Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

已知数列 \{a_n\},\{b_n\} 满足 a_1=2, b_1=0, a_2=2,b_2=2，且对 n\geq3，有 a_n=a_{n-1}a_{n-2}-b_{n-1}b_{n-2}，b_n=a_{n-1}b_{n-2}+a_{n-2}b_{n-1}. 求 |a_{2022}| 的末两位数字.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设实数 x_1,x_2,\cdots,x_{2018} 满足 x_1+x_2+\cdots+x_{2018}=1007, 且 |x_{i+1}-x_i|\leq1, i=1, 2, \cdots, 2017. 求 x_2+x_4 的最大值.（赵斌供题）

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

将一个 99\times101 的方格表中若干个小方格涂黑，其余小方格涂白. 如果每个黑格至多和一个黑格相邻（有公边），最多能涂黑多少个小方格？

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设实数 a,b 满足：集合 A=\{x\in\bR\ |\ x^2-10x+a\leq0\} 与 B=\{x\in\bR\ |\ bx\leq b^3\} 的交集为 [4,9]，则 a+b 的值为__________.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设 f(x) 是定义域为 \bR、最小正周期为 5 的函数. 若函数 g(x)=f(2^x) 在区间 [0,5) 上的零点个数为 25，则 g(x) 在区间 [1,4) 上的零点个数为__________.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

称集合 A 为无和的，是指：若 a,b\in A，则 a+b\not\in A. 设 A_1, A_2 是 U=\{1,2,\cdots,999\} 的两个无和子集，求 |A_1\cup A_2| 的最大可能值.（吴茁供题）

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

棋盘上有一只羚羊棋子，它可以从格 (x_1,y_1) 跳到格 (x_2,y_2)，当且仅当 \{|x_1-x_2|,|y_1-y_2|\}=\{3,4\}. 将 1\sim10^{12} 任意填在一个 10^6\times10^6 的方格表中，每个数恰填在一个格中. 用 D 表示形如 |a-b| 的数的集合，其中羚羊棋子能从 a,b 所在格中的一个跳到另一个. 求填数的方法数，使得 D 中恰有 4 个元素.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设 a_1<a_2<\cdots<a_{2024} 是正整数等差数列，b_1<b_2<\cdots<b_{2024} 是正整数等比数列. 求同时出现在两个数列中的整数个数的最大可能值.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

若方程 \log_2\left(k\left(x-\df{1}{2}\right)\right)=2\log_2\left(x+\df{1}{2}\right) 仅有一个实根，则满足条件的 k 的最大值等于__________.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

满足 x^4-x^3-2\sqrt{5}x^2-7x^2+\sqrt{5}x+3x+7\sqrt{5}+17=0 的所有正实数 x 的整数部分之和是__________.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

有__________个不超过 2020 的正整数 k，满足对任意的正整数 n，均有3^{(k-1)n+1}\nmid\left(\df{(kn)!}{n!}\right)^2.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

求最大的正整数 n，使得存在正整数数列 a_1, a_2, \cdots, a_n，满足对2\leq k\leq n-1，有 a_{k+1}=\df{a_k^2+1}{a_{k-1}+1}-1.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设集合 A=\{2,3,4,\cdots,4050\}，集合 B=\{(a,b)\ |\ \log_ab+8\log_ba=6,a\in A,b\in A\}，则集合 B 的元素个数为__________.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

已知正整数 n 的所有正因数排列为：1=d_1<d_2<d_3<\cdots，则在 1, 2, 3, \cdots, 2024 中使得 d_{10}=88 的所有数之和为__________.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

实数 a,b,c 满足 ab+bc+ca=44，求 (a^2+4)(b^2+4)(c^2+4) 的最小值.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

若 3 个整数 a,b,c 满足 a^2+b^2+c^2+3<ab+3b+3c，则这样的有序整数组 (a,b,c) 共有__________组.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

