2021 年全国高中数学联赛 B1 卷（云南、内蒙古专用）（学生版）

books/national_high_school_league_students/papers/competition-national-league-2021-b1.pdf · HS-MATH-1024-v2.1-solution-aware

15 题0 个小问/题组

一试（B1 卷） · 一试试题（B1 卷） · 代数

已知 $\htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{x}\htmlData{tutor-start=3,tutor-end=4}{)}$ 是定义在 $\mathbb{\htmlData{tutor-start=8,tutor-end=9}{R}}$ 上的奇函数， $\htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{1}\htmlData{tutor-start=3,tutor-end=4}{)}\htmlData{tutor-start=4,tutor-end=5}{=}\htmlData{tutor-start=5,tutor-end=6}{2}$ ， $\htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{2}\htmlData{tutor-start=3,tutor-end=4}{)}\htmlData{tutor-start=4,tutor-end=5}{=}\htmlData{tutor-start=5,tutor-end=6}{3}$ ，则 $\htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{f}\htmlData{tutor-start=3,tutor-end=4}{(}\htmlData{tutor-start=4,tutor-end=5}{-}\htmlData{tutor-start=5,tutor-end=6}{1}\htmlData{tutor-start=6,tutor-end=7}{)}\htmlData{tutor-start=7,tutor-end=8}{)}\htmlData{tutor-start=8,tutor-end=9}{=}$ ________.

题解状态：标准答案与规范题解待补充

题目标签：FIRST-B1-PROBLEM-P1

解题过程

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

一试（B1 卷） · 一试试题（B1 卷） · 数学竞赛/待细分

已知复数 $\htmlData{tutor-start=0,tutor-end=1}{z}\htmlData{tutor-start=1,tutor-end=2}{=}\htmlData{tutor-start=2,tutor-end=3}{1}\htmlData{tutor-start=3,tutor-end=4}{+}\mathrm{\htmlData{tutor-start=12,tutor-end=13}{i}}\htmlData{tutor-start=14,tutor-end=15}{+}\frac{\htmlData{tutor-start=21,tutor-end=22}{m}}{\htmlData{tutor-start=24,tutor-end=25}{1}\htmlData{tutor-start=25,tutor-end=26}{+}\mathrm{\htmlData{tutor-start=34,tutor-end=35}{i}}}$ （ $\htmlData{tutor-start=0,tutor-end=1}{m}$ 为实数）的实部和虚部均大于 0，则 $\htmlData{tutor-start=0,tutor-end=1}{m}$ 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-B1-PROBLEM-P2

解题过程

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

一试（B1 卷） · 一试试题（B1 卷） · 组合数学

设集合 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{=}\htmlData{tutor-start=2,tutor-end=4}{\{}\htmlData{tutor-start=4,tutor-end=5}{1}\htmlData{tutor-start=5,tutor-end=6}{,}\htmlData{tutor-start=6,tutor-end=7}{2}\htmlData{tutor-start=7,tutor-end=8}{,}\htmlData{tutor-start=8,tutor-end=9}{3}\htmlData{tutor-start=9,tutor-end=11}{\}}$ ， $\htmlData{tutor-start=0,tutor-end=1}{X}\htmlData{tutor-start=1,tutor-end=2}{=}\htmlData{tutor-start=2,tutor-end=4}{\{}\htmlData{tutor-start=4,tutor-end=5}{4}\htmlData{tutor-start=5,tutor-end=6}{x}\htmlData{tutor-start=6,tutor-end=7}{+}\htmlData{tutor-start=7,tutor-end=8}{y} \htmlData{tutor-start=9,tutor-end=14}{\mid }\htmlData{tutor-start=14,tutor-end=15}{x}\htmlData{tutor-start=15,tutor-end=16}{,}\htmlData{tutor-start=16,tutor-end=17}{y} \htmlData{tutor-start=18,tutor-end=22}{\in }\htmlData{tutor-start=22,tutor-end=23}{A}\htmlData{tutor-start=23,tutor-end=25}{\}}$ ， $\htmlData{tutor-start=0,tutor-end=1}{Y}\htmlData{tutor-start=1,tutor-end=2}{=}\htmlData{tutor-start=2,tutor-end=4}{\{}\htmlData{tutor-start=4,tutor-end=5}{4}\htmlData{tutor-start=5,tutor-end=6}{x}\htmlData{tutor-start=6,tutor-end=7}{-}\htmlData{tutor-start=7,tutor-end=8}{y} \htmlData{tutor-start=9,tutor-end=14}{\mid }\htmlData{tutor-start=14,tutor-end=15}{x}\htmlData{tutor-start=15,tutor-end=16}{,}\htmlData{tutor-start=16,tutor-end=17}{y} \htmlData{tutor-start=18,tutor-end=22}{\in }\htmlData{tutor-start=22,tutor-end=23}{A}\htmlData{tutor-start=23,tutor-end=25}{\}}$ ，则 $\htmlData{tutor-start=0,tutor-end=1}{X} \htmlData{tutor-start=2,tutor-end=7}{\cap }\htmlData{tutor-start=7,tutor-end=8}{Y}$ 的元素之和为 ________.

