2005 年全国高中数学联赛（学生版）

books/national_high_school_league_students/papers/competition-national-league-2005.pdf · HS-MATH-1024-v2.1-solution-aware

18 题0 个小问/题组

一试 · 1. 选择题（本题满分 36 分，每小题 6 分） · 代数

使关于 $\htmlData{tutor-start=0,tutor-end=1}{x}$ 的不等式 $\sqrt{\htmlData{tutor-start=6,tutor-end=7}{x}\htmlData{tutor-start=7,tutor-end=8}{-}\htmlData{tutor-start=8,tutor-end=9}{3}}\htmlData{tutor-start=10,tutor-end=11}{+}\sqrt{\htmlData{tutor-start=17,tutor-end=18}{6}\htmlData{tutor-start=18,tutor-end=19}{-}\htmlData{tutor-start=19,tutor-end=20}{x}} \htmlData{tutor-start=22,tutor-end=26}{\ge }\htmlData{tutor-start=26,tutor-end=27}{k}$ 有解的实数 $\htmlData{tutor-start=0,tutor-end=1}{k}$ 的最大值是（） A. $\sqrt{\htmlData{tutor-start=6,tutor-end=7}{6}}\htmlData{tutor-start=8,tutor-end=9}{-}\sqrt{\htmlData{tutor-start=15,tutor-end=16}{3}}$ B. $\sqrt{\htmlData{tutor-start=6,tutor-end=7}{3}}$ C. [图片模糊] D. $\sqrt{\htmlData{tutor-start=6,tutor-end=7}{6}}$

题解状态：标准答案与规范题解待补充

题目标签：FIRST-CHOICE-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 · 1. 选择题（本题满分 36 分，每小题 6 分） · 数学竞赛/待细分

空间四点 A、B、C、D 满足 $\htmlData{tutor-start=0,tutor-end=1}{|}\overrightarrow{\htmlData{tutor-start=17,tutor-end=18}{A}\htmlData{tutor-start=18,tutor-end=19}{B}}\htmlData{tutor-start=20,tutor-end=21}{|}\htmlData{tutor-start=21,tutor-end=22}{=}\htmlData{tutor-start=22,tutor-end=23}{3}\htmlData{tutor-start=23,tutor-end=24}{,} \htmlData{tutor-start=25,tutor-end=26}{|}\overrightarrow{\htmlData{tutor-start=42,tutor-end=43}{B}\htmlData{tutor-start=43,tutor-end=44}{C}}\htmlData{tutor-start=45,tutor-end=46}{|}\htmlData{tutor-start=46,tutor-end=47}{=}\htmlData{tutor-start=47,tutor-end=48}{7}\htmlData{tutor-start=48,tutor-end=49}{,} \htmlData{tutor-start=50,tutor-end=51}{|}\overrightarrow{\htmlData{tutor-start=67,tutor-end=68}{C}\htmlData{tutor-start=68,tutor-end=69}{D}}\htmlData{tutor-start=70,tutor-end=71}{|}\htmlData{tutor-start=71,tutor-end=72}{=}\htmlData{tutor-start=72,tutor-end=73}{1}\htmlData{tutor-start=73,tutor-end=74}{1}\htmlData{tutor-start=74,tutor-end=75}{,} \htmlData{tutor-start=76,tutor-end=77}{|}\overrightarrow{\htmlData{tutor-start=93,tutor-end=94}{D}\htmlData{tutor-start=94,tutor-end=95}{A}}\htmlData{tutor-start=96,tutor-end=97}{|}\htmlData{tutor-start=97,tutor-end=98}{=}\htmlData{tutor-start=98,tutor-end=99}{9}$ ，则 $\overrightarrow{\htmlData{tutor-start=16,tutor-end=17}{A}\htmlData{tutor-start=17,tutor-end=18}{C}} \htmlData{tutor-start=20,tutor-end=26}{\cdot }\overrightarrow{\htmlData{tutor-start=42,tutor-end=43}{B}\htmlData{tutor-start=43,tutor-end=44}{D}}$ 的取值（） A. 只有一个 B. 有二个 C. 有四个 D. 有无穷多个

题解状态：标准答案与规范题解待补充

题目标签：FIRST-CHOICE-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 · 1. 选择题（本题满分 36 分，每小题 6 分） · 平面几何

$\htmlData{tutor-start=0,tutor-end=10}{\triangle }\htmlData{tutor-start=10,tutor-end=11}{A}\htmlData{tutor-start=11,tutor-end=12}{B}\htmlData{tutor-start=12,tutor-end=13}{C}$ 内接于单位圆，三个内角 A、B、C 的平分线延长后分别交此圆于 $\htmlData{tutor-start=0,tutor-end=1}{A}_{\htmlData{tutor-start=3,tutor-end=4}{1}}$ 、 $\htmlData{tutor-start=0,tutor-end=1}{B}_{\htmlData{tutor-start=3,tutor-end=4}{1}}$ 、 $\htmlData{tutor-start=0,tutor-end=1}{C}_{\htmlData{tutor-start=3,tutor-end=4}{1}}$ 。则 $\frac{\htmlData{tutor-start=6,tutor-end=7}{A}\htmlData{tutor-start=7,tutor-end=8}{A}_{\htmlData{tutor-start=10,tutor-end=11}{1}} \htmlData{tutor-start=13,tutor-end=19}{\cdot }\cos \frac{\htmlData{tutor-start=30,tutor-end=31}{A}}{\htmlData{tutor-start=33,tutor-end=34}{2}} \htmlData{tutor-start=36,tutor-end=37}{+} \htmlData{tutor-start=38,tutor-end=39}{B}\htmlData{tutor-start=39,tutor-end=40}{B}_{\htmlData{tutor-start=42,tutor-end=43}{1}} \htmlData{tutor-start=45,tutor-end=51}{\cdot }\cos \frac{\htmlData{tutor-start=62,tutor-end=63}{B}}{\htmlData{tutor-start=65,tutor-end=66}{2}} \htmlData{tutor-start=68,tutor-end=69}{+} \htmlData{tutor-start=70,tutor-end=71}{C}\htmlData{tutor-start=71,tutor-end=72}{C}_{\htmlData{tutor-start=74,tutor-end=75}{1}} \htmlData{tutor-start=77,tutor-end=83}{\cdot }\cos \frac{\htmlData{tutor-start=94,tutor-end=95}{C}}{\htmlData{tutor-start=97,tutor-end=98}{2}}}{\sin \htmlData{tutor-start=106,tutor-end=107}{A} \htmlData{tutor-start=108,tutor-end=109}{+} \sin \htmlData{tutor-start=115,tutor-end=116}{B} \htmlData{tutor-start=117,tutor-end=118}{+} \sin \htmlData{tutor-start=124,tutor-end=125}{C}}$ 的值为（） A. 2 B. 4 C. 6 D. 8

