-start=76,tutor-end=77}{9}\htmlData{tutor-start=77,tutor-end=78}{8}}\htmlData{tutor-start=79,tutor-end=80}{)}\htmlData{tutor-start=80,tutor-end=81}{+}\cdots\htmlData{tutor-start=87,tutor-end=88}{+}\htmlData{tutor-start=88,tutor-end=89}{f}\htmlData{tutor-start=89,tutor-end=90}{(}\frac{\htmlData{tutor-start=96,tutor-end=97}{1}}{\htmlData{tutor-start=99,tutor-end=100}{5}\htmlData{tutor-start=100,tutor-end=101}{0}}\htmlData{tutor-start=102,tutor-end=103}{)}\htmlData{tutor-start=103,tutor-end=104}{f}\htmlData{tutor-start=104,tutor-end=105}{(}\frac{\htmlData{tutor-start=111,tutor-end=112}{1}}{\htmlData{tutor-start=114,tutor-end=115}{5}\htmlData{tutor-start=115,tutor-end=116}{1}}\htmlData{tutor-start=117,tutor-end=118}{)}$ 的值。

题解状态：标准答案与规范题解待补充

题目标签：FIRST-A-PROBLEM-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） · 一试 · 平面几何

（本题满分 20 分）如图所示，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， $\htmlData{tutor-start=0,tutor-end=1}{F}$ 是 $\htmlData{tutor-start=0,tutor-end=1}{x}$ 轴正半轴上的一个动点。以 $\htmlData{tutor-start=0,tutor-end=1}{F}$ 为焦点， $\htmlData{tutor-start=0,tutor-end=1}{O}$ 为顶点作抛物线 $\htmlData{tutor-start=0,tutor-end=1}{C}$ 。设 $\htmlData{tutor-start=0,tutor-end=1}{P}$ 是第一象限内 $\htmlData{tutor-start=0,tutor-end=1}{C}$ 上的一点， $\htmlData{tutor-start=0,tutor-end=1}{Q}$ 是 $\htmlData{tutor-start=0,tutor-end=1}{x}$ 轴负半轴上一点，使得 $\htmlData{tutor-start=0,tutor-end=1}{P}\htmlData{tutor-start=1,tutor-end=2}{Q}$ 为 $\htmlData{tutor-start=0,tutor-end=1}{C}$ 的切线，且 $\htmlData{tutor-start=0,tutor-end=1}{|}\htmlData{tutor-start=1,tutor-end=2}{P}\htmlData{tutor-start=2,tutor-end=3}{Q}\htmlData{tutor-start=3,tutor-end=4}{|}\htmlData{tutor-start=4,tutor-end=5}{=}\htmlData{tutor-start=5,tutor-end=6}{2}$ 。圆 $\htmlData{tutor-start=0,tutor-end=1}{C}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{C}_{\htmlData{tutor-start=10,tutor-end=11}{2}}$ 均与直线 $\htmlData{tutor-start=0,tutor-end=1}{O}\htmlData{tutor-start=1,tutor-end=2}{P}$ 相切于点 $\htmlData{tutor-start=0,tutor-end=1}{P}$ ，且均与轴相切。求点 $\htmlData{tutor-start=0,tutor-end=1}{F}$ 的坐标，使圆 $\htmlData{tutor-start=0,tutor-end=1}{C}_{\htmlData{tutor-start=3,tutor-end=4}{1}}$ 与 $\htmlData{tutor-start=0,tutor-end=1}{C}_{\htmlData{tutor-start=3,tutor-end=4}{2}}$ 的面积之和取到最小值。

**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-A-PROBLEM-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） · 二试 · 数学竞赛/待细分