将正整数 1,2,\cdots,100 填入 10\times10 方格表中，每个小方格恰好填 1 个数，要求每行从左到右 10 个数依次递减，对 1\leq i\leq 10，记第 i 行的 10 个数之和为 S_i. 设 n\in\{1,2,\cdots,10\} 满足：存在一种填法，使得 S_1, S_2, \cdots, S_{10} 均大于第 n 列上的 10 个数之和，求 n 的最小值.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

如果关于 x 的三次方程 x^3=ax+a+1 有满足 |x|<1000 的偶数根，求实数 a 的个数.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

求最小的正整数 m，使得对任意整数 A,B，存在多项式 f(x)，满足 f(-2)=A,f(-3)=B，且 f(x) 的各项系数都是不超过 m 的非负整数.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

集合 \{0,1,2,\cdots,2012\} 中有多少个元素 k，使得 \text{C}_{2012}^k 是 2012 的倍数？

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

求所有的正整数 n，使得存在正整数 a,b，满足 a^2+b+3 不能被任何质数的立方整除，且n=\df{ab+3b+8}{a^2+b+3}.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

在直径为 \sqrt{2}+1，高为 8 的圆柱形桶内装直径为 1 的球，最多可装__________个.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

数列 \{a_n\} 为：1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 1, 1, 2, 3, 4, \cdots. 即先取 a_1=1，接着复制该项粘贴在后面作为 a_2，并添加后继数 2 作为 a_3，再复制所有项 1, 1, 2 并粘贴在后面作为 a_4, a_5, a_6，并添加后继数 3 作为 a_7, \cdots. 依次继续下去，则 a_{2021}=__________.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

已知不等式 |ax^2+bx+a|\leq x 对 x\in[1,2] 恒成立，则 3a+b 的最大值为__________.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设 S=\{1,2,\cdots,98\}. 求最小正整数 n，使得 S 的任一 n 元子集中都可以选出 10 个数，无论怎样将这 10 个数均分成两组，总有一组中存在一个数与另外 4 个数都互质，而另一组中总有一个数与另外 4 个数都不互质.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设数列 \{a_n\} 满足：(1) a_1 是完全平方数；(2) 对任意正整数 n，a_{n+1} 是使 2^na_1+2^{n-1}a_2+\cdots+2a_n+a_{n+1} 为完全平方数的最小的正整数. 若存在正整数 s，使得 a_s=a_{s+1}=t，求 t 的最小可能值.（1a2b3c）

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设 A 是平面内 6 个点组成的集合，记 n(A) 为经过 A 中至少 3 个点的单位圆的个数. 求 n(A) 的最大可能值.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设 P 是一个凸 2019 边形，任意三条对角线在形内不交于一点. 称 P 的对角线在形内的交点为"结点"，问：最多可以选取多少个结点，使得它们之间不存在一个圈？这里一个圈的任意两个相邻结点必须位于同一条对角线上，且在每条对角线上最多有一个圈上的两个结点.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设 p 是质数且 \df{28^p-1}{2p^2+2p+1} 是整数. 求 2p^2+2p+1 的正因数个数的所有可能值.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

关于 x,y 的方程 x^2+y^2=2023 的整数解所构成的集合 \{x,y\} 的个数为__________.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

若 z=\df{(3i-4)^2(i-\sqrt{7})^8}{(i-1)^{12}}，则 |z|=__________.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

记集合 A=\{1,2,\cdots,2023\}，存在集合 B\subseteq A，且 B 中任意两个元素之和不是 5 的倍数；存在集合 C\subseteq B，且 C 中元素均为完全平方数. 则 |C| 的最大值为__________.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

有 8 块不同的积木，每块积木的形状为方形或者圆形，颜色为红色或者黄色，印有城市名为南昌或者北京. 从 8 块不同的积木中按顺序任取 n 块积木（2\leq n\leq 8），D_n 表示后一次与前一次取出的积木在形状、颜色、城市名三个方面恰好有两个相同的所有不同取法总数，则 \displaystyle\max_{2\leq n\leq 8}D_n=__________.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

