2018 年全国高中数学联赛 B 卷（学生版）

books/national_high_school_league_students/papers/competition-national-league-2018-b.pdf · HS-MATH-1024-v2.1-solution-aware

15 题0 个小问/题组

一试（B 卷） · 一、填空题：本大题共 8 小题，每小题 8 分，满分 64 分。 · 组合数学

设集合 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{=}\htmlData{tutor-start=2,tutor-end=4}{\{}\htmlData{tutor-start=4,tutor-end=5}{2}\htmlData{tutor-start=5,tutor-end=6}{,}\htmlData{tutor-start=6,tutor-end=7}{0}\htmlData{tutor-start=7,tutor-end=8}{,}\htmlData{tutor-start=8,tutor-end=9}{1}\htmlData{tutor-start=9,tutor-end=10}{,}\htmlData{tutor-start=10,tutor-end=11}{8}\htmlData{tutor-start=11,tutor-end=13}{\}}$ ， $\htmlData{tutor-start=0,tutor-end=1}{B}\htmlData{tutor-start=1,tutor-end=2}{=}\htmlData{tutor-start=2,tutor-end=4}{\{}\htmlData{tutor-start=4,tutor-end=5}{2}\htmlData{tutor-start=5,tutor-end=6}{a} \htmlData{tutor-start=7,tutor-end=12}{\mid }\htmlData{tutor-start=12,tutor-end=13}{a} \htmlData{tutor-start=14,tutor-end=18}{\in }\htmlData{tutor-start=18,tutor-end=19}{A}\htmlData{tutor-start=19,tutor-end=21}{\}}$ ，则 $\htmlData{tutor-start=0,tutor-end=1}{A} \htmlData{tutor-start=2,tutor-end=7}{\cup }\htmlData{tutor-start=7,tutor-end=8}{B}$ 的所有元素之和是______。

题解状态：标准答案与规范题解待补充

题目标签：FIRST-B-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（B 卷） · 一、填空题：本大题共 8 小题，每小题 8 分，满分 64 分。 · 平面几何

已知圆锥的顶点为 $\htmlData{tutor-start=0,tutor-end=1}{P}$ ，底面半径长为 2，高为 $\htmlData{tutor-start=0,tutor-end=1}{l}$ ，在圆锥底面上取一点 $\htmlData{tutor-start=0,tutor-end=1}{Q}$ ，使得直线 $\htmlData{tutor-start=0,tutor-end=1}{P}\htmlData{tutor-start=1,tutor-end=2}{Q}$ 与底面所成角不大于 $45^\circ$ ，则满足条件的点 $\htmlData{tutor-start=0,tutor-end=1}{Q}$ 所构成的区域的面积为______。

题解状态：标准答案与规范题解待补充

题目标签：FIRST-B-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（B 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 1，2，3，4，5，6 随机排成一行，记为 $\htmlData{tutor-start=0,tutor-end=1}{a}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{b}\htmlData{tutor-start=4,tutor-end=5}{,} \htmlData{tutor-start=6,tutor-end=7}{c}\htmlData{tutor-start=7,tutor-end=8}{,} \htmlData{tutor-start=9,tutor-end=10}{d}\htmlData{tutor-start=10,tutor-end=11}{,} \htmlData{tutor-start=12,tutor-end=13}{e}\htmlData{tutor-start=13,tutor-end=14}{,} \htmlData{tutor-start=15,tutor-end=16}{f}$ ，则 $\htmlData{tutor-start=0,tutor-end=1}{a}\htmlData{tutor-start=1,tutor-end=2}{b}\htmlData{tutor-start=2,tutor-end=3}{c}\htmlData{tutor-start=3,tutor-end=4}{+}\htmlData{tutor-start=4,tutor-end=5}{d}\htmlData{tutor-start=5,tutor-end=6}{e}\htmlData{tutor-start=6,tutor-end=7}{f}$ 是奇数数的概率为______。