（本题满分 40 分）设实数 $\htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{a}_{\htmlData{tutor-start=10,tutor-end=11}{2}}\htmlData{tutor-start=12,tutor-end=13}{,} \cdots\htmlData{tutor-start=20,tutor-end=21}{,} \htmlData{tutor-start=22,tutor-end=23}{a}_{\htmlData{tutor-start=25,tutor-end=26}{2}\htmlData{tutor-start=26,tutor-end=27}{0}\htmlData{tutor-start=27,tutor-end=28}{1}\htmlData{tutor-start=28,tutor-end=29}{6}}$ 满足 $\htmlData{tutor-start=0,tutor-end=1}{9}\htmlData{tutor-start=1,tutor-end=2}{a}_{\htmlData{tutor-start=4,tutor-end=5}{i}} \htmlData{tutor-start=7,tutor-end=8}{>} \htmlData{tutor-start=9,tutor-end=10}{1}\htmlData{tutor-start=10,tutor-end=11}{1}\htmlData{tutor-start=11,tutor-end=12}{a}_{\htmlData{tutor-start=14,tutor-end=15}{i}\htmlData{tutor-start=15,tutor-end=16}{+}\htmlData{tutor-start=16,tutor-end=17}{1}}^{\htmlData{tutor-start=20,tutor-end=21}{2}} \htmlData{tutor-start=23,tutor-end=24}{(}\htmlData{tutor-start=24,tutor-end=25}{i}\htmlData{tutor-start=25,tutor-end=26}{=}\htmlData{tutor-start=26,tutor-end=27}{1}\htmlData{tutor-start=27,tutor-end=28}{,} \htmlData{tutor-start=29,tutor-end=30}{2}\htmlData{tutor-start=30,tutor-end=31}{,} \cdots\htmlData{tutor-start=38,tutor-end=39}{,} \htmlData{tutor-start=40,tutor-end=41}{2}\htmlData{tutor-start=41,tutor-end=42}{0}\htmlData{tutor-start=42,tutor-end=43}{1}\htmlData{tutor-start=43,tutor-end=44}{5}\htmlData{tutor-start=44,tutor-end=45}{)}$ 。求 $\htmlData{tutor-start=0,tutor-end=1}{(}\htmlData{tutor-start=1,tutor-end=2}{a}_{\htmlData{tutor-start=4,tutor-end=5}{1}}\htmlData{tutor-start=6,tutor-end=7}{-}\htmlData{tutor-start=7,tutor-end=8}{a}_{\htmlData{tutor-start=10,tutor-end=11}{2}}^{\htmlData{tutor-start=14,tutor-end=15}{2}}\htmlData{tutor-start=16,tutor-end=17}{)}\htmlData{tutor-start=17,tutor-end=18}{(}\htmlData{tutor-start=18,tutor-end=19}{a}_{\htmlData{tutor-start=21,tutor-end=22}{2}}\htmlData{tutor-start=23,tutor-end=24}{-}\htmlData{tutor-start=24,tutor-end=25}{a}_{\htmlData{tutor-start=27,tutor-end=28}{3}}^{\htmlData{tutor-start=31,tutor-end=32}{2}}\htmlData{tutor-start=33,tutor-end=34}{)}\cdots\htmlData{tutor-start=40,tutor-end=41}{(}\htmlData{tutor-start=41,tutor-end=42}{a}_{\htmlData{tutor-start=44,tutor-end=45}{2}\htmlData{tutor-start=45,tutor-end=46}{0}\htmlData{tutor-start=46,tutor-end=47}{1}\htmlData{tutor-start=47,tutor-end=48}{5}}\htmlData{tutor-start=49,tutor-end=50}{-}\htmlData{tutor-start=50,tutor-end=51}{a}_{\htmlData{tutor-start=53,tutor-end=54}{2}\htmlData{tutor-start=54,tutor-end=55}{0}\htmlData{tutor-start=55,tutor-end=56}{1}\htmlData{tutor-start=56,tutor-end=57}{6}}^{\htmlData{tutor-start=60,tutor-end=61}{2}}\htmlData{tutor-start=62,tutor-end=63}{)}\htmlData{tutor-start=63,tutor-end=64}{(}\htmlData{tutor-start=64,tutor-end=65}{a}_{\htmlData{tutor-start=67,tutor-end=68}{2}\htmlData{tutor-start=68,tutor-end=69}{0}\htmlData{tutor-start=69,tutor-end=70}{1}\htmlData{tutor-start=70,tutor-end=71}{6}}\htmlData{tutor-start=72,tutor-end=73}{-}\htmlData{tutor-start=73,tutor-end=74}{a}_{\htmlData{tutor-start=76,tutor-end=77}{1}}^{\htmlData{tutor-start=80,tutor-end=81}{2}}\htmlData{tutor-start=82,tutor-end=83}{)}$ 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-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 · 平面几何