题解状态：标准答案与规范题解待补充

题目标签：FIRST-B1-PROBLEM-P3

解题过程

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

一试（B1 卷） · 一试试题（B1 卷） · 平面几何

已知平面直角坐标系 $\htmlData{tutor-start=0,tutor-end=1}{x}\htmlData{tutor-start=1,tutor-end=2}{O}\htmlData{tutor-start=2,tutor-end=3}{y}$ 中，抛物线 $\htmlData{tutor-start=0,tutor-end=1}{y}^{\htmlData{tutor-start=3,tutor-end=4}{2}}\htmlData{tutor-start=5,tutor-end=6}{=}\htmlData{tutor-start=6,tutor-end=7}{2}\htmlData{tutor-start=7,tutor-end=8}{p}\htmlData{tutor-start=8,tutor-end=9}{x}$ 与一条倾斜角为 $\frac{\htmlData{tutor-start=6,tutor-end=9}{\pi}}{\htmlData{tutor-start=11,tutor-end=12}{4}}$ 的直线相交于 $\htmlData{tutor-start=0,tutor-end=1}{P}$ ， $\htmlData{tutor-start=0,tutor-end=1}{Q}$ 两点，且点 $\htmlData{tutor-start=0,tutor-end=1}{P}$ 的坐标为 $\htmlData{tutor-start=0,tutor-end=1}{(}\htmlData{tutor-start=1,tutor-end=2}{3}\htmlData{tutor-start=2,tutor-end=3}{,}\htmlData{tutor-start=3,tutor-end=4}{2}\htmlData{tutor-start=4,tutor-end=5}{)}$ ，则 $\htmlData{tutor-start=0,tutor-end=10}{\triangle }\htmlData{tutor-start=10,tutor-end=11}{O}\htmlData{tutor-start=11,tutor-end=12}{P}\htmlData{tutor-start=12,tutor-end=13}{Q}$ 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-B1-PROBLEM-P4

解题过程

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

一试（B1 卷） · 一试试题（B1 卷） · 代数

数列 $\htmlData{tutor-start=0,tutor-end=2}{\{}\htmlData{tutor-start=2,tutor-end=3}{a}_{\htmlData{tutor-start=5,tutor-end=6}{n}}\htmlData{tutor-start=7,tutor-end=9}{\}}$ 满足 $\htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{=}\htmlData{tutor-start=6,tutor-end=7}{2}$ ， $\htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{n}\htmlData{tutor-start=4,tutor-end=5}{+}\htmlData{tutor-start=5,tutor-end=6}{1}}\htmlData{tutor-start=7,tutor-end=8}{=}\htmlData{tutor-start=8,tutor-end=9}{(}\htmlData{tutor-start=9,tutor-end=10}{n}\htmlData{tutor-start=10,tutor-end=11}{+}\htmlData{tutor-start=11,tutor-end=12}{1}\htmlData{tutor-start=12,tutor-end=13}{)}\htmlData{tutor-start=13,tutor-end=14}{a}_{\htmlData{tutor-start=16,tutor-end=17}{n}}$ ( $\htmlData{tutor-start=0,tutor-end=1}{n} \htmlData{tutor-start=2,tutor-end=6}{\ge }\htmlData{tutor-start=6,tutor-end=7}{2}$ )，则 $\htmlData{tutor-start=0,tutor-end=2}{\{}\htmlData{tutor-start=2,tutor-end=3}{a}_{\htmlData{tutor-start=5,tutor-end=6}{n}}\htmlData{tutor-start=7,tutor-end=9}{\}}$ 的通项公式为 ________.

