2019 年全国高中数学联赛 B 卷（学生版）

books/national_high_school_league_students/papers/competition-national-league-2019-b.pdf · HS-MATH-1024-v2.1-solution-aware

15 题0 个小问/题组

一试（B 卷） · 一试 · 组合数学

已知实数集合 $\htmlData{tutor-start=0,tutor-end=2}{\{}\htmlData{tutor-start=2,tutor-end=3}{1}\htmlData{tutor-start=3,tutor-end=4}{,} \htmlData{tutor-start=5,tutor-end=6}{2}\htmlData{tutor-start=6,tutor-end=7}{,} \htmlData{tutor-start=8,tutor-end=9}{3}\htmlData{tutor-start=9,tutor-end=10}{,} \htmlData{tutor-start=11,tutor-end=12}{x}\htmlData{tutor-start=12,tutor-end=14}{\}}$ 的最大元素等于该集合的所有元素之和，则 $\htmlData{tutor-start=0,tutor-end=1}{x}$ 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-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（B 卷） · 一试 · 数学竞赛/待细分

若平面向量 $\vec{\htmlData{tutor-start=5,tutor-end=6}{a}}\htmlData{tutor-start=7,tutor-end=8}{=}\htmlData{tutor-start=8,tutor-end=9}{(}\htmlData{tutor-start=9,tutor-end=10}{2}^{\htmlData{tutor-start=12,tutor-end=13}{m}}\htmlData{tutor-start=14,tutor-end=15}{,} \htmlData{tutor-start=16,tutor-end=17}{-}\htmlData{tutor-start=17,tutor-end=18}{1}\htmlData{tutor-start=18,tutor-end=19}{)}$ 与 $\vec{\htmlData{tutor-start=5,tutor-end=6}{b}}\htmlData{tutor-start=7,tutor-end=8}{=}\htmlData{tutor-start=8,tutor-end=9}{(}\htmlData{tutor-start=9,tutor-end=10}{2}^{\htmlData{tutor-start=12,tutor-end=13}{m}}\htmlData{tutor-start=14,tutor-end=15}{-}\htmlData{tutor-start=15,tutor-end=16}{1}\htmlData{tutor-start=16,tutor-end=17}{,} \htmlData{tutor-start=18,tutor-end=19}{2}^{\htmlData{tutor-start=21,tutor-end=22}{m}\htmlData{tutor-start=22,tutor-end=23}{+}\htmlData{tutor-start=23,tutor-end=24}{1}}\htmlData{tutor-start=25,tutor-end=26}{)}$ 垂直，其中 $\htmlData{tutor-start=0,tutor-end=1}{m}$ 为实数，则 $\vec{\htmlData{tutor-start=5,tutor-end=6}{a}}$ 模为________

题解状态：标准答案与规范题解待补充

题目标签：FIRST-B-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（B 卷） · 一试 · 数学竞赛/待细分

设 $\htmlData{tutor-start=0,tutor-end=6}{\alpha}\htmlData{tutor-start=6,tutor-end=7}{,} \htmlData{tutor-start=8,tutor-end=14}{\beta }\htmlData{tutor-start=14,tutor-end=18}{\in }\htmlData{tutor-start=18,tutor-end=19}{(}\htmlData{tutor-start=19,tutor-end=20}{0}\htmlData{tutor-start=20,tutor-end=21}{,} \htmlData{tutor-start=22,tutor-end=25}{\pi}\htmlData{tutor-start=25,tutor-end=26}{)}$ ， $\cos\htmlData{tutor-start=4,tutor-end=10}{\alpha}\htmlData{tutor-start=10,tutor-end=11}{,} \cos\htmlData{tutor-start=16,tutor-end=21}{\beta}$ 是方程 $\htmlData{tutor-start=0,tutor-end=1}{5}\htmlData{tutor-start=1,tutor-end=2}{x}^{\htmlData{tutor-start=4,tutor-end=5}{2}}\htmlData{tutor-start=6,tutor-end=7}{-}\htmlData{tutor-start=7,tutor-end=8}{3}\htmlData{tutor-start=8,tutor-end=9}{x}\htmlData{tutor-start=9,tutor-end=10}{-}\htmlData{tutor-start=10,tutor-end=11}{1}\htmlData{tutor-start=11,tutor-end=12}{=}\htmlData{tutor-start=12,tutor-end=13}{0}$ 的两根，则 $\sin\htmlData{tutor-start=4,tutor-end=10}{\alpha}\sin\htmlData{tutor-start=14,tutor-end=19}{\beta}$ 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-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（B 卷） · 一试 · 数学竞赛/待细分