**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：FIRST-CHOICE-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 · 1. 选择题（本题满分 36 分，每小题 6 分） · 数学竞赛/待细分

如图， $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{B}\htmlData{tutor-start=2,tutor-end=3}{C}\htmlData{tutor-start=3,tutor-end=4}{D}\htmlData{tutor-start=4,tutor-end=5}{-}\htmlData{tutor-start=5,tutor-end=6}{A}'\htmlData{tutor-start=7,tutor-end=8}{B}'\htmlData{tutor-start=9,tutor-end=10}{C}'\htmlData{tutor-start=11,tutor-end=12}{D}'$ 为正方体。任作平面 $\htmlData{tutor-start=0,tutor-end=6}{\alpha}$ 与对角线 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{C}'$ 垂直，使得 $\htmlData{tutor-start=0,tutor-end=6}{\alpha}$ 与正方体的每个面都有公共点，记这样得到的截面多边形的面积为 $\htmlData{tutor-start=0,tutor-end=1}{S}$ ，周长为 $\htmlData{tutor-start=0,tutor-end=1}{l}$ 。则（） A. $\htmlData{tutor-start=0,tutor-end=1}{S}$ 为定值， $\htmlData{tutor-start=0,tutor-end=1}{l}$ 不为定值 B. $\htmlData{tutor-start=0,tutor-end=1}{S}$ 不为定值， $\htmlData{tutor-start=0,tutor-end=1}{l}$ 为定值 C. $\htmlData{tutor-start=0,tutor-end=1}{S}$ 与 $\htmlData{tutor-start=0,tutor-end=1}{l}$ 均为定值 D. $\htmlData{tutor-start=0,tutor-end=1}{S}$ 与 $\htmlData{tutor-start=0,tutor-end=1}{l}$ 均不为定值

题解状态：标准答案与规范题解待补充

题目标签：FIRST-CHOICE-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 · 1. 选择题（本题满分 36 分，每小题 6 分） · 平面几何

方程 $\frac{\htmlData{tutor-start=6,tutor-end=7}{x}^{\htmlData{tutor-start=9,tutor-end=10}{2}}}{\sin\sqrt{\htmlData{tutor-start=23,tutor-end=24}{2}}\htmlData{tutor-start=25,tutor-end=26}{-}\sin\sqrt{\htmlData{tutor-start=36,tutor-end=37}{3}}} \htmlData{tutor-start=40,tutor-end=41}{+} \frac{\htmlData{tutor-start=48,tutor-end=49}{y}^{\htmlData{tutor-start=51,tutor-end=52}{2}}}{\cos\sqrt{\htmlData{tutor-start=65,tutor-end=66}{2}}\htmlData{tutor-start=67,tutor-end=68}{-}\cos\sqrt{\htmlData{tutor-start=78,tutor-end=79}{3}}} \htmlData{tutor-start=82,tutor-end=83}{=} \htmlData{tutor-start=84,tutor-end=85}{1}$ 表示的曲线是（） A. 焦点在 $\htmlData{tutor-start=0,tutor-end=1}{x}$ 轴上的椭圆 B. 焦点在 $\htmlData{tutor-start=0,tutor-end=1}{x}$ 轴上的双曲线 C. 焦点在 $\htmlData{tutor-start=0,tutor-end=1}{y}$ 轴上的椭圆 D. 焦点在 $\htmlData{tutor-start=0,tutor-end=1}{y}$ 轴上的双曲线

题解状态：标准答案与规范题解待补充

题目标签：FIRST-CHOICE-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 · 1. 选择题（本题满分 36 分，每小题 6 分） · 组合数学