（本题满分 40 分）如图所示，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， $\htmlData{tutor-start=0,tutor-end=1}{X}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{Y}$ 是直线 $\htmlData{tutor-start=0,tutor-end=1}{B}\htmlData{tutor-start=1,tutor-end=2}{C}$ 上两点（ $\htmlData{tutor-start=0,tutor-end=1}{X}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{B}\htmlData{tutor-start=4,tutor-end=5}{,} \htmlData{tutor-start=6,tutor-end=7}{C}\htmlData{tutor-start=7,tutor-end=8}{,} \htmlData{tutor-start=9,tutor-end=10}{Y}$ 顺次排列），使得 $\htmlData{tutor-start=0,tutor-end=1}{B}\htmlData{tutor-start=1,tutor-end=2}{X} \htmlData{tutor-start=3,tutor-end=9}{\cdot }\htmlData{tutor-start=9,tutor-end=10}{A}\htmlData{tutor-start=10,tutor-end=11}{C} \htmlData{tutor-start=12,tutor-end=13}{=} \htmlData{tutor-start=14,tutor-end=15}{C}\htmlData{tutor-start=15,tutor-end=16}{Y} \htmlData{tutor-start=17,tutor-end=23}{\cdot }\htmlData{tutor-start=23,tutor-end=24}{A}\htmlData{tutor-start=24,tutor-end=25}{B}$ 。设 $\htmlData{tutor-start=0,tutor-end=10}{\triangle }\htmlData{tutor-start=10,tutor-end=11}{A}\htmlData{tutor-start=11,tutor-end=12}{C}\htmlData{tutor-start=12,tutor-end=13}{X}$ ， $\htmlData{tutor-start=0,tutor-end=10}{\triangle }\htmlData{tutor-start=10,tutor-end=11}{A}\htmlData{tutor-start=11,tutor-end=12}{B}\htmlData{tutor-start=12,tutor-end=13}{Y}$ 的外心分别为 $\htmlData{tutor-start=0,tutor-end=1}{O}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{O}_{\htmlData{tutor-start=10,tutor-end=11}{2}}$ ，直线 $\htmlData{tutor-start=0,tutor-end=1}{O}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{O}_{\htmlData{tutor-start=8,tutor-end=9}{2}}$ 与 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{B}\htmlData{tutor-start=2,tutor-end=3}{,} \htmlData{tutor-start=4,tutor-end=5}{A}\htmlData{tutor-start=5,tutor-end=6}{C}$ 分别交于点 $\htmlData{tutor-start=0,tutor-end=1}{U}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{V}$ 。证明： $\htmlData{tutor-start=0,tutor-end=10}{\triangle }\htmlData{tutor-start=10,tutor-end=11}{A}\htmlData{tutor-start=11,tutor-end=12}{U}\htmlData{tutor-start=12,tutor-end=13}{V}$ 是等腰三角形。

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 · 平面几何

（本题满分 50 分）给定空间中 10 个点，其中任意四点不在一个平面上，将某些点之间的线段相连，若得到的图形中没有三角形也没有空间四边形，试确定所连线段数目的最大值.