题解状态：标准答案与规范题解待补充

题目标签：FIRST-B-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（B 卷） · 一、填空题：本大题共 8 小题，每小题 8 分，满分 64 分。 · 平面几何

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，直线 $\htmlData{tutor-start=0,tutor-end=1}{l}$ 通过原点， $\vec{\htmlData{tutor-start=5,tutor-end=6}{n}}\htmlData{tutor-start=7,tutor-end=8}{=}\htmlData{tutor-start=8,tutor-end=9}{(}\htmlData{tutor-start=9,tutor-end=10}{3}\htmlData{tutor-start=10,tutor-end=11}{,}\htmlData{tutor-start=11,tutor-end=12}{1}\htmlData{tutor-start=12,tutor-end=13}{)}$ 是 $\htmlData{tutor-start=0,tutor-end=1}{l}$ 的一个法向量，已知数列 $\htmlData{tutor-start=0,tutor-end=2}{\{}\htmlData{tutor-start=2,tutor-end=3}{a}_{\htmlData{tutor-start=5,tutor-end=6}{n}}\htmlData{tutor-start=7,tutor-end=9}{\}}$ 满足：对任意正整数 $\htmlData{tutor-start=0,tutor-end=1}{n}$ ，点 $\htmlData{tutor-start=0,tutor-end=1}{(}\htmlData{tutor-start=1,tutor-end=2}{a}_{\htmlData{tutor-start=4,tutor-end=5}{n}\htmlData{tutor-start=5,tutor-end=6}{+}\htmlData{tutor-start=6,tutor-end=7}{1}}\htmlData{tutor-start=8,tutor-end=9}{,} \htmlData{tutor-start=10,tutor-end=11}{a}_{\htmlData{tutor-start=13,tutor-end=14}{n}}\htmlData{tutor-start=15,tutor-end=16}{)}$ 均在 $\htmlData{tutor-start=0,tutor-end=1}{l}$ 上。若 $\htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{2}}\htmlData{tutor-start=5,tutor-end=6}{=}\htmlData{tutor-start=6,tutor-end=7}{6}$ ，则 $\htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{1}} \htmlData{tutor-start=6,tutor-end=7}{a}_{\htmlData{tutor-start=9,tutor-end=10}{2}} \htmlData{tutor-start=12,tutor-end=13}{a}_{\htmlData{tutor-start=15,tutor-end=16}{3}} \htmlData{tutor-start=18,tutor-end=19}{a}_{\htmlData{tutor-start=21,tutor-end=22}{4}} \htmlData{tutor-start=24,tutor-end=25}{a}_{\htmlData{tutor-start=27,tutor-end=28}{5}}$ 的值为______。

题解状态：标准答案与规范题解待补充

题目标签：FIRST-B-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（B 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

设 $\htmlData{tutor-start=0,tutor-end=6}{\alpha}\htmlData{tutor-start=6,tutor-end=7}{,} \htmlData{tutor-start=8,tutor-end=13}{\beta}$ 满足 $\tan\left(\htmlData{tutor-start=10,tutor-end=16}{\alpha}\htmlData{tutor-start=16,tutor-end=17}{+}\frac{\htmlData{tutor-start=23,tutor-end=26}{\pi}}{\htmlData{tutor-start=28,tutor-end=29}{3}}\right)\htmlData{tutor-start=37,tutor-end=38}{=}\htmlData{tutor-start=38,tutor-end=39}{-}\htmlData{tutor-start=39,tutor-end=40}{3}$ ， $\tan\left(\htmlData{tutor-start=10,tutor-end=15}{\beta}\htmlData{tutor-start=15,tutor-end=16}{-}\frac{\htmlData{tutor-start=22,tutor-end=25}{\pi}}{\htmlData{tutor-start=27,tutor-end=28}{6}}\right)\htmlData{tutor-start=36,tutor-end=37}{=}\htmlData{tutor-start=37,tutor-end=38}{5}$ 则 $\tan\htmlData{tutor-start=4,tutor-end=5}{(}\htmlData{tutor-start=5,tutor-end=11}{\alpha}\htmlData{tutor-start=11,tutor-end=12}{-}\htmlData{tutor-start=12,tutor-end=17}{\beta}\htmlData{tutor-start=17,tutor-end=18}{)}$ 的值为______。