题解状态：标准答案与规范题解待补充

题目标签：FIRST-B1-PROBLEM-P5

解题过程

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

一试（B1 卷） · 一试试题（B1 卷） · 平面几何

正四棱锥 $\htmlData{tutor-start=0,tutor-end=1}{P}\htmlData{tutor-start=1,tutor-end=2}{-}\htmlData{tutor-start=2,tutor-end=3}{A}\htmlData{tutor-start=3,tutor-end=4}{B}\htmlData{tutor-start=4,tutor-end=5}{C}\htmlData{tutor-start=5,tutor-end=6}{D}$ 中， $\htmlData{tutor-start=0,tutor-end=1}{G}$ 是 $\htmlData{tutor-start=0,tutor-end=10}{\triangle }\htmlData{tutor-start=10,tutor-end=11}{P}\htmlData{tutor-start=11,tutor-end=12}{B}\htmlData{tutor-start=12,tutor-end=13}{C}$ 的重心，则 $\frac{\htmlData{tutor-start=6,tutor-end=7}{V}_{\htmlData{tutor-start=9,tutor-end=10}{G}\htmlData{tutor-start=10,tutor-end=11}{P}\htmlData{tutor-start=11,tutor-end=12}{B}\htmlData{tutor-start=12,tutor-end=13}{C}}}{\htmlData{tutor-start=16,tutor-end=17}{V}_{\htmlData{tutor-start=19,tutor-end=20}{G}\htmlData{tutor-start=20,tutor-end=21}{P}\htmlData{tutor-start=21,tutor-end=22}{A}\htmlData{tutor-start=22,tutor-end=23}{B}}}\htmlData{tutor-start=25,tutor-end=26}{=}$ ________.

题解状态：标准答案与规范题解待补充

题目标签：FIRST-B1-PROBLEM-P6

解题过程

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

一试（B1 卷） · 一试试题（B1 卷） · 数学竞赛/待细分

已知 $\htmlData{tutor-start=0,tutor-end=6}{\alpha}\htmlData{tutor-start=6,tutor-end=7}{,} \htmlData{tutor-start=8,tutor-end=14}{\beta }\htmlData{tutor-start=14,tutor-end=18}{\ge }\htmlData{tutor-start=18,tutor-end=19}{0}$ ， $\htmlData{tutor-start=0,tutor-end=6}{\alpha}\htmlData{tutor-start=6,tutor-end=7}{+}\htmlData{tutor-start=7,tutor-end=13}{\beta }\htmlData{tutor-start=13,tutor-end=17}{\le }\htmlData{tutor-start=17,tutor-end=18}{2}\htmlData{tutor-start=18,tutor-end=21}{\pi}$ ，则 $\sin\htmlData{tutor-start=4,tutor-end=10}{\alpha}\htmlData{tutor-start=10,tutor-end=11}{+}\htmlData{tutor-start=11,tutor-end=12}{2}\cos\htmlData{tutor-start=16,tutor-end=21}{\beta}$ 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-B1-PROBLEM-P7