题解状态：标准答案与规范题解待补充

题目标签：SECOND-A-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） · 四 · 数论

（本题满分 50 分）设 $\htmlData{tutor-start=0,tutor-end=1}{p}$ 与 $\htmlData{tutor-start=0,tutor-end=1}{p}\htmlData{tutor-start=1,tutor-end=2}{+}\htmlData{tutor-start=2,tutor-end=3}{2}$ 均是素数， $\htmlData{tutor-start=0,tutor-end=1}{p}\htmlData{tutor-start=1,tutor-end=2}{>}\htmlData{tutor-start=2,tutor-end=3}{3}$ ，数列 $\htmlData{tutor-start=0,tutor-end=2}{\{}\htmlData{tutor-start=2,tutor-end=3}{a}_{\htmlData{tutor-start=5,tutor-end=6}{n}}\htmlData{tutor-start=7,tutor-end=9}{\}}$ 定义为 $\htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{1}} \htmlData{tutor-start=6,tutor-end=7}{=} \htmlData{tutor-start=8,tutor-end=9}{2}$ ， $\htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{n}} \htmlData{tutor-start=6,tutor-end=7}{=} \htmlData{tutor-start=8,tutor-end=9}{a}_{\htmlData{tutor-start=11,tutor-end=12}{n}\htmlData{tutor-start=12,tutor-end=13}{-}\htmlData{tutor-start=13,tutor-end=14}{1}} \htmlData{tutor-start=16,tutor-end=17}{+} \htmlData{tutor-start=18,tutor-end=19}{[}\frac{\htmlData{tutor-start=25,tutor-end=26}{p}\htmlData{tutor-start=26,tutor-end=27}{a}_{\htmlData{tutor-start=29,tutor-end=30}{n}\htmlData{tutor-start=30,tutor-end=31}{-}\htmlData{tutor-start=31,tutor-end=32}{1}}}{\htmlData{tutor-start=35,tutor-end=36}{a}_{\htmlData{tutor-start=38,tutor-end=39}{n}}}\htmlData{tutor-start=41,tutor-end=42}{]}\htmlData{tutor-start=42,tutor-end=43}{,} \htmlData{tutor-start=44,tutor-end=45}{n} \htmlData{tutor-start=46,tutor-end=47}{=} \htmlData{tutor-start=48,tutor-end=49}{2}\htmlData{tutor-start=49,tutor-end=50}{,} \htmlData{tutor-start=51,tutor-end=52}{3}\htmlData{tutor-start=52,tutor-end=53}{,} \cdots$ . 这里 $\htmlData{tutor-start=0,tutor-end=1}{[}\htmlData{tutor-start=1,tutor-end=2}{x}\htmlData{tutor-start=2,tutor-end=3}{]}$ 表示不小于实数 $\htmlData{tutor-start=0,tutor-end=1}{x}$ 的最小整数. 证明：对 $\htmlData{tutor-start=0,tutor-end=1}{n} \htmlData{tutor-start=2,tutor-end=3}{=} \htmlData{tutor-start=4,tutor-end=5}{3}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{4}\htmlData{tutor-start=8,tutor-end=9}{,} \cdots\htmlData{tutor-start=16,tutor-end=17}{,} \htmlData{tutor-start=18,tutor-end=19}{p}\htmlData{tutor-start=19,tutor-end=20}{-}\htmlData{tutor-start=20,tutor-end=21}{1}$ 均 $\htmlData{tutor-start=0,tutor-end=1}{n} \htmlData{tutor-start=2,tutor-end=7}{\mid }\htmlData{tutor-start=7,tutor-end=8}{p}\htmlData{tutor-start=8,tutor-end=9}{a}_{\htmlData{tutor-start=11,tutor-end=12}{n}\htmlData{tutor-start=12,tutor-end=13}{-}\htmlData{tutor-start=13,tutor-end=14}{1}} \htmlData{tutor-start=16,tutor-end=17}{+} \htmlData{tutor-start=18,tutor-end=19}{1}$ 有成立

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。