记集合 $\htmlData{tutor-start=0,tutor-end=1}{T}\htmlData{tutor-start=1,tutor-end=2}{=}\htmlData{tutor-start=2,tutor-end=4}{\{}\htmlData{tutor-start=4,tutor-end=5}{0}\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=8,tutor-end=9}{2}\htmlData{tutor-start=9,tutor-end=10}{,}\htmlData{tutor-start=10,tutor-end=11}{3}\htmlData{tutor-start=11,tutor-end=12}{,}\htmlData{tutor-start=12,tutor-end=13}{4}\htmlData{tutor-start=13,tutor-end=14}{,}\htmlData{tutor-start=14,tutor-end=15}{5}\htmlData{tutor-start=15,tutor-end=16}{,}\htmlData{tutor-start=16,tutor-end=17}{6}\htmlData{tutor-start=17,tutor-end=19}{\}}\htmlData{tutor-start=19,tutor-end=20}{,} \htmlData{tutor-start=21,tutor-end=22}{M}\htmlData{tutor-start=22,tutor-end=23}{=}\htmlData{tutor-start=23,tutor-end=25}{\{}\frac{\htmlData{tutor-start=31,tutor-end=32}{a}_{\htmlData{tutor-start=34,tutor-end=35}{1}}}{\htmlData{tutor-start=38,tutor-end=39}{7}}\htmlData{tutor-start=40,tutor-end=41}{+}\frac{\htmlData{tutor-start=47,tutor-end=48}{a}_{\htmlData{tutor-start=50,tutor-end=51}{2}}}{\htmlData{tutor-start=54,tutor-end=55}{7}^{\htmlData{tutor-start=57,tutor-end=58}{2}}}\htmlData{tutor-start=60,tutor-end=61}{+}\frac{\htmlData{tutor-start=67,tutor-end=68}{a}_{\htmlData{tutor-start=70,tutor-end=71}{3}}}{\htmlData{tutor-start=74,tutor-end=75}{7}^{\htmlData{tutor-start=77,tutor-end=78}{3}}}\htmlData{tutor-start=80,tutor-end=81}{+}\frac{\htmlData{tutor-start=87,tutor-end=88}{a}_{\htmlData{tutor-start=90,tutor-end=91}{4}}}{\htmlData{tutor-start=94,tutor-end=95}{7}^{\htmlData{tutor-start=97,tutor-end=98}{4}}} \htmlData{tutor-start=101,tutor-end=106}{\mid }\htmlData{tutor-start=106,tutor-end=107}{a}_{\htmlData{tutor-start=109,tutor-end=110}{i}} \htmlData{tutor-start=112,tutor-end=116}{\in }\htmlData{tutor-start=116,tutor-end=117}{T}\htmlData{tutor-start=117,tutor-end=118}{,} \htmlData{tutor-start=119,tutor-end=120}{i}\htmlData{tutor-start=120,tutor-end=121}{=}\htmlData{tutor-start=121,tutor-end=122}{1}\htmlData{tutor-start=122,tutor-end=123}{,}\htmlData{tutor-start=123,tutor-end=124}{2}\htmlData{tutor-start=124,tutor-end=125}{,}\htmlData{tutor-start=125,tutor-end=126}{3}\htmlData{tutor-start=126,tutor-end=127}{,}\htmlData{tutor-start=127,tutor-end=128}{4}\htmlData{tutor-start=128,tutor-end=130}{\}}$ ，将 M 中的元素按从大到小的顺序排列，则第 2005 个数是（） A. $\frac{\htmlData{tutor-start=6,tutor-end=7}{5}}{\htmlData{tutor-start=9,tutor-end=10}{7}}\htmlData{tutor-start=11,tutor-end=12}{+}\frac{\htmlData{tutor-start=18,tutor-end=19}{5}}{\htmlData{tutor-start=21,tutor-end=22}{7}^{\htmlData{tutor-start=24,tutor-end=25}{2}}}\htmlData{tutor-start=27,tutor-end=28}{+}\frac{\htmlData{tutor-start=34,tutor-end=35}{6}}{\htmlData{tutor-start=37,tutor-end=38}{7}^{\htmlData{tutor-start=40,tutor-end=41}{3}}}\htmlData{tutor-start=43,tutor-end=44}{+}\frac{\htmlData{tutor-start=50,tutor-end=51}{3}}{\htmlData{tutor-start=53,tutor-end=54}{7}^{\htmlData{tutor-start=56,tutor-end=57}{4}}}$ B. $\frac{\htmlData{tutor-start=6,tutor-end=7}{5}}{\htmlData{tutor-start=9,tutor-end=10}{7}}\htmlData{tutor-start=11,tutor-end=12}{+}\frac{\htmlData{tutor-start=18,tutor-end=19}{5}}{\htmlData{tutor-start=21,tutor-end=22}{7}^{\htmlData{tutor-start=24,tutor-end=25}{2}}}\htmlData{tutor-start=27,tutor-end=28}{+}\frac{\htmlData{tutor-start=34,tutor-end=35}{6}}{\htmlData{tutor-start=37,tutor-end=38}{7}^{\htmlData{tutor-start=40,tutor-end=41}{3}}}\htmlData{tutor-start=43,tutor-end=44}{+}\frac{\htmlData{tutor-start=50,tutor-end=51}{2}}{\htmlData{tutor-start=53,tutor-end=54}{7}^{\htmlData{tutor-start=56,tutor-end=57}{4}}}$ C. $\frac{\htmlData{tutor-start=6,tutor-end=7}{1}}{\htmlData{tutor-start=9,tutor-end=10}{7}}\htmlData{tutor-start=11,tutor-end=12}{+}\frac{\htmlData{tutor-start=18,tutor-end=19}{1}}{\htmlData{tutor-start=21,tutor-end=22}{7}^{\htmlData{tutor-start=24,tutor-end=25}{2}}}\htmlData{tutor-start=27,tutor-end=28}{+}\frac{\htmlData{tutor-start=34,tutor-end=35}{0}}{\htmlData{tutor-start=37,tutor-end=38}{7}^{\htmlData{tutor-start=40,tutor-end=41}{3}}}\htmlData{tutor-start=43,tutor-end=44}{+}\frac{\htmlData{tutor-start=50,tutor-end=51}{4}}{\htmlData{tutor-start=53,tutor-end=54}{7}^{\htmlData{tutor-start=56,tutor-end=57}{4}}}$ D. $\frac{\htmlData{tutor-start=6,tutor-end=7}{1}}{\htmlData{tutor-start=9,tutor-end=10}{7}}\htmlData{tutor-start=11,tutor-end=12}{+}\frac{\htmlData{tutor-start=18,tutor-end=19}{1}}{\htmlData{tutor-start=21,tutor-end=22}{7}^{\htmlData{tutor-start=24,tutor-end=25}{2}}}\htmlData{tutor-start=27,tutor-end=28}{+}\frac{\htmlData{tutor-start=34,tutor-end=35}{0}}{\htmlData{tutor-start=37,tutor-end=38}{7}^{\htmlData{tutor-start=40,tutor-end=41}{3}}}\htmlData{tutor-start=43,tutor-end=44}{+}\frac{\htmlData{tutor-start=50,tutor-end=51}{3}}{\htmlData{tutor-start=53,tutor-end=54}{7}^{\htmlData{tutor-start=56,tutor-end=57}{4}}}$

