返回特征解读

2010 年全国高中数学联赛 B 卷（学生版）

books/national_high_school_league_students/papers/competition-national-league-2010-b.pdf · HS-MATH-1024-v2.1-solution-aware

15 题0 个小问/题组

1

一试（B 卷） · 一、填空题（本题满分 64 分，每小题 8 分） · 代数

填空题（本题满分 64 分，每小题 8 分）函数 $\htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{x}\htmlData{tutor-start=3,tutor-end=4}{)}\htmlData{tutor-start=4,tutor-end=5}{=}\sqrt{\htmlData{tutor-start=11,tutor-end=12}{x}\htmlData{tutor-start=12,tutor-end=13}{-}\htmlData{tutor-start=13,tutor-end=14}{5}}\htmlData{tutor-start=15,tutor-end=16}{-}\sqrt{\htmlData{tutor-start=22,tutor-end=23}{2}\htmlData{tutor-start=23,tutor-end=24}{4}\htmlData{tutor-start=24,tutor-end=25}{-}\htmlData{tutor-start=25,tutor-end=26}{3}\htmlData{tutor-start=26,tutor-end=27}{x}}$ 的值域是________.

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。

2

一试（B 卷） · 一、填空题（本题满分 64 分，每小题 8 分） · 代数

填空题（本题满分 64 分，每小题 8 分）已知函数 $\htmlData{tutor-start=0,tutor-end=1}{y}\htmlData{tutor-start=1,tutor-end=2}{=}\htmlData{tutor-start=2,tutor-end=3}{(}\htmlData{tutor-start=3,tutor-end=4}{a}\cos^{\htmlData{tutor-start=10,tutor-end=11}{2}} \htmlData{tutor-start=13,tutor-end=14}{x}\htmlData{tutor-start=14,tutor-end=15}{-}\htmlData{tutor-start=15,tutor-end=16}{3}\htmlData{tutor-start=16,tutor-end=17}{)}\sin \htmlData{tutor-start=22,tutor-end=23}{x}$ 的最小值为 $\htmlData{tutor-start=0,tutor-end=1}{-}\htmlData{tutor-start=1,tutor-end=2}{3}$ ，则实数 $\htmlData{tutor-start=0,tutor-end=1}{a}$ 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-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。

3

一试（B 卷） · 一、填空题（本题满分 64 分，每小题 8 分） · 平面几何

填空题（本题满分 64 分，每小题 8 分）双曲线 $\htmlData{tutor-start=0,tutor-end=1}{x}^{\htmlData{tutor-start=3,tutor-end=4}{2}}\htmlData{tutor-start=5,tutor-end=6}{-}\htmlData{tutor-start=6,tutor-end=7}{y}^{\htmlData{tutor-start=9,tutor-end=10}{2}}\htmlData{tutor-start=11,tutor-end=12}{=}\htmlData{tutor-start=12,tutor-end=13}{1}$ 的右半支与直线 $\htmlData{tutor-start=0,tutor-end=1}{x}\htmlData{tutor-start=1,tutor-end=2}{=}\htmlData{tutor-start=2,tutor-end=3}{1}\htmlData{tutor-start=3,tutor-end=4}{0}\htmlData{tutor-start=4,tutor-end=5}{0}$ 围成的区域内部（不含边界）整点（纵横坐标均为整数的点）的个数是________.