设三棱锥 P-ABC 满足 PA=PB=3, AB=BC=CA=2，则该三棱锥的体积的最大值为________

题解状态：标准答案与规范题解待补充

题目标签：FIRST-B-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（B 卷） · 一试 · 数学竞赛/待细分

将 5 个数 2, 0, 1, 9, 2019 按任意次序排成一行，拼成一个 8 位数（首位不为 0），则产生的不同的 8 位数的个数为________

题解状态：标准答案与规范题解待补充

题目标签：FIRST-B-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（B 卷） · 一试 · 数论

设整数 $\htmlData{tutor-start=0,tutor-end=1}{n}\htmlData{tutor-start=1,tutor-end=2}{>}\htmlData{tutor-start=2,tutor-end=3}{4}$ ， $\htmlData{tutor-start=0,tutor-end=1}{(}\htmlData{tutor-start=1,tutor-end=2}{x}\htmlData{tutor-start=2,tutor-end=3}{+}\htmlData{tutor-start=3,tutor-end=4}{2}\sqrt{\htmlData{tutor-start=10,tutor-end=11}{y}}\htmlData{tutor-start=12,tutor-end=13}{-}\htmlData{tutor-start=13,tutor-end=14}{1}\htmlData{tutor-start=14,tutor-end=15}{)}^{\htmlData{tutor-start=17,tutor-end=18}{n}}$ 的展开式中 $\htmlData{tutor-start=0,tutor-end=1}{x}^{\htmlData{tutor-start=3,tutor-end=4}{n}\htmlData{tutor-start=4,tutor-end=5}{-}\htmlData{tutor-start=5,tutor-end=6}{4}}$ 和 $\htmlData{tutor-start=0,tutor-end=1}{x}\htmlData{tutor-start=1,tutor-end=2}{y}$ 两项的系数相等，则 $\htmlData{tutor-start=0,tutor-end=1}{n}$ 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-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（B 卷） · 一试 · 平面几何

在平面直角坐标系中，若以 $\htmlData{tutor-start=0,tutor-end=1}{(}\htmlData{tutor-start=1,tutor-end=2}{r}\htmlData{tutor-start=2,tutor-end=3}{+}\htmlData{tutor-start=3,tutor-end=4}{1}\htmlData{tutor-start=4,tutor-end=5}{,} \htmlData{tutor-start=6,tutor-end=7}{0}\htmlData{tutor-start=7,tutor-end=8}{)}$ 为圆心、 $\htmlData{tutor-start=0,tutor-end=1}{r}$ 为半径的圆上存在一点 $\htmlData{tutor-start=0,tutor-end=1}{(}\htmlData{tutor-start=1,tutor-end=2}{a}\htmlData{tutor-start=2,tutor-end=3}{,} \htmlData{tutor-start=4,tutor-end=5}{b}\htmlData{tutor-start=5,tutor-end=6}{)}$ 满足 $\htmlData{tutor-start=0,tutor-end=1}{b}^{\htmlData{tutor-start=3,tutor-end=4}{2}} \htmlData{tutor-start=6,tutor-end=10}{\ge }\htmlData{tutor-start=10,tutor-end=11}{4}\htmlData{tutor-start=11,tutor-end=12}{a}$ ，则 $\htmlData{tutor-start=0,tutor-end=1}{r}$ 的最小值是________

题解状态：标准答案与规范题解待补充

题目标签：FIRST-B-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（B 卷） · 一试 · 数论