解题过程

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

一试（B1 卷） · 一试试题（B1 卷） · 数学竞赛/待细分

设 $\htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{a}_{\htmlData{tutor-start=10,tutor-end=11}{2}}\htmlData{tutor-start=12,tutor-end=13}{,} \cdots\htmlData{tutor-start=20,tutor-end=21}{,} \htmlData{tutor-start=22,tutor-end=23}{a}_{\htmlData{tutor-start=25,tutor-end=26}{2}\htmlData{tutor-start=26,tutor-end=27}{1}}$ 为 $\htmlData{tutor-start=0,tutor-end=1}{1}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{2}\htmlData{tutor-start=4,tutor-end=5}{,} \cdots\htmlData{tutor-start=12,tutor-end=13}{,} \htmlData{tutor-start=14,tutor-end=15}{2}\htmlData{tutor-start=15,tutor-end=16}{1}$ 的排列，满足 $\htmlData{tutor-start=0,tutor-end=1}{|}\htmlData{tutor-start=1,tutor-end=2}{a}_{\htmlData{tutor-start=4,tutor-end=5}{2}\htmlData{tutor-start=5,tutor-end=6}{0}}\htmlData{tutor-start=7,tutor-end=8}{-}\htmlData{tutor-start=8,tutor-end=9}{a}_{\htmlData{tutor-start=11,tutor-end=12}{2}\htmlData{tutor-start=12,tutor-end=13}{1}}\htmlData{tutor-start=14,tutor-end=15}{|} \htmlData{tutor-start=16,tutor-end=20}{\ge }\htmlData{tutor-start=20,tutor-end=21}{|}\htmlData{tutor-start=21,tutor-end=22}{a}_{\htmlData{tutor-start=24,tutor-end=25}{1}\htmlData{tutor-start=25,tutor-end=26}{9}}\htmlData{tutor-start=27,tutor-end=28}{-}\htmlData{tutor-start=28,tutor-end=29}{a}_{\htmlData{tutor-start=31,tutor-end=32}{2}\htmlData{tutor-start=32,tutor-end=33}{1}}\htmlData{tutor-start=34,tutor-end=35}{|} \htmlData{tutor-start=36,tutor-end=40}{\ge }\htmlData{tutor-start=40,tutor-end=41}{|}\htmlData{tutor-start=41,tutor-end=42}{a}_{\htmlData{tutor-start=44,tutor-end=45}{1}\htmlData{tutor-start=45,tutor-end=46}{8}}\htmlData{tutor-start=47,tutor-end=48}{-}\htmlData{tutor-start=48,tutor-end=49}{a}_{\htmlData{tutor-start=51,tutor-end=52}{2}\htmlData{tutor-start=52,tutor-end=53}{1}}\htmlData{tutor-start=54,tutor-end=55}{|} \htmlData{tutor-start=56,tutor-end=60}{\ge }\cdots \htmlData{tutor-start=67,tutor-end=71}{\ge }\htmlData{tutor-start=71,tutor-end=72}{|}\htmlData{tutor-start=72,tutor-end=73}{a}_{\htmlData{tutor-start=75,tutor-end=76}{1}}\htmlData{tutor-start=77,tutor-end=78}{-}\htmlData{tutor-start=78,tutor-end=79}{a}_{\htmlData{tutor-start=81,tutor-end=82}{2}\htmlData{tutor-start=82,tutor-end=83}{1}}\htmlData{tutor-start=84,tutor-end=85}{|}\htmlData{tutor-start=85,tutor-end=86}{.}$ 这样的排列的个数为 ________.