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。

4

一试（B 卷） · 一、填空题（本题满分 64 分，每小题 8 分） · 代数

填空题（本题满分 64 分，每小题 8 分）已知 $\htmlData{tutor-start=0,tutor-end=2}{\{}\htmlData{tutor-start=2,tutor-end=3}{a}_{\htmlData{tutor-start=5,tutor-end=6}{n}}\htmlData{tutor-start=7,tutor-end=9}{\}}$ 是公差不为 0 的等差数列， $\htmlData{tutor-start=0,tutor-end=2}{\{}\htmlData{tutor-start=2,tutor-end=3}{b}_{\htmlData{tutor-start=5,tutor-end=6}{n}}\htmlData{tutor-start=7,tutor-end=9}{\}}$ 是等比数列，其中 $\htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{=}\htmlData{tutor-start=6,tutor-end=7}{3}\htmlData{tutor-start=7,tutor-end=8}{,} \htmlData{tutor-start=9,tutor-end=10}{b}_{\htmlData{tutor-start=12,tutor-end=13}{1}}\htmlData{tutor-start=14,tutor-end=15}{=}\htmlData{tutor-start=15,tutor-end=16}{1}\htmlData{tutor-start=16,tutor-end=17}{,} \htmlData{tutor-start=18,tutor-end=19}{a}_{\htmlData{tutor-start=21,tutor-end=22}{2}}\htmlData{tutor-start=23,tutor-end=24}{=}\htmlData{tutor-start=24,tutor-end=25}{b}_{\htmlData{tutor-start=27,tutor-end=28}{2}}\htmlData{tutor-start=29,tutor-end=30}{,} \htmlData{tutor-start=31,tutor-end=32}{3}\htmlData{tutor-start=32,tutor-end=33}{a}_{\htmlData{tutor-start=35,tutor-end=36}{5}}\htmlData{tutor-start=37,tutor-end=38}{=}\htmlData{tutor-start=38,tutor-end=39}{b}_{\htmlData{tutor-start=41,tutor-end=42}{3}}$ ，且存在常数 $\htmlData{tutor-start=0,tutor-end=6}{\alpha}\htmlData{tutor-start=6,tutor-end=7}{,} \htmlData{tutor-start=8,tutor-end=13}{\beta}$ 使得对每一个正整数 $\htmlData{tutor-start=0,tutor-end=1}{n}$ 都有 $a_{n}=\log_\alpha b_{n}+\beta$ ，则 $\htmlData{tutor-start=0,tutor-end=6}{\alpha}\htmlData{tutor-start=6,tutor-end=7}{+}\htmlData{tutor-start=7,tutor-end=12}{\beta}\htmlData{tutor-start=12,tutor-end=13}{=}$ ________.

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。

5

一试（B 卷） · 一、填空题（本题满分 64 分，每小题 8 分） · 代数

填空题（本题满分 64 分，每小题 8 分）函数 $\htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{x}\htmlData{tutor-start=3,tutor-end=4}{)}\htmlData{tutor-start=4,tutor-end=5}{=}\htmlData{tutor-start=5,tutor-end=6}{a}^{\htmlData{tutor-start=8,tutor-end=9}{2}\htmlData{tutor-start=9,tutor-end=10}{x}}\htmlData{tutor-start=11,tutor-end=12}{+}\htmlData{tutor-start=12,tutor-end=13}{3}\htmlData{tutor-start=13,tutor-end=14}{a}^{\htmlData{tutor-start=16,tutor-end=17}{x}}\htmlData{tutor-start=18,tutor-end=19}{-}\htmlData{tutor-start=19,tutor-end=20}{2} \htmlData{tutor-start=21,tutor-end=22}{(}\htmlData{tutor-start=22,tutor-end=23}{a}\htmlData{tutor-start=23,tutor-end=24}{>}\htmlData{tutor-start=24,tutor-end=25}{0}\htmlData{tutor-start=25,tutor-end=26}{,} \htmlData{tutor-start=27,tutor-end=28}{a}\neq \htmlData{tutor-start=33,tutor-end=34}{1}\htmlData{tutor-start=34,tutor-end=35}{)}$ 在区间 $\htmlData{tutor-start=0,tutor-end=1}{x}\htmlData{tutor-start=1,tutor-end=4}{\in}\htmlData{tutor-start=4,tutor-end=5}{[}\htmlData{tutor-start=5,tutor-end=6}{-}\htmlData{tutor-start=6,tutor-end=7}{1}\htmlData{tutor-start=7,tutor-end=8}{,} \htmlData{tutor-start=9,tutor-end=10}{1}\htmlData{tutor-start=10,tutor-end=11}{]}$ 上的最大值为 8，则它在这个区间上的最小值是________.