题解状态：标准答案与规范题解待补充

题目标签：FIRST-B-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（B 卷） · 一、填空题：本大题共 8 小题，每小题 8 分，满分 64 分。 · 平面几何

设抛物线 $\htmlData{tutor-start=0,tutor-end=1}{C}\htmlData{tutor-start=1,tutor-end=2}{:} \htmlData{tutor-start=3,tutor-end=4}{y}^{\htmlData{tutor-start=6,tutor-end=7}{2}}\htmlData{tutor-start=8,tutor-end=9}{=}\htmlData{tutor-start=9,tutor-end=10}{2}\htmlData{tutor-start=10,tutor-end=11}{x}$ 的准线与 $\htmlData{tutor-start=0,tutor-end=1}{x}$ 轴交于点 $\htmlData{tutor-start=0,tutor-end=1}{A}$ ，过点 $\htmlData{tutor-start=0,tutor-end=1}{B}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{-}\htmlData{tutor-start=3,tutor-end=4}{1}\htmlData{tutor-start=4,tutor-end=5}{,}\htmlData{tutor-start=5,tutor-end=6}{0}\htmlData{tutor-start=6,tutor-end=7}{)}$ 作一直线 $\htmlData{tutor-start=0,tutor-end=1}{l}$ 与抛物线 $\htmlData{tutor-start=0,tutor-end=1}{C}$ 相切于点 $\htmlData{tutor-start=0,tutor-end=1}{K}$ ，过点 $\htmlData{tutor-start=0,tutor-end=1}{A}$ 作 $\htmlData{tutor-start=0,tutor-end=1}{l}$ 的平行线，与抛物线 $\htmlData{tutor-start=0,tutor-end=1}{C}$ 交于点 $\htmlData{tutor-start=0,tutor-end=1}{M}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{N}$ ，则 $\htmlData{tutor-start=0,tutor-end=10}{\triangle }\htmlData{tutor-start=10,tutor-end=11}{K}\htmlData{tutor-start=11,tutor-end=12}{M}\htmlData{tutor-start=12,tutor-end=13}{N}$ 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-B-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（B 卷） · 一、填空题：本大题共 8 小题，每小题 8 分，满分 64 分。 · 代数

设 $\htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{x}\htmlData{tutor-start=3,tutor-end=4}{)}$ 是定义在 $\htmlData{tutor-start=0,tutor-end=1}{R}$ 上的以 2 为周期的偶函数，在区间 $\htmlData{tutor-start=0,tutor-end=1}{[}\htmlData{tutor-start=1,tutor-end=2}{0}\htmlData{tutor-start=2,tutor-end=3}{,}\htmlData{tutor-start=3,tutor-end=4}{1}\htmlData{tutor-start=4,tutor-end=5}{]}$ 上严格递减，且满足 $\htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=5}{\pi}\htmlData{tutor-start=5,tutor-end=6}{)}\htmlData{tutor-start=6,tutor-end=7}{=}\htmlData{tutor-start=7,tutor-end=8}{1}$ ， $\htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{2}\htmlData{tutor-start=3,tutor-end=6}{\pi}\htmlData{tutor-start=6,tutor-end=7}{)}\htmlData{tutor-start=7,tutor-end=8}{=}\htmlData{tutor-start=8,tutor-end=9}{1}$ ，则不等式组 $\begin{cases} \htmlData{tutor-start=14,tutor-end=15}{1} \htmlData{tutor-start=16,tutor-end=20}{\le }\htmlData{tutor-start=20,tutor-end=21}{x} \htmlData{tutor-start=22,tutor-end=26}{\le }\htmlData{tutor-start=26,tutor-end=27}{2} \\ \htmlData{tutor-start=31,tutor-end=32}{1} \htmlData{tutor-start=33,tutor-end=37}{\le }\htmlData{tutor-start=37,tutor-end=38}{f}\htmlData{tutor-start=38,tutor-end=39}{(}\htmlData{tutor-start=39,tutor-end=40}{x}\htmlData{tutor-start=40,tutor-end=41}{)} \htmlData{tutor-start=42,tutor-end=46}{\le }\htmlData{tutor-start=46,tutor-end=47}{2} \end{cases}$ 的解集为______。