设等差数列 $\htmlData{tutor-start=0,tutor-end=2}{\{}\htmlData{tutor-start=2,tutor-end=3}{a}_{\htmlData{tutor-start=5,tutor-end=6}{n}}\htmlData{tutor-start=7,tutor-end=9}{\}}$ 的各项均为整数，首项 $\htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{=}\htmlData{tutor-start=6,tutor-end=7}{2}\htmlData{tutor-start=7,tutor-end=8}{0}\htmlData{tutor-start=8,tutor-end=9}{1}\htmlData{tutor-start=9,tutor-end=10}{9}$ ，且对任意正整数 $\htmlData{tutor-start=0,tutor-end=1}{n}$ ，总存在正整数 $\htmlData{tutor-start=0,tutor-end=1}{m}$ ，使 $\htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{+}\htmlData{tutor-start=6,tutor-end=7}{a}_{\htmlData{tutor-start=9,tutor-end=10}{2}}\htmlData{tutor-start=11,tutor-end=12}{+}\dots\htmlData{tutor-start=17,tutor-end=18}{+}\htmlData{tutor-start=18,tutor-end=19}{a}_{\htmlData{tutor-start=21,tutor-end=22}{n}}\htmlData{tutor-start=23,tutor-end=24}{=}\htmlData{tutor-start=24,tutor-end=25}{a}_{\htmlData{tutor-start=27,tutor-end=28}{m}}$ ，这样的数列 $\htmlData{tutor-start=0,tutor-end=2}{\{}\htmlData{tutor-start=2,tutor-end=3}{a}_{\htmlData{tutor-start=5,tutor-end=6}{n}}\htmlData{tutor-start=7,tutor-end=9}{\}}$ 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-B-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（B 卷） · 一试 · 平面几何

(本题满分 16 分) 在椭圆 $\htmlData{tutor-start=0,tutor-end=6}{\Gamma}$ 中，F 为一个焦点，A、B 为两个顶点，若 $\htmlData{tutor-start=0,tutor-end=1}{|}\htmlData{tutor-start=1,tutor-end=2}{F}\htmlData{tutor-start=2,tutor-end=3}{A}\htmlData{tutor-start=3,tutor-end=4}{|}\htmlData{tutor-start=4,tutor-end=5}{=}\htmlData{tutor-start=5,tutor-end=6}{3}\htmlData{tutor-start=6,tutor-end=7}{,} \htmlData{tutor-start=8,tutor-end=9}{|}\htmlData{tutor-start=9,tutor-end=10}{F}\htmlData{tutor-start=10,tutor-end=11}{B}\htmlData{tutor-start=11,tutor-end=12}{|}\htmlData{tutor-start=12,tutor-end=13}{=}\htmlData{tutor-start=13,tutor-end=14}{2}$ ，求 $\htmlData{tutor-start=0,tutor-end=1}{|}\htmlData{tutor-start=1,tutor-end=2}{A}\htmlData{tutor-start=2,tutor-end=3}{B}\htmlData{tutor-start=3,tutor-end=4}{|}$ 的所有可能值

题解状态：标准答案与规范题解待补充

题目标签：FIRST-B-PROBLEM-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） · 一试 · 数学竞赛/待细分

(本题满分 20 分) 设 $\htmlData{tutor-start=0,tutor-end=1}{a}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{b}\htmlData{tutor-start=4,tutor-end=5}{,} \htmlData{tutor-start=6,tutor-end=7}{c}$ 均大于 1，满足 $\begin{cases} \lg \htmlData{tutor-start=18,tutor-end=19}{a} \htmlData{tutor-start=20,tutor-end=21}{+} \log_{\htmlData{tutor-start=28,tutor-end=29}{b}} \htmlData{tutor-start=31,tutor-end=32}{c} \htmlData{tutor-start=33,tutor-end=34}{=} \htmlData{tutor-start=35,tutor-end=36}{3} \\ \lg \htmlData{tutor-start=44,tutor-end=45}{b} \htmlData{tutor-start=46,tutor-end=47}{+} \log_{\htmlData{tutor-start=54,tutor-end=55}{a}} \htmlData{tutor-start=57,tutor-end=58}{c} \htmlData{tutor-start=59,tutor-end=60}{=} \htmlData{tutor-start=61,tutor-end=62}{4} \end{cases}$ 求 $\lg \htmlData{tutor-start=4,tutor-end=5}{a} \htmlData{tutor-start=6,tutor-end=12}{\cdot }\lg \htmlData{tutor-start=16,tutor-end=17}{c}$ 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-B-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（B 卷） · 一试 · 数论