题解状态：标准答案与规范题解待补充

题目标签：FIRST-B-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。

6

一试（B 卷） · 一、填空题（本题满分 64 分，每小题 8 分） · 数学竞赛/待细分

填空题（本题满分 64 分，每小题 8 分）两人轮流投掷骰子，每人每次投掷两颗，第一个使两颗骰子点数和大于 6 者为胜，否则轮由另一人投掷.先投掷人的获胜概率是________.

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。

7

一试（B 卷） · 一、填空题（本题满分 64 分，每小题 8 分） · 数学竞赛/待细分

填空题（本题满分 64 分，每小题 8 分）正三棱柱 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{B}\htmlData{tutor-start=2,tutor-end=3}{C}\htmlData{tutor-start=3,tutor-end=4}{-}\htmlData{tutor-start=4,tutor-end=5}{A}_{\htmlData{tutor-start=7,tutor-end=8}{1}}\htmlData{tutor-start=9,tutor-end=10}{B}_{\htmlData{tutor-start=12,tutor-end=13}{1}}\htmlData{tutor-start=14,tutor-end=15}{C}_{\htmlData{tutor-start=17,tutor-end=18}{1}}$ 的 9 条棱长都相等， $\htmlData{tutor-start=0,tutor-end=1}{P}$ 是 $\htmlData{tutor-start=0,tutor-end=1}{C}\htmlData{tutor-start=1,tutor-end=2}{C}_{\htmlData{tutor-start=4,tutor-end=5}{1}}$ 的中点，二面角 $\htmlData{tutor-start=0,tutor-end=1}{B}\htmlData{tutor-start=1,tutor-end=2}{-}\htmlData{tutor-start=2,tutor-end=3}{A}_{\htmlData{tutor-start=5,tutor-end=6}{1}}\htmlData{tutor-start=7,tutor-end=8}{P}\htmlData{tutor-start=8,tutor-end=9}{-}\htmlData{tutor-start=9,tutor-end=10}{B}_{\htmlData{tutor-start=12,tutor-end=13}{1}}\htmlData{tutor-start=14,tutor-end=15}{=}\htmlData{tutor-start=15,tutor-end=21}{\alpha}$ ，则 $\sin\htmlData{tutor-start=4,tutor-end=10}{\alpha}\htmlData{tutor-start=10,tutor-end=11}{=}$ ________.

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。

8

一试（B 卷） · 一、填空题（本题满分 64 分，每小题 8 分） · 数论

填空题（本题满分 64 分，每小题 8 分）方程 $\htmlData{tutor-start=0,tutor-end=1}{x}\htmlData{tutor-start=1,tutor-end=2}{+}\htmlData{tutor-start=2,tutor-end=3}{y}\htmlData{tutor-start=3,tutor-end=4}{+}\htmlData{tutor-start=4,tutor-end=5}{z}\htmlData{tutor-start=5,tutor-end=6}{=}\htmlData{tutor-start=6,tutor-end=7}{2}\htmlData{tutor-start=7,tutor-end=8}{0}\htmlData{tutor-start=8,tutor-end=9}{1}\htmlData{tutor-start=9,tutor-end=10}{0}$ 满足 $\htmlData{tutor-start=0,tutor-end=1}{x}\htmlData{tutor-start=1,tutor-end=5}{\le }\htmlData{tutor-start=5,tutor-end=6}{y}\htmlData{tutor-start=6,tutor-end=10}{\le }\htmlData{tutor-start=10,tutor-end=11}{z}$ 的正整数解 (x,y,z) 的个数是________.