题解状态：标准答案与规范题解待补充

题目标签：FIRST-B1-PROBLEM-P8

解题过程

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

一试（B1 卷） · 二、解答题：本大题共 3 小题，满分 56 分. 解答应写出文字说明、证明过程或演算步骤. · 代数

（本题满分 16 分）平面向量 $\vec{\htmlData{tutor-start=5,tutor-end=6}{a}}$ ， $\vec{\htmlData{tutor-start=5,tutor-end=6}{b}}$ 的夹角为 $\frac{\htmlData{tutor-start=6,tutor-end=9}{\pi}}{\htmlData{tutor-start=11,tutor-end=12}{3}}$ ，若 $\htmlData{tutor-start=0,tutor-end=1}{|}\vec{\htmlData{tutor-start=6,tutor-end=7}{a}}\htmlData{tutor-start=8,tutor-end=9}{|}$ ， $\htmlData{tutor-start=0,tutor-end=1}{|}\vec{\htmlData{tutor-start=6,tutor-end=7}{b}}\htmlData{tutor-start=8,tutor-end=9}{|}$ ， $\htmlData{tutor-start=0,tutor-end=1}{|}\vec{\htmlData{tutor-start=6,tutor-end=7}{a}}\htmlData{tutor-start=8,tutor-end=9}{+}\vec{\htmlData{tutor-start=14,tutor-end=15}{b}}\htmlData{tutor-start=16,tutor-end=17}{|}$ 依次成等差数列，求 $\htmlData{tutor-start=0,tutor-end=1}{|}\vec{\htmlData{tutor-start=6,tutor-end=7}{a}}\htmlData{tutor-start=8,tutor-end=9}{|}\htmlData{tutor-start=9,tutor-end=10}{:}\htmlData{tutor-start=10,tutor-end=11}{|}\vec{\htmlData{tutor-start=16,tutor-end=17}{b}}\htmlData{tutor-start=18,tutor-end=19}{|}\htmlData{tutor-start=19,tutor-end=20}{:}\htmlData{tutor-start=20,tutor-end=21}{|}\vec{\htmlData{tutor-start=26,tutor-end=27}{a}}\htmlData{tutor-start=28,tutor-end=29}{+}\vec{\htmlData{tutor-start=34,tutor-end=35}{b}}\htmlData{tutor-start=36,tutor-end=37}{|}$ .

题解状态：标准答案与规范题解待补充

题目标签：FIRST-B1-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（B1 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）设函数 $\htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{x}\htmlData{tutor-start=3,tutor-end=4}{)} \htmlData{tutor-start=5,tutor-end=6}{=} \htmlData{tutor-start=7,tutor-end=8}{|}\htmlData{tutor-start=8,tutor-end=9}{2} \htmlData{tutor-start=10,tutor-end=11}{-} \log_{\htmlData{tutor-start=18,tutor-end=19}{3}} \htmlData{tutor-start=21,tutor-end=22}{x}\htmlData{tutor-start=22,tutor-end=23}{|}$ ，正实数 $\htmlData{tutor-start=0,tutor-end=1}{a}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{b}\htmlData{tutor-start=4,tutor-end=5}{,} \htmlData{tutor-start=6,tutor-end=7}{c}$ 满足 $\htmlData{tutor-start=0,tutor-end=1}{a} \htmlData{tutor-start=2,tutor-end=3}{<} \htmlData{tutor-start=4,tutor-end=5}{b} \htmlData{tutor-start=6,tutor-end=7}{<} \htmlData{tutor-start=8,tutor-end=9}{c}$ ，且 $\htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{a}\htmlData{tutor-start=3,tutor-end=4}{)} \htmlData{tutor-start=5,tutor-end=6}{=} \htmlData{tutor-start=7,tutor-end=8}{2}\htmlData{tutor-start=8,tutor-end=9}{f}\htmlData{tutor-start=9,tutor-end=10}{(}\htmlData{tutor-start=10,tutor-end=11}{b}\htmlData{tutor-start=11,tutor-end=12}{)} \htmlData{tutor-start=13,tutor-end=14}{=} \htmlData{tutor-start=15,tutor-end=16}{2}\htmlData{tutor-start=16,tutor-end=17}{f}\htmlData{tutor-start=17,tutor-end=18}{(}\htmlData{tutor-start=18,tutor-end=19}{c}\htmlData{tutor-start=19,tutor-end=20}{)}$ . 求 $\frac{\htmlData{tutor-start=6,tutor-end=7}{a}\htmlData{tutor-start=7,tutor-end=8}{c}}{\htmlData{tutor-start=10,tutor-end=11}{b}}$ 的值.

题解状态：标准答案与规范题解待补充

题目标签：FIRST-B1-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（B1 卷） · 二、解答题：本大题共 3 小题，满分 56 分. 解答应写出文字说明、证明过程或演算步骤. · 平面几何