题解状态：标准答案与规范题解待补充

题目标签：FIRST-B-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（B 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

已知复数 $\htmlData{tutor-start=0,tutor-end=1}{z}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{z}_{\htmlData{tutor-start=10,tutor-end=11}{2}}\htmlData{tutor-start=12,tutor-end=13}{,} \htmlData{tutor-start=14,tutor-end=15}{z}_{\htmlData{tutor-start=17,tutor-end=18}{3}}$ 满足 $\htmlData{tutor-start=0,tutor-end=1}{|}\htmlData{tutor-start=1,tutor-end=2}{z}_{\htmlData{tutor-start=4,tutor-end=5}{1}}\htmlData{tutor-start=6,tutor-end=7}{|}\htmlData{tutor-start=7,tutor-end=8}{=}\htmlData{tutor-start=8,tutor-end=9}{|}\htmlData{tutor-start=9,tutor-end=10}{z}_{\htmlData{tutor-start=12,tutor-end=13}{2}}\htmlData{tutor-start=14,tutor-end=15}{|}\htmlData{tutor-start=15,tutor-end=16}{=}\htmlData{tutor-start=16,tutor-end=17}{|}\htmlData{tutor-start=17,tutor-end=18}{z}_{\htmlData{tutor-start=20,tutor-end=21}{3}}\htmlData{tutor-start=22,tutor-end=23}{|}\htmlData{tutor-start=23,tutor-end=24}{=}\htmlData{tutor-start=24,tutor-end=25}{1}$ ， $\htmlData{tutor-start=0,tutor-end=1}{|}\htmlData{tutor-start=1,tutor-end=2}{z}_{\htmlData{tutor-start=4,tutor-end=5}{1}}\htmlData{tutor-start=6,tutor-end=7}{+}\htmlData{tutor-start=7,tutor-end=8}{z}_{\htmlData{tutor-start=10,tutor-end=11}{2}}\htmlData{tutor-start=12,tutor-end=13}{+}\htmlData{tutor-start=13,tutor-end=14}{z}_{\htmlData{tutor-start=16,tutor-end=17}{3}}\htmlData{tutor-start=18,tutor-end=19}{|}\htmlData{tutor-start=19,tutor-end=20}{=}\htmlData{tutor-start=20,tutor-end=21}{r}$ 。其中 $\htmlData{tutor-start=0,tutor-end=1}{r}$ 是给定实数，则 $\frac{\htmlData{tutor-start=6,tutor-end=7}{z}_{\htmlData{tutor-start=9,tutor-end=10}{1}}}{\htmlData{tutor-start=13,tutor-end=14}{z}_{\htmlData{tutor-start=16,tutor-end=17}{2}}} \htmlData{tutor-start=20,tutor-end=21}{+} \frac{\htmlData{tutor-start=28,tutor-end=29}{z}_{\htmlData{tutor-start=31,tutor-end=32}{2}}}{\htmlData{tutor-start=35,tutor-end=36}{z}_{\htmlData{tutor-start=38,tutor-end=39}{3}}} \htmlData{tutor-start=42,tutor-end=43}{+} \frac{\htmlData{tutor-start=50,tutor-end=51}{z}_{\htmlData{tutor-start=53,tutor-end=54}{3}}}{\htmlData{tutor-start=57,tutor-end=58}{z}_{\htmlData{tutor-start=60,tutor-end=61}{1}}}$ 的实部是______。（用含有 $\htmlData{tutor-start=0,tutor-end=1}{r}$ 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：FIRST-B-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（B 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。 · 数论