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。

9

一试（B 卷） · 二、解答题（本题满分 56 分） · 代数

（本小题满分 16 分）已知函数 $\htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{x}\htmlData{tutor-start=3,tutor-end=4}{)}\htmlData{tutor-start=4,tutor-end=5}{=}\htmlData{tutor-start=5,tutor-end=6}{a}\htmlData{tutor-start=6,tutor-end=7}{x}^{\htmlData{tutor-start=9,tutor-end=10}{3}}\htmlData{tutor-start=11,tutor-end=12}{+}\htmlData{tutor-start=12,tutor-end=13}{b}\htmlData{tutor-start=13,tutor-end=14}{x}^{\htmlData{tutor-start=16,tutor-end=17}{2}}\htmlData{tutor-start=18,tutor-end=19}{+}\htmlData{tutor-start=19,tutor-end=20}{c}\htmlData{tutor-start=20,tutor-end=21}{x}\htmlData{tutor-start=21,tutor-end=22}{+}\htmlData{tutor-start=22,tutor-end=23}{d}\htmlData{tutor-start=23,tutor-end=24}{(}\htmlData{tutor-start=24,tutor-end=25}{a}\\htmlData{tutor-start=26,tutor-end=27}{n}eq \htmlData{tutor-start=30,tutor-end=31}{0}\htmlData{tutor-start=31,tutor-end=32}{)}$ ，当 $\htmlData{tutor-start=0,tutor-end=1}{0}\htmlData{tutor-start=1,tutor-end=5}{\le }\htmlData{tutor-start=5,tutor-end=6}{x}\htmlData{tutor-start=6,tutor-end=10}{\le }\htmlData{tutor-start=10,tutor-end=11}{1}$ 时， $\htmlData{tutor-start=0,tutor-end=1}{|}\htmlData{tutor-start=1,tutor-end=2}{f}'\htmlData{tutor-start=3,tutor-end=4}{(}\htmlData{tutor-start=4,tutor-end=5}{x}\htmlData{tutor-start=5,tutor-end=6}{)}\htmlData{tutor-start=6,tutor-end=7}{|}\htmlData{tutor-start=7,tutor-end=11}{\le }\htmlData{tutor-start=11,tutor-end=12}{1}$ ，试求 $\htmlData{tutor-start=0,tutor-end=1}{a}$ 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-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。

10

一试（B 卷） · 二、解答题（本题满分 56 分） · 数学竞赛/待细分

（本小题满分 20 分）已知抛物线 $\htmlData{tutor-start=0,tutor-end=1}{y}^{\htmlData{tutor-start=3,tutor-end=4}{2}}\htmlData{tutor-start=5,tutor-end=6}{=}\htmlData{tutor-start=6,tutor-end=7}{6}\htmlData{tutor-start=7,tutor-end=8}{x}$ 上的两个动点 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{x}_{\htmlData{tutor-start=5,tutor-end=6}{1}}\htmlData{tutor-start=7,tutor-end=8}{,}\htmlData{tutor-start=8,tutor-end=9}{y}_{\htmlData{tutor-start=11,tutor-end=12}{1}}\htmlData{tutor-start=13,tutor-end=14}{)}$ 和 $\htmlData{tutor-start=0,tutor-end=1}{B}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{x}_{\htmlData{tutor-start=5,tutor-end=6}{2}}\htmlData{tutor-start=7,tutor-end=8}{,}\htmlData{tutor-start=8,tutor-end=9}{y}_{\htmlData{tutor-start=11,tutor-end=12}{2}}\htmlData{tutor-start=13,tutor-end=14}{)}$ ，其中 $\htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\neq \htmlData{tutor-start=10,tutor-end=11}{x}_{\htmlData{tutor-start=13,tutor-end=14}{2}}$ 且 $\htmlData{tutor-start=0,tutor-end=1}{x}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{+}\htmlData{tutor-start=6,tutor-end=7}{x}_{\htmlData{tutor-start=9,tutor-end=10}{2}}\htmlData{tutor-start=11,tutor-end=12}{=}\htmlData{tutor-start=12,tutor-end=13}{4}$ . 线段 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{B}$ 的垂直平分线与 $\htmlData{tutor-start=0,tutor-end=1}{x}$ 轴交于点 $\htmlData{tutor-start=0,tutor-end=1}{C}$ ，求 $\htmlData{tutor-start=0,tutor-end=7}{\Delta }\htmlData{tutor-start=7,tutor-end=8}{A}\htmlData{tutor-start=8,tutor-end=9}{B}\htmlData{tutor-start=9,tutor-end=10}{C}$ 面积的最大值.