（本题满分 20 分）如图所示，在平面直角坐标系中，椭圆 $\htmlData{tutor-start=0,tutor-end=6}{\Gamma}\htmlData{tutor-start=6,tutor-end=7}{:} \frac{\htmlData{tutor-start=14,tutor-end=15}{x}^{\htmlData{tutor-start=17,tutor-end=18}{2}}}{\htmlData{tutor-start=21,tutor-end=22}{2}} \htmlData{tutor-start=24,tutor-end=25}{+} \htmlData{tutor-start=26,tutor-end=27}{y}^{\htmlData{tutor-start=29,tutor-end=30}{2}} \htmlData{tutor-start=32,tutor-end=33}{=} \htmlData{tutor-start=34,tutor-end=35}{1}$ 的左、右焦点分别为 $\htmlData{tutor-start=0,tutor-end=1}{F}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{F}_{\htmlData{tutor-start=10,tutor-end=11}{2}}$ ，设 $\htmlData{tutor-start=0,tutor-end=1}{P}$ 是第一象限内 $\htmlData{tutor-start=0,tutor-end=6}{\Gamma}$ 上一点， $\htmlData{tutor-start=0,tutor-end=1}{P}\htmlData{tutor-start=1,tutor-end=2}{F}_{\htmlData{tutor-start=4,tutor-end=5}{1}}\htmlData{tutor-start=6,tutor-end=7}{,} \htmlData{tutor-start=8,tutor-end=9}{P}\htmlData{tutor-start=9,tutor-end=10}{F}_{\htmlData{tutor-start=12,tutor-end=13}{2}}$ 的延长线分别交 $\htmlData{tutor-start=0,tutor-end=6}{\Gamma}$ 于点 $\htmlData{tutor-start=0,tutor-end=1}{Q}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{(}\htmlData{tutor-start=6,tutor-end=7}{x}_{\htmlData{tutor-start=9,tutor-end=10}{1}}\htmlData{tutor-start=11,tutor-end=12}{,} \htmlData{tutor-start=13,tutor-end=14}{y}_{\htmlData{tutor-start=16,tutor-end=17}{1}}\htmlData{tutor-start=18,tutor-end=19}{)}\htmlData{tutor-start=19,tutor-end=20}{,} \htmlData{tutor-start=21,tutor-end=22}{Q}_{\htmlData{tutor-start=24,tutor-end=25}{2}}\htmlData{tutor-start=26,tutor-end=27}{(}\htmlData{tutor-start=27,tutor-end=28}{x}_{\htmlData{tutor-start=30,tutor-end=31}{2}}\htmlData{tutor-start=32,tutor-end=33}{,} \htmlData{tutor-start=34,tutor-end=35}{y}_{\htmlData{tutor-start=37,tutor-end=38}{2}}\htmlData{tutor-start=39,tutor-end=40}{)}$ . 求 $\htmlData{tutor-start=0,tutor-end=1}{y}_{\htmlData{tutor-start=3,tutor-end=4}{1}} \htmlData{tutor-start=6,tutor-end=7}{-} \htmlData{tutor-start=8,tutor-end=9}{y}_{\htmlData{tutor-start=11,tutor-end=12}{2}}$ 的最大值.

**原卷图示 1**原卷第 2 页 · qwen3.7-plus_layout_detection · 需复核

题解状态：标准答案与规范题解待补充

题目标签：FIRST-B1-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（B1 卷） · 加试试题 · 数学竞赛/待细分

（本题满分 40 分）已知 $\htmlData{tutor-start=0,tutor-end=1}{a}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{b}\htmlData{tutor-start=4,tutor-end=5}{,} \htmlData{tutor-start=6,tutor-end=7}{c}$ 是非负实数，且 $\htmlData{tutor-start=0,tutor-end=1}{S} \htmlData{tutor-start=2,tutor-end=3}{=} \htmlData{tutor-start=4,tutor-end=5}{a} \htmlData{tutor-start=6,tutor-end=7}{+} \htmlData{tutor-start=8,tutor-end=9}{2}\htmlData{tutor-start=9,tutor-end=10}{b} \htmlData{tutor-start=11,tutor-end=12}{+} \htmlData{tutor-start=13,tutor-end=14}{3}\htmlData{tutor-start=14,tutor-end=15}{c}$ , $\htmlData{tutor-start=0,tutor-end=1}{T} \htmlData{tutor-start=2,tutor-end=3}{=} \htmlData{tutor-start=4,tutor-end=5}{a} \htmlData{tutor-start=6,tutor-end=7}{+} \htmlData{tutor-start=8,tutor-end=9}{b}^{\htmlData{tutor-start=11,tutor-end=12}{2}} \htmlData{tutor-start=14,tutor-end=15}{+} \htmlData{tutor-start=16,tutor-end=17}{c}^{\htmlData{tutor-start=19,tutor-end=20}{3}}$ . (1) 求 $\htmlData{tutor-start=0,tutor-end=1}{T} \htmlData{tutor-start=2,tutor-end=3}{-} \htmlData{tutor-start=4,tutor-end=5}{S}$ 的最小值； (2) 若 $\htmlData{tutor-start=0,tutor-end=1}{S} \htmlData{tutor-start=2,tutor-end=3}{=} \htmlData{tutor-start=4,tutor-end=5}{4}$ ，求 $\htmlData{tutor-start=0,tutor-end=1}{T}$ 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-B1-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（B1 卷） · 加试试题 · 平面几何