题解状态：标准答案与规范题解待补充

题目标签：FIRST-CHOICE-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 · 二、填空题（本题满分 54 分，每小题 9 分） · 代数

将关于 $\htmlData{tutor-start=0,tutor-end=1}{x}$ 的多项式 $\htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{x}\htmlData{tutor-start=3,tutor-end=4}{)}\htmlData{tutor-start=4,tutor-end=5}{=}\htmlData{tutor-start=5,tutor-end=6}{1}\htmlData{tutor-start=6,tutor-end=7}{-}\htmlData{tutor-start=7,tutor-end=8}{x}\htmlData{tutor-start=8,tutor-end=9}{+}\htmlData{tutor-start=9,tutor-end=10}{x}^{\htmlData{tutor-start=12,tutor-end=13}{2}}\htmlData{tutor-start=14,tutor-end=15}{-}\htmlData{tutor-start=15,tutor-end=16}{x}^{\htmlData{tutor-start=18,tutor-end=19}{3}}\htmlData{tutor-start=20,tutor-end=21}{+}\cdots\htmlData{tutor-start=27,tutor-end=28}{-}\htmlData{tutor-start=28,tutor-end=29}{x}^{\htmlData{tutor-start=31,tutor-end=32}{1}\htmlData{tutor-start=32,tutor-end=33}{9}}\htmlData{tutor-start=34,tutor-end=35}{+}\htmlData{tutor-start=35,tutor-end=36}{x}^{\htmlData{tutor-start=38,tutor-end=39}{2}\htmlData{tutor-start=39,tutor-end=40}{0}}$ 表为关于 $\htmlData{tutor-start=0,tutor-end=1}{y}$ 的多项式 $\htmlData{tutor-start=0,tutor-end=1}{g}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{y}\htmlData{tutor-start=3,tutor-end=4}{)}\htmlData{tutor-start=4,tutor-end=5}{=}\htmlData{tutor-start=5,tutor-end=6}{a}_{\htmlData{tutor-start=8,tutor-end=9}{0}}\htmlData{tutor-start=10,tutor-end=11}{+}\htmlData{tutor-start=11,tutor-end=12}{a}_{\htmlData{tutor-start=14,tutor-end=15}{1}}\htmlData{tutor-start=16,tutor-end=17}{y}\htmlData{tutor-start=17,tutor-end=18}{+}\htmlData{tutor-start=18,tutor-end=19}{a}_{\htmlData{tutor-start=21,tutor-end=22}{2}}\htmlData{tutor-start=23,tutor-end=24}{y}^{\htmlData{tutor-start=26,tutor-end=27}{2}}\htmlData{tutor-start=28,tutor-end=29}{+}\cdots\htmlData{tutor-start=35,tutor-end=36}{+}\htmlData{tutor-start=36,tutor-end=37}{a}_{\htmlData{tutor-start=39,tutor-end=40}{1}\htmlData{tutor-start=40,tutor-end=41}{9}}\htmlData{tutor-start=42,tutor-end=43}{y}^{\htmlData{tutor-start=45,tutor-end=46}{1}\htmlData{tutor-start=46,tutor-end=47}{9}}\htmlData{tutor-start=48,tutor-end=49}{+}\htmlData{tutor-start=49,tutor-end=50}{a}_{\htmlData{tutor-start=52,tutor-end=53}{2}\htmlData{tutor-start=53,tutor-end=54}{0}}\htmlData{tutor-start=55,tutor-end=56}{y}^{\htmlData{tutor-start=58,tutor-end=59}{2}\htmlData{tutor-start=59,tutor-end=60}{0}}$ ，其中 $\htmlData{tutor-start=0,tutor-end=1}{y}\htmlData{tutor-start=1,tutor-end=2}{=}\htmlData{tutor-start=2,tutor-end=3}{x}\htmlData{tutor-start=3,tutor-end=4}{-}\htmlData{tutor-start=4,tutor-end=5}{4}$ . 则 $\htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{0}}\htmlData{tutor-start=5,tutor-end=6}{+}\htmlData{tutor-start=6,tutor-end=7}{a}_{\htmlData{tutor-start=9,tutor-end=10}{1}}\htmlData{tutor-start=11,tutor-end=12}{+}\cdots\htmlData{tutor-start=18,tutor-end=19}{+}\htmlData{tutor-start=19,tutor-end=20}{a}_{\htmlData{tutor-start=22,tutor-end=23}{2}\htmlData{tutor-start=23,tutor-end=24}{0}}\htmlData{tutor-start=25,tutor-end=26}{=}$ _________ .