题解状态：标准答案与规范题解待补充

题目标签：FIRST-B-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。

11

一试（B 卷） · 二、解答题（本题满分 56 分） · 代数

（本小题满分 20 分）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 $\htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{=}\frac{\htmlData{tutor-start=12,tutor-end=13}{1}}{\htmlData{tutor-start=15,tutor-end=16}{3}}\htmlData{tutor-start=17,tutor-end=18}{,} \htmlData{tutor-start=19,tutor-end=20}{a}_{\htmlData{tutor-start=22,tutor-end=23}{n}\htmlData{tutor-start=23,tutor-end=24}{+}\htmlData{tutor-start=24,tutor-end=25}{1}}\htmlData{tutor-start=26,tutor-end=27}{=}\frac{\htmlData{tutor-start=33,tutor-end=34}{a}_{\htmlData{tutor-start=36,tutor-end=37}{n}}^{\htmlData{tutor-start=40,tutor-end=41}{2}}}{\htmlData{tutor-start=44,tutor-end=45}{a}_{\htmlData{tutor-start=47,tutor-end=48}{n}}^{\htmlData{tutor-start=51,tutor-end=52}{2}}\htmlData{tutor-start=53,tutor-end=54}{-}\htmlData{tutor-start=54,tutor-end=55}{a}_{\htmlData{tutor-start=57,tutor-end=58}{n}}\htmlData{tutor-start=59,tutor-end=60}{+}\htmlData{tutor-start=60,tutor-end=61}{1}}\htmlData{tutor-start=62,tutor-end=63}{(}\htmlData{tutor-start=63,tutor-end=64}{n}\htmlData{tutor-start=64,tutor-end=65}{=}\htmlData{tutor-start=65,tutor-end=66}{1}\htmlData{tutor-start=66,tutor-end=67}{,}\htmlData{tutor-start=67,tutor-end=68}{2}\htmlData{tutor-start=68,tutor-end=69}{,}\cdots\htmlData{tutor-start=75,tutor-end=76}{)}$ . 求证： $\frac{1}{2}-\frac{1}{3^{2^{n-1}}} < a_{1}+a_{2}+\cdots+a_{n} < \frac{1}{2}-\frac{1}{3^{2^{n}}}$ . （1）

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。

12

二试/加试（B 卷） · 加试 · 平面几何

（本题满分 40 分）如图，锐角三角形 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{B}\htmlData{tutor-start=2,tutor-end=3}{C}$ 的外心为 $\htmlData{tutor-start=0,tutor-end=1}{O}$ ， $\htmlData{tutor-start=0,tutor-end=1}{K}$ 是边 $\htmlData{tutor-start=0,tutor-end=1}{B}\htmlData{tutor-start=1,tutor-end=2}{C}$ 上一点（不是边 $\htmlData{tutor-start=0,tutor-end=1}{B}\htmlData{tutor-start=1,tutor-end=2}{C}$ 的中点）， $\htmlData{tutor-start=0,tutor-end=1}{D}$ 是线段 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{K}$ 延长线上一点，直线 $\htmlData{tutor-start=0,tutor-end=1}{B}\htmlData{tutor-start=1,tutor-end=2}{D}$ 与 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{C}$ 交于点 $\htmlData{tutor-start=0,tutor-end=1}{N}$ ，直线 $\htmlData{tutor-start=0,tutor-end=1}{C}\htmlData{tutor-start=1,tutor-end=2}{D}$ 与 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{B}$ 交于点 $\htmlData{tutor-start=0,tutor-end=1}{M}$ . 求证：若 $\htmlData{tutor-start=0,tutor-end=1}{O}\htmlData{tutor-start=1,tutor-end=2}{K} \htmlData{tutor-start=3,tutor-end=9}{\perp }\htmlData{tutor-start=9,tutor-end=10}{M}\htmlData{tutor-start=10,tutor-end=11}{N}$ ，则 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{B}\htmlData{tutor-start=4,tutor-end=5}{,} \htmlData{tutor-start=6,tutor-end=7}{D}\htmlData{tutor-start=7,tutor-end=8}{,} \htmlData{tutor-start=9,tutor-end=10}{C}$ 四点共圆.