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

题解状态：标准答案与规范题解待补充

题目标签：SECOND-B1-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（B1 卷） · 加试试题 · 数论

（本题满分 50 分）已知 $\htmlData{tutor-start=0,tutor-end=1}{m}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{n}$ 是大于 1 的整数，且 $\htmlData{tutor-start=0,tutor-end=1}{n}$ 不是完全平方数，若 $\htmlData{tutor-start=0,tutor-end=1}{n}^{\htmlData{tutor-start=3,tutor-end=4}{2}} \htmlData{tutor-start=6,tutor-end=7}{+} \htmlData{tutor-start=8,tutor-end=9}{n} \htmlData{tutor-start=10,tutor-end=11}{+} \htmlData{tutor-start=12,tutor-end=13}{1}$ 被 $\htmlData{tutor-start=0,tutor-end=1}{m}$ 整除，求证： $\htmlData{tutor-start=0,tutor-end=1}{|}\htmlData{tutor-start=1,tutor-end=2}{m} \htmlData{tutor-start=3,tutor-end=4}{-} \htmlData{tutor-start=5,tutor-end=6}{n}\htmlData{tutor-start=6,tutor-end=7}{|} \htmlData{tutor-start=8,tutor-end=9}{>} \sqrt{\htmlData{tutor-start=16,tutor-end=17}{3}\htmlData{tutor-start=17,tutor-end=18}{n}} \htmlData{tutor-start=20,tutor-end=21}{-} \htmlData{tutor-start=22,tutor-end=23}{2}\htmlData{tutor-start=23,tutor-end=24}{.}$

题解状态：标准答案与规范题解待补充

题目标签：SECOND-B1-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（B1 卷） · 加试试题 · 数学竞赛/待细分

（本题满分 50 分）现有 $\htmlData{tutor-start=0,tutor-end=1}{T}_{\htmlData{tutor-start=3,tutor-end=4}{1}}$ 至 $\htmlData{tutor-start=0,tutor-end=1}{T}_{\htmlData{tutor-start=3,tutor-end=4}{9}}$ 九支球队进行单循环比赛（即每两队之间进行一场比赛），每场比赛仅有胜、平、负三种情况，且胜者得 3 分，打平则两队各得 1 分，负者得 0 分，已知这九支球队根据分数由多到少的顺序排列，且得分各不相同， $\htmlData{tutor-start=0,tutor-end=1}{T}_{\htmlData{tutor-start=3,tutor-end=4}{1}}$ 胜 3 场，平 4 场，负 1 场； $\htmlData{tutor-start=0,tutor-end=1}{T}_{\htmlData{tutor-start=3,tutor-end=4}{9}}$ 胜 0 场，平 5 场，负 3 场. 试问：(1) $\htmlData{tutor-start=0,tutor-end=1}{T}_{\htmlData{tutor-start=3,tutor-end=4}{3}}$ 是否可能胜 $\htmlData{tutor-start=0,tutor-end=1}{T}_{\htmlData{tutor-start=3,tutor-end=4}{4}}$ ？(2) $\htmlData{tutor-start=0,tutor-end=1}{T}_{\htmlData{tutor-start=3,tutor-end=4}{4}}$ 是否有可能胜 $\htmlData{tutor-start=0,tutor-end=1}{T}_{\htmlData{tutor-start=3,tutor-end=4}{3}}$ ？

题解状态：标准答案与规范题解待补充

题目标签：SECOND-B1-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。