（本题满分 16 分）已知数列 $\htmlData{tutor-start=0,tutor-end=2}{\{}\htmlData{tutor-start=2,tutor-end=3}{a}_{\htmlData{tutor-start=5,tutor-end=6}{n}}\htmlData{tutor-start=7,tutor-end=9}{\}}$ ： $\htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{=}\htmlData{tutor-start=6,tutor-end=7}{7}$ ， $\frac{a_{n+1}}{a_{n}} = a_{n} + 2, n=1,2,3,\cdots$ ，求满足 $\htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{n}} \htmlData{tutor-start=6,tutor-end=7}{>} \htmlData{tutor-start=8,tutor-end=9}{4}^{\htmlData{tutor-start=11,tutor-end=12}{2}\htmlData{tutor-start=12,tutor-end=13}{0}\htmlData{tutor-start=13,tutor-end=14}{1}\htmlData{tutor-start=14,tutor-end=15}{8}}$ 的最小正整数 $\htmlData{tutor-start=0,tutor-end=1}{n}$ 。

题解状态：标准答案与规范题解待补充

题目标签：FIRST-B-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（B 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 20 分）已知定义在 $R^+$ 上的函数 $\htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{x}\htmlData{tutor-start=3,tutor-end=4}{)}$ 为 $\htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{x}\htmlData{tutor-start=3,tutor-end=4}{)}\htmlData{tutor-start=4,tutor-end=5}{=}\begin{cases} \htmlData{tutor-start=19,tutor-end=20}{|}\log_{\htmlData{tutor-start=26,tutor-end=27}{3}} \htmlData{tutor-start=29,tutor-end=30}{x}\htmlData{tutor-start=30,tutor-end=31}{-}\htmlData{tutor-start=31,tutor-end=32}{1}\htmlData{tutor-start=32,tutor-end=33}{|}\htmlData{tutor-start=33,tutor-end=34}{,} \htmlData{tutor-start=35,tutor-end=36}{0} \htmlData{tutor-start=37,tutor-end=38}{<} \htmlData{tutor-start=39,tutor-end=40}{x} \htmlData{tutor-start=41,tutor-end=42}{<} \htmlData{tutor-start=43,tutor-end=44}{9} \\ \htmlData{tutor-start=48,tutor-end=49}{4}\htmlData{tutor-start=49,tutor-end=50}{-}\sqrt{\htmlData{tutor-start=56,tutor-end=57}{x}}\htmlData{tutor-start=58,tutor-end=59}{,} \htmlData{tutor-start=60,tutor-end=61}{x} \htmlData{tutor-start=62,tutor-end=63}{>} \htmlData{tutor-start=64,tutor-end=65}{9} \end{cases}$ 设 $\htmlData{tutor-start=0,tutor-end=1}{a}\htmlData{tutor-start=1,tutor-end=2}{,}\htmlData{tutor-start=2,tutor-end=3}{b}\htmlData{tutor-start=3,tutor-end=4}{,}\htmlData{tutor-start=4,tutor-end=5}{c}$ 是三个互不相同的实数，满足 $\htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{a}\htmlData{tutor-start=3,tutor-end=4}{)}\htmlData{tutor-start=4,tutor-end=5}{=}\htmlData{tutor-start=5,tutor-end=6}{f}\htmlData{tutor-start=6,tutor-end=7}{(}\htmlData{tutor-start=7,tutor-end=8}{b}\htmlData{tutor-start=8,tutor-end=9}{)}\htmlData{tutor-start=9,tutor-end=10}{=}\htmlData{tutor-start=10,tutor-end=11}{f}\htmlData{tutor-start=11,tutor-end=12}{(}\htmlData{tutor-start=12,tutor-end=13}{c}\htmlData{tutor-start=13,tutor-end=14}{)}$ ，求 $\htmlData{tutor-start=0,tutor-end=1}{a}\htmlData{tutor-start=1,tutor-end=2}{b}\htmlData{tutor-start=2,tutor-end=3}{c}$ 的取值范围。