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。

13

二试/加试（B 卷） · 加试 · 数论

（本题满分 40 分）设 $\htmlData{tutor-start=0,tutor-end=1}{m}$ 和 $\htmlData{tutor-start=0,tutor-end=1}{n}$ 是大于 1 的整数，求证： $\htmlData{tutor-start=0,tutor-end=1}{1}^{\htmlData{tutor-start=3,tutor-end=4}{m}} \htmlData{tutor-start=6,tutor-end=7}{+} \htmlData{tutor-start=8,tutor-end=9}{2}^{\htmlData{tutor-start=11,tutor-end=12}{m}} \htmlData{tutor-start=14,tutor-end=15}{+} \cdots \htmlData{tutor-start=23,tutor-end=24}{+} \htmlData{tutor-start=25,tutor-end=26}{n}^{\htmlData{tutor-start=28,tutor-end=29}{m}} \htmlData{tutor-start=31,tutor-end=32}{=} \frac{\htmlData{tutor-start=39,tutor-end=40}{1}}{\htmlData{tutor-start=42,tutor-end=43}{m}\htmlData{tutor-start=43,tutor-end=44}{+}\htmlData{tutor-start=44,tutor-end=45}{1}} \left\{ \htmlData{tutor-start=55,tutor-end=56}{(}\htmlData{tutor-start=56,tutor-end=57}{n}\htmlData{tutor-start=57,tutor-end=58}{+}\htmlData{tutor-start=58,tutor-end=59}{1}\htmlData{tutor-start=59,tutor-end=60}{)} \sum_{\htmlData{tutor-start=67,tutor-end=68}{k}\htmlData{tutor-start=68,tutor-end=69}{=}\htmlData{tutor-start=69,tutor-end=70}{1}}^{\htmlData{tutor-start=73,tutor-end=74}{m}} \htmlData{tutor-start=76,tutor-end=77}{C}_{\htmlData{tutor-start=79,tutor-end=80}{m}}^{\htmlData{tutor-start=83,tutor-end=84}{k}} \htmlData{tutor-start=86,tutor-end=87}{n}^{\htmlData{tutor-start=89,tutor-end=90}{k}} \htmlData{tutor-start=92,tutor-end=93}{-} \sum_{\htmlData{tutor-start=100,tutor-end=101}{j}\htmlData{tutor-start=101,tutor-end=102}{=}\htmlData{tutor-start=102,tutor-end=103}{1}}^{\htmlData{tutor-start=106,tutor-end=107}{m}\htmlData{tutor-start=107,tutor-end=108}{-}\htmlData{tutor-start=108,tutor-end=109}{1}} \htmlData{tutor-start=111,tutor-end=112}{(}\htmlData{tutor-start=112,tutor-end=113}{C}_{\htmlData{tutor-start=115,tutor-end=116}{m}\htmlData{tutor-start=116,tutor-end=117}{+}\htmlData{tutor-start=117,tutor-end=118}{1}}^{\htmlData{tutor-start=121,tutor-end=122}{j}} \sum_{\htmlData{tutor-start=130,tutor-end=131}{i}\htmlData{tutor-start=131,tutor-end=132}{=}\htmlData{tutor-start=132,tutor-end=133}{1}}^{\htmlData{tutor-start=136,tutor-end=137}{n}} \htmlData{tutor-start=139,tutor-end=140}{i}^{\htmlData{tutor-start=142,tutor-end=143}{j}}\htmlData{tutor-start=144,tutor-end=145}{)} \right\}$ .