(本题满分 20 分) 设复数数列 $\htmlData{tutor-start=0,tutor-end=2}{\{}\htmlData{tutor-start=2,tutor-end=3}{z}_{\htmlData{tutor-start=5,tutor-end=6}{n}}\htmlData{tutor-start=7,tutor-end=9}{\}}$ 满足： $\htmlData{tutor-start=0,tutor-end=1}{|}\htmlData{tutor-start=1,tutor-end=2}{z}_{\htmlData{tutor-start=4,tutor-end=5}{1}}\htmlData{tutor-start=6,tutor-end=7}{|}\htmlData{tutor-start=7,tutor-end=8}{=}\htmlData{tutor-start=8,tutor-end=9}{1}$ ，且对任意正整数 $\htmlData{tutor-start=0,tutor-end=1}{n}$ ，均有 $\htmlData{tutor-start=0,tutor-end=1}{4}\htmlData{tutor-start=1,tutor-end=2}{z}_{\htmlData{tutor-start=4,tutor-end=5}{n}\htmlData{tutor-start=5,tutor-end=6}{+}\htmlData{tutor-start=6,tutor-end=7}{1}}^{\htmlData{tutor-start=10,tutor-end=11}{2}} \htmlData{tutor-start=13,tutor-end=14}{+} \htmlData{tutor-start=15,tutor-end=16}{2}\htmlData{tutor-start=16,tutor-end=17}{z}_{\htmlData{tutor-start=19,tutor-end=20}{n}} \htmlData{tutor-start=22,tutor-end=23}{z}_{\htmlData{tutor-start=25,tutor-end=26}{n}\htmlData{tutor-start=26,tutor-end=27}{+}\htmlData{tutor-start=27,tutor-end=28}{1}} \htmlData{tutor-start=30,tutor-end=31}{+} \htmlData{tutor-start=32,tutor-end=33}{z}_{\htmlData{tutor-start=35,tutor-end=36}{n}}^{\htmlData{tutor-start=39,tutor-end=40}{2}} \htmlData{tutor-start=42,tutor-end=43}{=} \htmlData{tutor-start=44,tutor-end=45}{0}$ ，证明：对任意正整数 $\htmlData{tutor-start=0,tutor-end=1}{m}$ ，均有 $\htmlData{tutor-start=0,tutor-end=1}{|}\htmlData{tutor-start=1,tutor-end=2}{z}_{\htmlData{tutor-start=4,tutor-end=5}{1}}\htmlData{tutor-start=6,tutor-end=7}{+}\htmlData{tutor-start=7,tutor-end=8}{z}_{\htmlData{tutor-start=10,tutor-end=11}{2}}\htmlData{tutor-start=12,tutor-end=13}{+}\dots\htmlData{tutor-start=18,tutor-end=19}{+}\htmlData{tutor-start=19,tutor-end=20}{z}_{\htmlData{tutor-start=22,tutor-end=23}{m}}\htmlData{tutor-start=24,tutor-end=25}{|} \htmlData{tutor-start=26,tutor-end=27}{<} \frac{\htmlData{tutor-start=34,tutor-end=35}{2}\sqrt{\htmlData{tutor-start=41,tutor-end=42}{3}}}{\htmlData{tutor-start=45,tutor-end=46}{3}}$

题解状态：标准答案与规范题解待补充

题目标签：FIRST-B-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（B 卷） · 加试 · 数学竞赛/待细分