题解状态：标准答案与规范题解待补充

题目标签：FIRST-B-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（B 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 $\htmlData{tutor-start=0,tutor-end=1}{x}\htmlData{tutor-start=1,tutor-end=2}{O}\htmlData{tutor-start=2,tutor-end=3}{y}$ 中， $\htmlData{tutor-start=0,tutor-end=1}{A}$ ， $\htmlData{tutor-start=0,tutor-end=1}{B}$ 与 $\htmlData{tutor-start=0,tutor-end=1}{C}$ 、 $\htmlData{tutor-start=0,tutor-end=1}{D}$ 分别是椭圆 $\htmlData{tutor-start=0,tutor-end=6}{\Gamma}\htmlData{tutor-start=6,tutor-end=7}{:} \frac{\htmlData{tutor-start=14,tutor-end=15}{x}^{\htmlData{tutor-start=17,tutor-end=18}{2}}}{\htmlData{tutor-start=21,tutor-end=22}{a}^{\htmlData{tutor-start=24,tutor-end=25}{2}}} \htmlData{tutor-start=28,tutor-end=29}{+} \frac{\htmlData{tutor-start=36,tutor-end=37}{y}^{\htmlData{tutor-start=39,tutor-end=40}{2}}}{\htmlData{tutor-start=43,tutor-end=44}{b}^{\htmlData{tutor-start=46,tutor-end=47}{2}}} \htmlData{tutor-start=50,tutor-end=51}{=} \htmlData{tutor-start=52,tutor-end=53}{1} \htmlData{tutor-start=54,tutor-end=55}{(}\htmlData{tutor-start=55,tutor-end=56}{a} \htmlData{tutor-start=57,tutor-end=58}{>} \htmlData{tutor-start=59,tutor-end=60}{b} \htmlData{tutor-start=61,tutor-end=62}{>} \htmlData{tutor-start=63,tutor-end=64}{0}\htmlData{tutor-start=64,tutor-end=65}{)}$ 的左、右顶点与上、下顶点，设 $\htmlData{tutor-start=0,tutor-end=1}{P}$ ， $\htmlData{tutor-start=0,tutor-end=1}{Q}$ 是 $\htmlData{tutor-start=0,tutor-end=6}{\Gamma}$ 上且位于第一象限的两点，满足 $\htmlData{tutor-start=0,tutor-end=1}{O}\htmlData{tutor-start=1,tutor-end=2}{Q} \htmlData{tutor-start=3,tutor-end=4}{/}\htmlData{tutor-start=4,tutor-end=5}{/} \htmlData{tutor-start=6,tutor-end=7}{A}\htmlData{tutor-start=7,tutor-end=8}{P}$ ， $\htmlData{tutor-start=0,tutor-end=1}{M}$ 是线段 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{P}$ 的中点，射线 $\htmlData{tutor-start=0,tutor-end=1}{O}\htmlData{tutor-start=1,tutor-end=2}{M}$ 与椭圆交于点 $\htmlData{tutor-start=0,tutor-end=1}{R}$ 。证明：线段 $\htmlData{tutor-start=0,tutor-end=1}{O}\htmlData{tutor-start=1,tutor-end=2}{Q}$ ， $\htmlData{tutor-start=0,tutor-end=1}{O}\htmlData{tutor-start=1,tutor-end=2}{R}$ ， $\htmlData{tutor-start=0,tutor-end=1}{B}\htmlData{tutor-start=1,tutor-end=2}{C}$ 能构成一个直角三角形。

**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-B-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（B 卷） · 一、（本题满分 40 分） · 代数