题解状态：标准答案与规范题解待补充

题目标签：FIRST-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 · 二、填空题（本题满分 54 分，每小题 9 分） · 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 $\htmlData{tutor-start=0,tutor-end=1}{(}\htmlData{tutor-start=1,tutor-end=2}{0}\htmlData{tutor-start=2,tutor-end=3}{,}\htmlData{tutor-start=3,tutor-end=4}{+}\htmlData{tutor-start=4,tutor-end=10}{\infty}\htmlData{tutor-start=10,tutor-end=11}{)}$ 上的减函数，若 $\htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{2}\htmlData{tutor-start=3,tutor-end=4}{a}^{\htmlData{tutor-start=6,tutor-end=7}{2}}\htmlData{tutor-start=8,tutor-end=9}{+}\htmlData{tutor-start=9,tutor-end=10}{a}\htmlData{tutor-start=10,tutor-end=11}{+}\htmlData{tutor-start=11,tutor-end=12}{1}\htmlData{tutor-start=12,tutor-end=13}{)}\htmlData{tutor-start=13,tutor-end=14}{<}\htmlData{tutor-start=14,tutor-end=15}{f}\htmlData{tutor-start=15,tutor-end=16}{(}\htmlData{tutor-start=16,tutor-end=17}{3}\htmlData{tutor-start=17,tutor-end=18}{a}^{\htmlData{tutor-start=20,tutor-end=21}{2}}\htmlData{tutor-start=22,tutor-end=23}{-}\htmlData{tutor-start=23,tutor-end=24}{4}\htmlData{tutor-start=24,tutor-end=25}{a}\htmlData{tutor-start=25,tutor-end=26}{+}\htmlData{tutor-start=26,tutor-end=27}{1}\htmlData{tutor-start=27,tutor-end=28}{)}$ 成立，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-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 · 二、填空题（本题满分 54 分，每小题 9 分） · 数学竞赛/待细分

设 $\htmlData{tutor-start=0,tutor-end=6}{\alpha}$ 、 $\htmlData{tutor-start=0,tutor-end=5}{\beta}$ 、 $\htmlData{tutor-start=0,tutor-end=6}{\gamma}$ 满足 $\htmlData{tutor-start=0,tutor-end=1}{0}\htmlData{tutor-start=1,tutor-end=2}{<}\htmlData{tutor-start=2,tutor-end=8}{\alpha}\htmlData{tutor-start=8,tutor-end=9}{<}\htmlData{tutor-start=9,tutor-end=14}{\beta}\htmlData{tutor-start=14,tutor-end=15}{<}\htmlData{tutor-start=15,tutor-end=21}{\gamma}\htmlData{tutor-start=21,tutor-end=22}{<}\htmlData{tutor-start=22,tutor-end=23}{2}\htmlData{tutor-start=23,tutor-end=26}{\pi}$ ，若对于任意 $\htmlData{tutor-start=0,tutor-end=1}{x}\htmlData{tutor-start=1,tutor-end=5}{\in }\htmlData{tutor-start=5,tutor-end=6}{R}\htmlData{tutor-start=6,tutor-end=7}{,} \cos\htmlData{tutor-start=12,tutor-end=13}{(}\htmlData{tutor-start=13,tutor-end=14}{x}\htmlData{tutor-start=14,tutor-end=15}{+}\htmlData{tutor-start=15,tutor-end=21}{\alpha}\htmlData{tutor-start=21,tutor-end=22}{)}\htmlData{tutor-start=22,tutor-end=23}{+}\cos\htmlData{tutor-start=27,tutor-end=28}{(}\htmlData{tutor-start=28,tutor-end=29}{x}\htmlData{tutor-start=29,tutor-end=30}{+}\htmlData{tutor-start=30,tutor-end=35}{\beta}\htmlData{tutor-start=35,tutor-end=36}{)}\htmlData{tutor-start=36,tutor-end=37}{+}\cos\htmlData{tutor-start=41,tutor-end=42}{(}\htmlData{tutor-start=42,tutor-end=43}{x}\htmlData{tutor-start=43,tutor-end=44}{+}\htmlData{tutor-start=44,tutor-end=50}{\gamma}\htmlData{tutor-start=50,tutor-end=51}{)}\htmlData{tutor-start=51,tutor-end=52}{=}\htmlData{tutor-start=52,tutor-end=53}{0}$ ，则 $\htmlData{tutor-start=0,tutor-end=6}{\gamma}\htmlData{tutor-start=6,tutor-end=7}{-}\htmlData{tutor-start=7,tutor-end=13}{\alpha}\htmlData{tutor-start=13,tutor-end=14}{=}$ ___________.

题解状态：标准答案与规范题解待补充

题目标签：FIRST-FILL-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 · 二、填空题（本题满分 54 分，每小题 9 分） · 平面几何

如图，四面体 DABC 的体积为 $\frac{\htmlData{tutor-start=6,tutor-end=7}{1}}{\htmlData{tutor-start=9,tutor-end=10}{6}}$ ，且满足 $\angle ACB=45^\circ, AD+BC+\frac{AC}{\sqrt{2}}=3$ ，则 $\htmlData{tutor-start=0,tutor-end=1}{C}\htmlData{tutor-start=1,tutor-end=2}{D}\htmlData{tutor-start=2,tutor-end=3}{=}$ _________.

题解状态：标准答案与规范题解待补充

题目标签：FIRST-FILL-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 · 二、填空题（本题满分 54 分，每小题 9 分） · 平面几何

若正方形 ABCD 的一条边在直线 $\htmlData{tutor-start=0,tutor-end=1}{y}\htmlData{tutor-start=1,tutor-end=2}{=}\htmlData{tutor-start=2,tutor-end=3}{2}\htmlData{tutor-start=3,tutor-end=4}{x}\htmlData{tutor-start=4,tutor-end=5}{-}\htmlData{tutor-start=5,tutor-end=6}{1}\htmlData{tutor-start=6,tutor-end=7}{7}$ 上，另外两个顶点在抛物线 $\htmlData{tutor-start=0,tutor-end=1}{y}\htmlData{tutor-start=1,tutor-end=2}{=}\htmlData{tutor-start=2,tutor-end=3}{x}^{\htmlData{tutor-start=5,tutor-end=6}{2}}$ 上. 则该正方形面积的最小值为_________.