(本题满分 40 分) 设正实数 $\htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{,} \htmlData{tutor-start=7,tutor-end=8}{a}_{\htmlData{tutor-start=10,tutor-end=11}{2}}\htmlData{tutor-start=12,tutor-end=13}{,} \dots\htmlData{tutor-start=19,tutor-end=20}{,} \htmlData{tutor-start=21,tutor-end=22}{a}_{\htmlData{tutor-start=24,tutor-end=25}{1}\htmlData{tutor-start=25,tutor-end=26}{0}\htmlData{tutor-start=26,tutor-end=27}{0}}$ 满足 $\htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{i}} \htmlData{tutor-start=6,tutor-end=7}{>} \htmlData{tutor-start=8,tutor-end=9}{a}_{\htmlData{tutor-start=11,tutor-end=12}{1}\htmlData{tutor-start=12,tutor-end=13}{0}\htmlData{tutor-start=13,tutor-end=14}{1}\htmlData{tutor-start=14,tutor-end=15}{-}\htmlData{tutor-start=15,tutor-end=16}{i}} \htmlData{tutor-start=18,tutor-end=20}{\ }\htmlData{tutor-start=20,tutor-end=21}{(}\htmlData{tutor-start=21,tutor-end=22}{i}\htmlData{tutor-start=22,tutor-end=23}{=}\htmlData{tutor-start=23,tutor-end=24}{1}\htmlData{tutor-start=24,tutor-end=25}{,} \htmlData{tutor-start=26,tutor-end=27}{2}\htmlData{tutor-start=27,tutor-end=28}{,} \dots\htmlData{tutor-start=34,tutor-end=35}{,} \htmlData{tutor-start=36,tutor-end=37}{5}\htmlData{tutor-start=37,tutor-end=38}{0}\htmlData{tutor-start=38,tutor-end=39}{)}$ ，记 $\htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{k}} \htmlData{tutor-start=6,tutor-end=7}{=} \frac{\htmlData{tutor-start=14,tutor-end=15}{k} \htmlData{tutor-start=16,tutor-end=17}{a}_{\htmlData{tutor-start=19,tutor-end=20}{k}\htmlData{tutor-start=20,tutor-end=21}{+}\htmlData{tutor-start=21,tutor-end=22}{1}}}{\htmlData{tutor-start=25,tutor-end=26}{a}_{\htmlData{tutor-start=28,tutor-end=29}{1}}\htmlData{tutor-start=30,tutor-end=31}{+}\htmlData{tutor-start=31,tutor-end=32}{a}_{\htmlData{tutor-start=34,tutor-end=35}{2}}\htmlData{tutor-start=36,tutor-end=37}{+}\dots\htmlData{tutor-start=42,tutor-end=43}{+}\htmlData{tutor-start=43,tutor-end=44}{a}_{\htmlData{tutor-start=46,tutor-end=47}{k}}} \htmlData{tutor-start=50,tutor-end=51}{(}\htmlData{tutor-start=51,tutor-end=52}{k}\htmlData{tutor-start=52,tutor-end=53}{=}\htmlData{tutor-start=53,tutor-end=54}{1}\htmlData{tutor-start=54,tutor-end=55}{,} \htmlData{tutor-start=56,tutor-end=57}{2}\htmlData{tutor-start=57,tutor-end=58}{,} \dots\htmlData{tutor-start=64,tutor-end=65}{,} \htmlData{tutor-start=66,tutor-end=67}{9}\htmlData{tutor-start=67,tutor-end=68}{9}\htmlData{tutor-start=68,tutor-end=69}{)}$ 。证明： $\htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{1}} \htmlData{tutor-start=6,tutor-end=7}{x}_{\htmlData{tutor-start=9,tutor-end=10}{2}} \dots \htmlData{tutor-start=18,tutor-end=19}{x}_{\htmlData{tutor-start=21,tutor-end=22}{9}\htmlData{tutor-start=22,tutor-end=23}{9}} \htmlData{tutor-start=25,tutor-end=29}{\le }\htmlData{tutor-start=29,tutor-end=30}{1}$

题解状态：标准答案与规范题解待补充

题目标签：SECOND-B-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（B 卷） · 加试 · 数论