设 $\htmlData{tutor-start=0,tutor-end=1}{a}\htmlData{tutor-start=1,tutor-end=2}{,}\htmlData{tutor-start=2,tutor-end=3}{b}$ 是实数，函数 $\htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{x}\htmlData{tutor-start=3,tutor-end=4}{)} \htmlData{tutor-start=5,tutor-end=6}{=} \htmlData{tutor-start=7,tutor-end=8}{a}\htmlData{tutor-start=8,tutor-end=9}{x} \htmlData{tutor-start=10,tutor-end=11}{+} \htmlData{tutor-start=12,tutor-end=13}{b} \htmlData{tutor-start=14,tutor-end=15}{+} \frac{\htmlData{tutor-start=22,tutor-end=23}{9}}{\htmlData{tutor-start=25,tutor-end=26}{x}}$ 。证明：存在 $\htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{0}} \htmlData{tutor-start=6,tutor-end=10}{\in }\htmlData{tutor-start=10,tutor-end=11}{[}\htmlData{tutor-start=11,tutor-end=12}{1}\htmlData{tutor-start=12,tutor-end=13}{,}\htmlData{tutor-start=13,tutor-end=14}{9}\htmlData{tutor-start=14,tutor-end=15}{]}$ ，使得 $\htmlData{tutor-start=0,tutor-end=1}{|}\htmlData{tutor-start=1,tutor-end=2}{f}\htmlData{tutor-start=2,tutor-end=3}{(}\htmlData{tutor-start=3,tutor-end=4}{x}_{\htmlData{tutor-start=6,tutor-end=7}{0}}\htmlData{tutor-start=8,tutor-end=9}{)}\htmlData{tutor-start=9,tutor-end=10}{|} \htmlData{tutor-start=11,tutor-end=15}{\ge }\htmlData{tutor-start=15,tutor-end=16}{2}$

题解状态：标准答案与规范题解待补充

题目标签：SECOND-B-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（B 卷） · 二、（本题满分 40 分） · 平面几何

如图所示，在等腰 $\htmlData{tutor-start=0,tutor-end=10}{\triangle }\htmlData{tutor-start=10,tutor-end=11}{A}\htmlData{tutor-start=11,tutor-end=12}{B}\htmlData{tutor-start=12,tutor-end=13}{C}$ 中， $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{B}\htmlData{tutor-start=2,tutor-end=3}{=}\htmlData{tutor-start=3,tutor-end=4}{A}\htmlData{tutor-start=4,tutor-end=5}{C}$ ，边 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{C}$ 的上一点 $\htmlData{tutor-start=0,tutor-end=1}{D}$ 及 $\htmlData{tutor-start=0,tutor-end=1}{B}\htmlData{tutor-start=1,tutor-end=2}{C}$ 延长线上一点 $\htmlData{tutor-start=0,tutor-end=1}{E}$ 分满足 $\frac{\htmlData{tutor-start=6,tutor-end=7}{A}\htmlData{tutor-start=7,tutor-end=8}{D}}{\htmlData{tutor-start=10,tutor-end=11}{D}\htmlData{tutor-start=11,tutor-end=12}{C}} \htmlData{tutor-start=14,tutor-end=15}{=} \frac{\htmlData{tutor-start=22,tutor-end=23}{B}\htmlData{tutor-start=23,tutor-end=24}{C}}{\htmlData{tutor-start=26,tutor-end=27}{2}\htmlData{tutor-start=27,tutor-end=28}{C}\htmlData{tutor-start=28,tutor-end=29}{E}}$ ，以 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{B}$ 为直径的圆 $\htmlData{tutor-start=0,tutor-end=6}{\omega}$ 与线段 $\htmlData{tutor-start=0,tutor-end=1}{D}\htmlData{tutor-start=1,tutor-end=2}{E}$ 交于一点 $\htmlData{tutor-start=0,tutor-end=1}{F}$ 。证明： $\htmlData{tutor-start=0,tutor-end=1}{B}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{C}\htmlData{tutor-start=4,tutor-end=5}{,} \htmlData{tutor-start=6,tutor-end=7}{F}\htmlData{tutor-start=7,tutor-end=8}{,} \htmlData{tutor-start=9,tutor-end=10}{D}$ 四点共圆（答题时请将图画在答卷纸上）