题解状态：标准答案与规范题解待补充

题目标签：FIRST-FILL-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。

一试 · 二、填空题（本题满分 54 分，每小题 9 分） · 数学竞赛/待细分

如果自然数 $\htmlData{tutor-start=0,tutor-end=1}{a}$ 的各位数字之和等于 7，那么称 $\htmlData{tutor-start=0,tutor-end=1}{a}$ 为“吉祥数”. 将所有“吉祥数”从小到大排成一列 $\htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{1}}\htmlData{tutor-start=5,tutor-end=6}{,}\htmlData{tutor-start=6,tutor-end=7}{a}_{\htmlData{tutor-start=9,tutor-end=10}{2}}\htmlData{tutor-start=11,tutor-end=12}{,}\htmlData{tutor-start=12,tutor-end=13}{a}_{\htmlData{tutor-start=15,tutor-end=16}{3}}\htmlData{tutor-start=17,tutor-end=18}{,}\cdots$ ，若 $\htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{n}}\htmlData{tutor-start=5,tutor-end=6}{=}\htmlData{tutor-start=6,tutor-end=7}{2}\htmlData{tutor-start=7,tutor-end=8}{0}\htmlData{tutor-start=8,tutor-end=9}{0}\htmlData{tutor-start=9,tutor-end=10}{5}$ ，则 $\htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{5}\htmlData{tutor-start=4,tutor-end=5}{n}}\htmlData{tutor-start=6,tutor-end=7}{=}$ _________.

题解状态：标准答案与规范题解待补充

题目标签：FIRST-FILL-P12

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 · 三、解答题（本题满分 60 分，每小题 20 分） · 数论

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： $\htmlData{tutor-start=0,tutor-end=1}{a}_{\htmlData{tutor-start=3,tutor-end=4}{0}}\htmlData{tutor-start=5,tutor-end=6}{=}\htmlData{tutor-start=6,tutor-end=7}{1}\htmlData{tutor-start=7,tutor-end=8}{,} \htmlData{tutor-start=9,tutor-end=10}{a}_{\htmlData{tutor-start=12,tutor-end=13}{n}\htmlData{tutor-start=13,tutor-end=14}{+}\htmlData{tutor-start=14,tutor-end=15}{1}}\htmlData{tutor-start=16,tutor-end=17}{=}\frac{\htmlData{tutor-start=23,tutor-end=24}{7}\htmlData{tutor-start=24,tutor-end=25}{a}_{\htmlData{tutor-start=27,tutor-end=28}{n}}\htmlData{tutor-start=29,tutor-end=30}{+}\sqrt{\htmlData{tutor-start=36,tutor-end=37}{4}\htmlData{tutor-start=37,tutor-end=38}{5}\htmlData{tutor-start=38,tutor-end=39}{a}_{\htmlData{tutor-start=41,tutor-end=42}{n}}^{\htmlData{tutor-start=45,tutor-end=46}{2}}\htmlData{tutor-start=47,tutor-end=48}{-}\htmlData{tutor-start=48,tutor-end=49}{3}\htmlData{tutor-start=49,tutor-end=50}{6}}}{\htmlData{tutor-start=53,tutor-end=54}{2}}\htmlData{tutor-start=55,tutor-end=56}{,} \htmlData{tutor-start=57,tutor-end=58}{n}\htmlData{tutor-start=58,tutor-end=62}{\in }\htmlData{tutor-start=62,tutor-end=63}{N}$ . 证明：(1) 对任意 $\htmlData{tutor-start=0,tutor-end=1}{n}\htmlData{tutor-start=1,tutor-end=5}{\in }\htmlData{tutor-start=5,tutor-end=6}{N}\htmlData{tutor-start=6,tutor-end=7}{,} \htmlData{tutor-start=8,tutor-end=9}{a}_{\htmlData{tutor-start=11,tutor-end=12}{n}}$ 为正整数；(2) 对任意 $\htmlData{tutor-start=0,tutor-end=1}{n}\htmlData{tutor-start=1,tutor-end=5}{\in }\htmlData{tutor-start=5,tutor-end=6}{N}\htmlData{tutor-start=6,tutor-end=7}{,} \htmlData{tutor-start=8,tutor-end=9}{a}_{\htmlData{tutor-start=11,tutor-end=12}{n}}\htmlData{tutor-start=13,tutor-end=14}{a}_{\htmlData{tutor-start=16,tutor-end=17}{n}\htmlData{tutor-start=17,tutor-end=18}{+}\htmlData{tutor-start=18,tutor-end=19}{1}}\htmlData{tutor-start=20,tutor-end=21}{-}\htmlData{tutor-start=21,tutor-end=22}{1}$ 为完全平方数。

题解状态：标准答案与规范题解待补充

题目标签：FIRST-SOLUTION-P13

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 · 三、解答题（本题满分 60 分，每小题 20 分） · 平面几何

将编号为 1, 2, $\cdots$ , 9 的九个小球随机放置在圆周的九个等分点上，每个等分点上各有一个小球. 设圆周上所有相邻两球号码之差的绝对值之和为要 S. 求使 S 达到最小值的放法的概率. (注：如果某种放法，经旋转或镜面反射后可与另一种放法重合，则认为是相同的放法)

题解状态：标准答案与规范题解待补充

题目标签：FIRST-SOLUTION-P14

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 · 三、解答题（本题满分 60 分，每小题 20 分） · 数学竞赛/待细分