求最小的正整数 k，使得对集合 S=\{1,2,\cdots,2012\} 的任意一个 k 元子集 A，都存在 S 的三个不同的元素 a,b,c，使得 a+b, b+c, c+a 均在 A 中.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设多项式 f(x)=x^{2020}+\ds{i=0}{2019}c_ix^i，其中 c_i\in\{-1,0,1\}. 记 N 为 f(x) 的正整数根的个数（含重根）. 若 f(x) 无负整数根，求 N 的最大值.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

某班有 47 个学生，所用教室有 6 排，每排有 8 个座位，用 (i,j) 表示位于第 i 排第 j 列的座位. 新学期准备调整座位，设某学生原来的座位为 (i,j)，如果调整后的座位为 (m,n)，则称该生作了移动 [a,b]=[i-m,j-n]，并称 a+b 为该生的位置数. 所有学生的位置数之和记为 S，求 S 的最大值与最小值之差.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设有理数 r=\df{p}{q}\in(0,1)，其中 p,q 为互质的正整数，且 pq 整除 3600. 这样的有理数 r 的个数为__________..

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

\displaystyle\prod_{k=1}^6(x^5-5x+k)=0 有__________个实根.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

给定 3\times3 整数元矩阵M=\begin{pmatrix} 11&11&20\\ 10&10&-8\\ -10&2&8\end{pmatrix},已知存在整数元矩阵 P,Q，满足 \det P, \det Q\in\{-1,1\}，且 PMQ 为对角矩阵，则 PMQ 的最大元素为__________.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

令 I=\displaystyle\sqrt{\df{2}{\pi}}\int_{-\infty}^{+\infty}e^{x-10\cosh(2x)}\dd x，则 [|\ln I|]=__________.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

一次会议共有 24 名代表参加，每两名代表之间要么握手一次，要么不握手. 会议结束后发现，总共出现了 216 次握手，且对任意相互握过手的两名代表 P,Q，在剩下的 22 名代表中，恰与 P,Q 之一握过手的代表不超过 10 名. 一个"朋友圈"指的是会议中三名两两之间握过手的代表所构成的集合. 求这 24 名代表中朋友圈个数的最小值.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

求最小的正整数 n，使得存在 n 个不同的正整数 s_1, s_2, \cdots, s_n，满足\left(1-\df{1}{s_1}\right)\left(1-\df{1}{s_2}\right)\cdots\left(1-\df{1}{s_n}\right)=\df{51}{2010}.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

对一个立方体，T 为连接两个顶点的线段的集合. 则 T 中垂直线段对的个数减去平行线段对的个数等于__________.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

某马戏团有 n 个小丑，他们每个人均用 12 种不同颜色中的至少 5 种装扮自己. 一天，马戏团经理要求任意两个小丑所用的颜色不能完全相同，并且每种颜色最多只能由 20 个小丑使用. 在满足经理要求的情况下，求 n 的最大值.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

某次会议共有 30 人参加，已知每个人在其余人中至多有 5 个熟人，且任 5 人中存在两人不是熟人. 求最大的正整数 k，使得在满足上述条件的 30 人中总存在 k 个两两不是熟人.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

展览馆中有 2015 幅画在展出. 每位观众选择一对画，并将它们的名字写在同一张纸上. 一个艺术家从这张纸中选择一些画对，并评判每对中哪幅画更好. 之后助手按下述规则继续评判：在每一步中，如果对某个正整数 n，在画对 (A_1,A_2) 中 A_1 比 A_2 好，在画对 (A_2,A_3) 中 A_2 比 A_3 好，\cdots\cdots，在画对 (A_{n-1},A_n) 中 A_{n-1} 比 A_n 好，则助手可以在画对 (A_1,A_n) 中评判 A_1 比 A_n 好. 求最小的正整数 k，使得无论纸上写了哪些画对，艺术家都可以对其中的 k 对评判，使得助手能在有限步内对余下所有的画对评判.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设实数 a,b,c,d 满足 b-d\geq5，实数 x_1,x_2,x_3,x_4 是多项式 P(x)=x^4+ax^3+bx^2+cx+d 的根，求 (x_1^2+1)(x_2^2+1)(x_3^2+1)(x_4^2+1) 的最小值.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