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。

14

二试/加试（B 卷） · 加试 · 数学竞赛/待细分

（本题满分 50 分）设 $\htmlData{tutor-start=0,tutor-end=1}{x}\htmlData{tutor-start=1,tutor-end=2}{,} \htmlData{tutor-start=3,tutor-end=4}{y}\htmlData{tutor-start=4,tutor-end=5}{,} \htmlData{tutor-start=6,tutor-end=7}{z}$ 为非负实数，求证： $\htmlData{tutor-start=0,tutor-end=1}{(}\frac{\htmlData{tutor-start=7,tutor-end=8}{x}\htmlData{tutor-start=8,tutor-end=9}{y}\htmlData{tutor-start=9,tutor-end=10}{+}\htmlData{tutor-start=10,tutor-end=11}{y}\htmlData{tutor-start=11,tutor-end=12}{z}\htmlData{tutor-start=12,tutor-end=13}{+}\htmlData{tutor-start=13,tutor-end=14}{z}\htmlData{tutor-start=14,tutor-end=15}{x}}{\htmlData{tutor-start=17,tutor-end=18}{3}}\htmlData{tutor-start=19,tutor-end=20}{)}^{\htmlData{tutor-start=22,tutor-end=23}{3}} \htmlData{tutor-start=25,tutor-end=29}{\le }\htmlData{tutor-start=29,tutor-end=30}{(}\htmlData{tutor-start=30,tutor-end=31}{x}^{\htmlData{tutor-start=33,tutor-end=34}{2}}\htmlData{tutor-start=35,tutor-end=36}{-}\htmlData{tutor-start=36,tutor-end=37}{x}\htmlData{tutor-start=37,tutor-end=38}{y}\htmlData{tutor-start=38,tutor-end=39}{+}\htmlData{tutor-start=39,tutor-end=40}{y}^{\htmlData{tutor-start=42,tutor-end=43}{2}}\htmlData{tutor-start=44,tutor-end=45}{)}\htmlData{tutor-start=45,tutor-end=46}{(}\htmlData{tutor-start=46,tutor-end=47}{y}^{\htmlData{tutor-start=49,tutor-end=50}{2}}\htmlData{tutor-start=51,tutor-end=52}{-}\htmlData{tutor-start=52,tutor-end=53}{y}\htmlData{tutor-start=53,tutor-end=54}{z}\htmlData{tutor-start=54,tutor-end=55}{+}\htmlData{tutor-start=55,tutor-end=56}{z}^{\htmlData{tutor-start=58,tutor-end=59}{2}}\htmlData{tutor-start=60,tutor-end=61}{)}\htmlData{tutor-start=61,tutor-end=62}{(}\htmlData{tutor-start=62,tutor-end=63}{z}^{\htmlData{tutor-start=65,tutor-end=66}{2}}\htmlData{tutor-start=67,tutor-end=68}{-}\htmlData{tutor-start=68,tutor-end=69}{z}\htmlData{tutor-start=69,tutor-end=70}{x}\htmlData{tutor-start=70,tutor-end=71}{+}\htmlData{tutor-start=71,tutor-end=72}{x}^{\htmlData{tutor-start=74,tutor-end=75}{2}}\htmlData{tutor-start=76,tutor-end=77}{)} \htmlData{tutor-start=78,tutor-end=82}{\le }\htmlData{tutor-start=82,tutor-end=83}{(}\frac{\htmlData{tutor-start=89,tutor-end=90}{x}^{\htmlData{tutor-start=92,tutor-end=93}{2}}\htmlData{tutor-start=94,tutor-end=95}{+}\htmlData{tutor-start=95,tutor-end=96}{y}^{\htmlData{tutor-start=98,tutor-end=99}{2}}\htmlData{tutor-start=100,tutor-end=101}{+}\htmlData{tutor-start=101,tutor-end=102}{z}^{\htmlData{tutor-start=104,tutor-end=105}{2}}}{\htmlData{tutor-start=108,tutor-end=109}{2}}\htmlData{tutor-start=110,tutor-end=111}{)}^{\htmlData{tutor-start=113,tutor-end=114}{3}}$ .