过抛物线 $\htmlData{tutor-start=0,tutor-end=1}{y}\htmlData{tutor-start=1,tutor-end=2}{=}\htmlData{tutor-start=2,tutor-end=3}{x}^{\htmlData{tutor-start=5,tutor-end=6}{2}}$ 上的一点 A (1, 1) 作抛物线的切线，分别交 $\htmlData{tutor-start=0,tutor-end=1}{x}$ 轴于 D，交 $\htmlData{tutor-start=0,tutor-end=1}{y}$ 轴于 B. 点 C 在抛物线上，点 E 在线段 AC 上，满足 $\frac{\htmlData{tutor-start=6,tutor-end=7}{A}\htmlData{tutor-start=7,tutor-end=8}{E}}{\htmlData{tutor-start=10,tutor-end=11}{E}\htmlData{tutor-start=11,tutor-end=12}{C}}\htmlData{tutor-start=13,tutor-end=14}{=}\htmlData{tutor-start=14,tutor-end=21}{\lambda}_{\htmlData{tutor-start=23,tutor-end=24}{1}}$ ；点 F 在线段 BC 上，满足 $\frac{\htmlData{tutor-start=6,tutor-end=7}{B}\htmlData{tutor-start=7,tutor-end=8}{F}}{\htmlData{tutor-start=10,tutor-end=11}{F}\htmlData{tutor-start=11,tutor-end=12}{C}}\htmlData{tutor-start=13,tutor-end=14}{=}\htmlData{tutor-start=14,tutor-end=21}{\lambda}_{\htmlData{tutor-start=23,tutor-end=24}{2}}$ ，且 $\htmlData{tutor-start=0,tutor-end=7}{\lambda}_{\htmlData{tutor-start=9,tutor-end=10}{1}}\htmlData{tutor-start=11,tutor-end=12}{+}\htmlData{tutor-start=12,tutor-end=19}{\lambda}_{\htmlData{tutor-start=21,tutor-end=22}{2}}\htmlData{tutor-start=23,tutor-end=24}{=}\htmlData{tutor-start=24,tutor-end=25}{1}$ ，线段 CD 与 EF 交于点 P. 当点 C 在抛物线上移动时，求点 P 的轨迹方程.

题解状态：标准答案与规范题解待补充

题目标签：FIRST-SOLUTION-P15

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 · 三、解答题（本题满分 60 分，每小题 20 分） · 平面几何

（本题满分 50 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，设 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{B}\htmlData{tutor-start=2,tutor-end=3}{>}\htmlData{tutor-start=3,tutor-end=4}{A}\htmlData{tutor-start=4,tutor-end=5}{C}$ ，过 $\htmlData{tutor-start=0,tutor-end=1}{A}$ 作 $\htmlData{tutor-start=0,tutor-end=10}{\triangle }\htmlData{tutor-start=10,tutor-end=11}{A}\htmlData{tutor-start=11,tutor-end=12}{B}\htmlData{tutor-start=12,tutor-end=13}{C}$ 的外接圆的切线 $\htmlData{tutor-start=0,tutor-end=1}{l}$ ，又以 $\htmlData{tutor-start=0,tutor-end=1}{A}$ 为圆心， $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{C}$ 为半径作圆分别交线段 $\htmlData{tutor-start=0,tutor-end=1}{A}\htmlData{tutor-start=1,tutor-end=2}{B}$ 于 $\htmlData{tutor-start=0,tutor-end=1}{D}$ ；交直线 $\htmlData{tutor-start=0,tutor-end=1}{l}$ 于 $\htmlData{tutor-start=0,tutor-end=1}{E}$ 、 $\htmlData{tutor-start=0,tutor-end=1}{F}$ 。证明：直线 $\htmlData{tutor-start=0,tutor-end=1}{D}\htmlData{tutor-start=1,tutor-end=2}{E}$ 、 $\htmlData{tutor-start=0,tutor-end=1}{D}\htmlData{tutor-start=1,tutor-end=2}{F}$ 分别通过 $\htmlData{tutor-start=0,tutor-end=10}{\triangle }\htmlData{tutor-start=10,tutor-end=11}{A}\htmlData{tutor-start=11,tutor-end=12}{B}\htmlData{tutor-start=12,tutor-end=13}{C}$ 的内心与一个旁心。（注：与三角形的一边及另两边的延长线均相切的圆称为三角形的旁切圆，旁切圆的圆心称为旁心。）

题解状态：标准答案与规范题解待补充

题目标签：SECOND-GENERAL-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 · 三、解答题（本题满分 60 分，每小题 20 分） · 代数