(本题满分 40 分) 求满足以下条件的所有正整数 n：①n 至少有 4 个正约数； ②若 $\htmlData{tutor-start=0,tutor-end=1}{d}_{\htmlData{tutor-start=3,tutor-end=4}{1}} \htmlData{tutor-start=6,tutor-end=7}{<} \htmlData{tutor-start=8,tutor-end=9}{d}_{\htmlData{tutor-start=11,tutor-end=12}{2}} \htmlData{tutor-start=14,tutor-end=15}{<} \dots \htmlData{tutor-start=22,tutor-end=23}{<} \htmlData{tutor-start=24,tutor-end=25}{d}_{\htmlData{tutor-start=27,tutor-end=28}{k}}$ 是 n 的所有正约数，则 $\htmlData{tutor-start=0,tutor-end=1}{d}_{\htmlData{tutor-start=3,tutor-end=4}{2}}\htmlData{tutor-start=5,tutor-end=6}{-}\htmlData{tutor-start=6,tutor-end=7}{d}_{\htmlData{tutor-start=9,tutor-end=10}{1}}\htmlData{tutor-start=11,tutor-end=12}{,} \htmlData{tutor-start=13,tutor-end=14}{d}_{\htmlData{tutor-start=16,tutor-end=17}{3}}\htmlData{tutor-start=18,tutor-end=19}{-}\htmlData{tutor-start=19,tutor-end=20}{d}_{\htmlData{tutor-start=22,tutor-end=23}{2}}\htmlData{tutor-start=24,tutor-end=25}{,} \dots\htmlData{tutor-start=31,tutor-end=32}{,} \htmlData{tutor-start=33,tutor-end=34}{d}_{\htmlData{tutor-start=36,tutor-end=37}{k}}\htmlData{tutor-start=38,tutor-end=39}{-}\htmlData{tutor-start=39,tutor-end=40}{d}_{\htmlData{tutor-start=42,tutor-end=43}{k}\htmlData{tutor-start=43,tutor-end=44}{-}\htmlData{tutor-start=44,tutor-end=45}{1}}$ 构成等比数列

题解状态：标准答案与规范题解待补充

题目标签：SECOND-B-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（B 卷） · 加试 · 平面几何

(本题满分 50 分) 如图，点 A, B, C, D, E 在一条直线上顺次排列，满足 $\htmlData{tutor-start=0,tutor-end=1}{B}\htmlData{tutor-start=1,tutor-end=2}{C}\htmlData{tutor-start=2,tutor-end=3}{=}\htmlData{tutor-start=3,tutor-end=4}{C}\htmlData{tutor-start=4,tutor-end=5}{D}\htmlData{tutor-start=5,tutor-end=6}{=}\sqrt{\htmlData{tutor-start=12,tutor-end=13}{A}\htmlData{tutor-start=13,tutor-end=14}{B} \htmlData{tutor-start=15,tutor-end=21}{\cdot }\htmlData{tutor-start=21,tutor-end=22}{D}\htmlData{tutor-start=22,tutor-end=23}{E}}$ 。点 P 在该直线外，满足 PB=PD。点 K, L 分别在线段 PB, PD 上，满足 KC 平分 $\htmlData{tutor-start=0,tutor-end=7}{\angle }\htmlData{tutor-start=7,tutor-end=8}{B}\htmlData{tutor-start=8,tutor-end=9}{K}\htmlData{tutor-start=9,tutor-end=10}{E}$ ，LC 平分 $\htmlData{tutor-start=0,tutor-end=7}{\angle }\htmlData{tutor-start=7,tutor-end=8}{A}\htmlData{tutor-start=8,tutor-end=9}{L}\htmlData{tutor-start=9,tutor-end=10}{D}$ 。证明：A, K, L, E 四点共圆（答题时请将图画在答卷纸上）

**原卷图示 1**原卷第 1 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-B-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（B 卷） · 加试 · 平面几何

(本题满分 50 分) 将一个凸 2019 边形的每条边任意染红、黄、蓝三种颜色之一，每种颜色的边各 673 条。证明：可作这个凸 2019 边形的 2016 条边在内部互不相交的对角线将其剖分为 2017 个三角形，并将所作的每条对角线也染为红、黄、蓝三种颜色之一，使得每个三角形的三条边或者颜色全部相同，或者颜色互不相同。

题解状态：标准答案与规范题解待补充

题目标签：SECOND-B-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。