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。

15

二试/加试（B 卷） · 加试 · 数论

（本题满分 50 分）设 $\htmlData{tutor-start=0,tutor-end=1}{k}$ 是给定的正整数， $\htmlData{tutor-start=0,tutor-end=1}{r} \htmlData{tutor-start=2,tutor-end=3}{=} \htmlData{tutor-start=4,tutor-end=5}{k} \htmlData{tutor-start=6,tutor-end=7}{+} \frac{\htmlData{tutor-start=14,tutor-end=15}{1}}{\htmlData{tutor-start=17,tutor-end=18}{2}}$ . 记 $\htmlData{tutor-start=0,tutor-end=1}{f}^{\htmlData{tutor-start=3,tutor-end=4}{(}\htmlData{tutor-start=4,tutor-end=5}{1}\htmlData{tutor-start=5,tutor-end=6}{)}}\htmlData{tutor-start=7,tutor-end=8}{(}\htmlData{tutor-start=8,tutor-end=9}{r}\htmlData{tutor-start=9,tutor-end=10}{)} \htmlData{tutor-start=11,tutor-end=12}{=} \htmlData{tutor-start=13,tutor-end=14}{f}\htmlData{tutor-start=14,tutor-end=15}{(}\htmlData{tutor-start=15,tutor-end=16}{r}\htmlData{tutor-start=16,tutor-end=17}{)} \htmlData{tutor-start=18,tutor-end=19}{=} \htmlData{tutor-start=20,tutor-end=21}{r} \htmlData{tutor-start=22,tutor-end=29}{\lceil }\htmlData{tutor-start=29,tutor-end=30}{r} \htmlData{tutor-start=31,tutor-end=37}{\rceil}$ , $\htmlData{tutor-start=0,tutor-end=1}{f}^{\htmlData{tutor-start=3,tutor-end=4}{(}\htmlData{tutor-start=4,tutor-end=5}{l}\htmlData{tutor-start=5,tutor-end=6}{)}}\htmlData{tutor-start=7,tutor-end=8}{(}\htmlData{tutor-start=8,tutor-end=9}{r}\htmlData{tutor-start=9,tutor-end=10}{)} \htmlData{tutor-start=11,tutor-end=12}{=} \htmlData{tutor-start=13,tutor-end=14}{f}\htmlData{tutor-start=14,tutor-end=15}{(}\htmlData{tutor-start=15,tutor-end=16}{f}^{\htmlData{tutor-start=18,tutor-end=19}{(}\htmlData{tutor-start=19,tutor-end=20}{l}\htmlData{tutor-start=20,tutor-end=21}{-}\htmlData{tutor-start=21,tutor-end=22}{1}\htmlData{tutor-start=22,tutor-end=23}{)}}\htmlData{tutor-start=24,tutor-end=25}{(}\htmlData{tutor-start=25,tutor-end=26}{r}\htmlData{tutor-start=26,tutor-end=27}{)}\htmlData{tutor-start=27,tutor-end=28}{)}$ , $\htmlData{tutor-start=0,tutor-end=1}{l} \htmlData{tutor-start=2,tutor-end=6}{\ge }\htmlData{tutor-start=6,tutor-end=7}{2}$ . 证明：存在正整数 $\htmlData{tutor-start=0,tutor-end=1}{m}$ ，使得 $\htmlData{tutor-start=0,tutor-end=1}{f}^{\htmlData{tutor-start=3,tutor-end=4}{(}\htmlData{tutor-start=4,tutor-end=5}{m}\htmlData{tutor-start=5,tutor-end=6}{)}}\htmlData{tutor-start=7,tutor-end=8}{(}\htmlData{tutor-start=8,tutor-end=9}{r}\htmlData{tutor-start=9,tutor-end=10}{)}$ 为一个整数. 这里， $\htmlData{tutor-start=0,tutor-end=7}{\lceil }\htmlData{tutor-start=7,tutor-end=8}{x} \htmlData{tutor-start=9,tutor-end=15}{\rceil}$ 表示不小于实数 $\htmlData{tutor-start=0,tutor-end=1}{x}$ 的最小整数，例如： $\left\lceil \frac{\htmlData{tutor-start=18,tutor-end=19}{1}}{\htmlData{tutor-start=21,tutor-end=22}{2}} \right\rceil \htmlData{tutor-start=37,tutor-end=38}{=} \htmlData{tutor-start=39,tutor-end=40}{1}$ ， $\htmlData{tutor-start=0,tutor-end=7}{\lceil }\htmlData{tutor-start=7,tutor-end=8}{1} \htmlData{tutor-start=9,tutor-end=16}{\rceil }\htmlData{tutor-start=16,tutor-end=17}{=} \htmlData{tutor-start=18,tutor-end=19}{1}$ 。

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。