（本题满分 50 分）设正数 $\htmlData{tutor-start=0,tutor-end=1}{a}$ 、 $\htmlData{tutor-start=0,tutor-end=1}{b}$ 、 $\htmlData{tutor-start=0,tutor-end=1}{c}$ 、 $\htmlData{tutor-start=0,tutor-end=1}{x}$ 、 $\htmlData{tutor-start=0,tutor-end=1}{y}$ 、 $\htmlData{tutor-start=0,tutor-end=1}{z}$ 满足 $\htmlData{tutor-start=0,tutor-end=1}{c}\htmlData{tutor-start=1,tutor-end=2}{y}\htmlData{tutor-start=2,tutor-end=3}{+}\htmlData{tutor-start=3,tutor-end=4}{b}\htmlData{tutor-start=4,tutor-end=5}{z}\htmlData{tutor-start=5,tutor-end=6}{=}\htmlData{tutor-start=6,tutor-end=7}{a}\htmlData{tutor-start=7,tutor-end=8}{,} \htmlData{tutor-start=9,tutor-end=10}{a}\htmlData{tutor-start=10,tutor-end=11}{z}\htmlData{tutor-start=11,tutor-end=12}{+}\htmlData{tutor-start=12,tutor-end=13}{c}\htmlData{tutor-start=13,tutor-end=14}{x}\htmlData{tutor-start=14,tutor-end=15}{=}\htmlData{tutor-start=15,tutor-end=16}{b}\htmlData{tutor-start=16,tutor-end=17}{;} \htmlData{tutor-start=18,tutor-end=19}{b}\htmlData{tutor-start=19,tutor-end=20}{x}\htmlData{tutor-start=20,tutor-end=21}{+}\htmlData{tutor-start=21,tutor-end=22}{a}\htmlData{tutor-start=22,tutor-end=23}{y}\htmlData{tutor-start=23,tutor-end=24}{=}\htmlData{tutor-start=24,tutor-end=25}{c}$ 。求函数 $\htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{x}\htmlData{tutor-start=3,tutor-end=4}{,}\htmlData{tutor-start=4,tutor-end=5}{y}\htmlData{tutor-start=5,tutor-end=6}{,}\htmlData{tutor-start=6,tutor-end=7}{z}\htmlData{tutor-start=7,tutor-end=8}{)} \htmlData{tutor-start=9,tutor-end=10}{=} \frac{\htmlData{tutor-start=17,tutor-end=18}{x}^{\htmlData{tutor-start=20,tutor-end=21}{2}}}{\htmlData{tutor-start=24,tutor-end=25}{1}\htmlData{tutor-start=25,tutor-end=26}{+}\htmlData{tutor-start=26,tutor-end=27}{x}} \htmlData{tutor-start=29,tutor-end=30}{+} \frac{\htmlData{tutor-start=37,tutor-end=38}{y}^{\htmlData{tutor-start=40,tutor-end=41}{2}}}{\htmlData{tutor-start=44,tutor-end=45}{1}\htmlData{tutor-start=45,tutor-end=46}{+}\htmlData{tutor-start=46,tutor-end=47}{y}} \htmlData{tutor-start=49,tutor-end=50}{+} \frac{\htmlData{tutor-start=57,tutor-end=58}{z}^{\htmlData{tutor-start=60,tutor-end=61}{2}}}{\htmlData{tutor-start=64,tutor-end=65}{1}\htmlData{tutor-start=65,tutor-end=66}{+}\htmlData{tutor-start=66,tutor-end=67}{z}}$ 的最小值。

题解状态：标准答案与规范题解待补充

题目标签：SECOND-GENERAL-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 · 三、解答题（本题满分 60 分，每小题 20 分） · 数论

（本题满分 50 分）对每个正整数 $\htmlData{tutor-start=0,tutor-end=1}{n}$ ，定义函数 $\htmlData{tutor-start=0,tutor-end=1}{f}\htmlData{tutor-start=1,tutor-end=2}{(}\htmlData{tutor-start=2,tutor-end=3}{n}\htmlData{tutor-start=3,tutor-end=4}{)} \htmlData{tutor-start=5,tutor-end=6}{=} \begin{cases} \htmlData{tutor-start=21,tutor-end=22}{0} & \text{\htmlData{tutor-start=31,tutor-end=32}{当}} \htmlData{tutor-start=34,tutor-end=35}{n} \text{\htmlData{tutor-start=42,tutor-end=43}{为}\htmlData{tutor-start=43,tutor-end=44}{平}\htmlData{tutor-start=44,tutor-end=45}{方}\htmlData{tutor-start=45,tutor-end=46}{数}}\htmlData{tutor-start=47,tutor-end=48}{,} \\ \htmlData{tutor-start=52,tutor-end=53}{[}\frac{\htmlData{tutor-start=59,tutor-end=60}{1}}{\htmlData{tutor-start=62,tutor-end=64}{\{}\sqrt{\htmlData{tutor-start=70,tutor-end=71}{n}}\htmlData{tutor-start=72,tutor-end=74}{\}}}\htmlData{tutor-start=75,tutor-end=76}{]} & \text{\htmlData{tutor-start=85,tutor-end=86}{当}} \htmlData{tutor-start=88,tutor-end=89}{n} \text{\htmlData{tutor-start=96,tutor-end=97}{不}\htmlData{tutor-start=97,tutor-end=98}{为}\htmlData{tutor-start=98,tutor-end=99}{平}\htmlData{tutor-start=99,tutor-end=100}{方}\htmlData{tutor-start=100,tutor-end=101}{数}}\htmlData{tutor-start=102,tutor-end=103}{.} \end{cases}$ （其中 $\htmlData{tutor-start=0,tutor-end=1}{[}\htmlData{tutor-start=1,tutor-end=2}{x}\htmlData{tutor-start=2,tutor-end=3}{]}$ 表示不超过 $\htmlData{tutor-start=0,tutor-end=1}{x}$ 的最大整数， $\htmlData{tutor-start=0,tutor-end=2}{\{}\htmlData{tutor-start=2,tutor-end=3}{x}\htmlData{tutor-start=3,tutor-end=5}{\}} \htmlData{tutor-start=6,tutor-end=7}{=} \htmlData{tutor-start=8,tutor-end=9}{x} \htmlData{tutor-start=10,tutor-end=11}{-} \htmlData{tutor-start=12,tutor-end=13}{[}\htmlData{tutor-start=13,tutor-end=14}{x}\htmlData{tutor-start=14,tutor-end=15}{]}$ ）。试求： $\sum_{\htmlData{tutor-start=6,tutor-end=7}{k}\htmlData{tutor-start=7,tutor-end=8}{=}\htmlData{tutor-start=8,tutor-end=9}{1}}^{\htmlData{tutor-start=12,tutor-end=13}{2}\htmlData{tutor-start=13,tutor-end=14}{4}\htmlData{tutor-start=14,tutor-end=15}{0}} \htmlData{tutor-start=17,tutor-end=18}{f}\htmlData{tutor-start=18,tutor-end=19}{(}\htmlData{tutor-start=19,tutor-end=20}{k}\htmlData{tutor-start=20,tutor-end=21}{)}$ 的值。

题解状态：标准答案与规范题解待补充

题目标签：SECOND-GENERAL-SOLUTION-P三

解题过程

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