题解状态：标准答案与规范题解待补充

题目标签：SECOND-B-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（B 卷） · 三、（本题满分 50 分） · 组合数学

设集合 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{=}\htmlData{tutor-start=2,tutor-end=4}{\{}\htmlData{tutor-start=4,tutor-end=5}{1}\htmlData{tutor-start=5,tutor-end=6}{,}\htmlData{tutor-start=6,tutor-end=7}{2}\htmlData{tutor-start=7,tutor-end=8}{,}\cdots\htmlData{tutor-start=14,tutor-end=15}{,}\htmlData{tutor-start=15,tutor-end=16}{n}\htmlData{tutor-start=16,tutor-end=18}{\}}$ ， $\htmlData{tutor-start=0,tutor-end=1}{X}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{Y}$ 均为 $\htmlData{tutor-start=0,tutor-end=1}{A}$ 的非空子集（允许 $\htmlData{tutor-start=0,tutor-end=1}{X}\htmlData{tutor-start=1,tutor-end=2}{=}\htmlData{tutor-start=2,tutor-end=3}{Y}$ ）。 $\htmlData{tutor-start=0,tutor-end=1}{X}$ 中的最大元与 $\htmlData{tutor-start=0,tutor-end=1}{Y}$ 中的最小元分别记为 $\max \htmlData{tutor-start=5,tutor-end=6}{X}\htmlData{tutor-start=6,tutor-end=7}{,} \min \htmlData{tutor-start=13,tutor-end=14}{Y}$ 求满足 $\max \htmlData{tutor-start=5,tutor-end=6}{X} \htmlData{tutor-start=7,tutor-end=8}{>} \min \htmlData{tutor-start=14,tutor-end=15}{Y}$ 的有序集合对 $\htmlData{tutor-start=0,tutor-end=1}{(}\htmlData{tutor-start=1,tutor-end=2}{X}\htmlData{tutor-start=2,tutor-end=3}{,}\htmlData{tutor-start=3,tutor-end=4}{Y}\htmlData{tutor-start=4,tutor-end=5}{)}$ 的数目。

题解状态：标准答案与规范题解待补充

题目标签：SECOND-B-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（B 卷） · 四、（本题满分 50 分） · 数论

给定整数 $\htmlData{tutor-start=0,tutor-end=1}{a} \htmlData{tutor-start=2,tutor-end=6}{\ge }\htmlData{tutor-start=6,tutor-end=7}{2}$ 。证明：对任意正整数 $\htmlData{tutor-start=0,tutor-end=1}{n}$ ，存在正整数 $\htmlData{tutor-start=0,tutor-end=1}{k}$ ，使得连续 $\htmlData{tutor-start=0,tutor-end=1}{n}$ 个数 $\htmlData{tutor-start=0,tutor-end=1}{a}^{\htmlData{tutor-start=3,tutor-end=4}{k}}\htmlData{tutor-start=5,tutor-end=6}{+}\htmlData{tutor-start=6,tutor-end=7}{1}\htmlData{tutor-start=7,tutor-end=8}{,} \htmlData{tutor-start=9,tutor-end=10}{a}^{\htmlData{tutor-start=12,tutor-end=13}{k}}\htmlData{tutor-start=14,tutor-end=15}{+}\htmlData{tutor-start=15,tutor-end=16}{2}\htmlData{tutor-start=16,tutor-end=17}{,} \cdots\htmlData{tutor-start=24,tutor-end=25}{,} \htmlData{tutor-start=26,tutor-end=27}{a}^{\htmlData{tutor-start=29,tutor-end=30}{k}}\htmlData{tutor-start=31,tutor-end=32}{+}\htmlData{tutor-start=32,tutor-end=33}{n}$ 均是合数。

题解状态：标准答案与规范题解待补充

题目标签：SECOND-B-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。