已知一个凸 2010 边形的任意三条对角线不在其内部交于一点. 考虑经过 2010 边形的每个顶点恰一次的闭折线，求该闭折线自交点的数目的最大值.注：闭折线是指由点 P_1,P_2,\cdots,P_n 及线段 P_1P_2, P_2P_3, \cdots,P_nP_1 构成的图形.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

有 10 个互不相同的正实数，考虑它们两两的和与积（各 45 个）. 已知这些和中有 5 个相等，求最大的正整数 k，使得这些积中有 k 个相等.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设数列 \{a_n\} 满足 a_1=8，且对任意正整数 n，均有a_{n+1}=\left(1+\df{3}{n}\right)a_n+\left(1-\df{3}{n}\right)\cdot 2^n,则 a_{2023} 除以 5 的余数是__________.（李纪琛供题）

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设 f(x)\in C^{\infty}(\bR)，满足 f''(x)=xf(x)，且 f(0)=18，f'(0)=3. 设 f(x) 的泰勒展开式为 \ds{n=0}{\infty}a_nx^n，求 \left[\df{a_9}{a_{10}}\right].

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设 S=\int_{-\infty}^{+\infty}\df{\dd x}{1+x^2+x^4}，求 [100S^2].

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

令 f(x)=[x]+[2x]+[3x]+[4x]+[5x]，当 x 取遍 [0,50] 中的实数时，求 f(x) 能取到的整数值的个数.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设S=\{(x,y,z,w)\in\bZ^4\ |\ x,y,z,w\in[0,18],x^2+y^2\equiv z^2+w^2\pmod{19}\},求 |S|.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设 V 为所有次数不超过 100 的实系数多项式构成的线性空间，令V_1=\{f\in V\ |\ f^{(2k-1)}(0)=0,k=1,2,\cdots,50\},V_2=\{f\in V\ |\ f^{(k)}(-1)=f^{(k)}(1)=0,k=70,71,\cdots,98\}.求 \dim(V_1+V_2).

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

对空间直角坐标系 xyz，先以 z 轴为中心轴旋转 \df{\pi}{3}，再以 x 轴为中心轴旋转 \df{\pi}{3}，得到一个以某条直线为中心轴的旋转. 设新的旋转角为 \theta\in(0,\pi]. 求 [\tan^2\theta].

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

对 \bR^{100} 中的点 x=(x_1,x_2,\cdots,x_{100}), y=(y_1,y_2,\cdots,y_{100})，令 f(x,y)=\sqrt{\ds{i=1}{100}(x_i-y_i)^2}. 对 x\neq y，定义 100 阶方阵 A_{x,y}=(a_{ij})，其中 a_{ij}=\df{\partial}{\partial x_i}\df{\partial}{\partial y_j}f(x,y). 求 A_{x,y} 的秩的最小可能值与最大可能值之和，即\min\{\text{rank}(A_{x,y})\ |\ x,y\in\bR^{100},x\neq y\}+\max\{\text{rank}(A_{x,y})\ |\ x,y\in\bR^{100},x\neq y\}.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设函数 f(x)=\ln|x|+|x|-2 的零点都在区间 [a,b] (a,b\in\bZ, a<b) 内，则 b-a 的最小值为__________.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

用 f(X,\Gamma) 表示点 X 与曲线 \Gamma 上任意一点距离的最小值. 已知 \odot O:x^2+y^2=1 及 \odot O_1:(x-4)^2+y^2=4. 设 P 为 \odot O 上的动点，则 f(P,\odot O_1) 的最大值为__________.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

若 T=\ds{k=1}{100}2^{99+k}\cdot3^{101-k}，则 T 的末尾数字 0 的个数为__________.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

记 I=120\displaystyle\int_0^1\sqrt{\df{1-x}{1+x}}\ \dd x，则 [I]=__________.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

记 I=\ds{m=1}{\infty}\ds{n=1}{\infty}\df{1}{n!n!+m!n!}，则 [10I]=__________.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

一个 10\times10 的数表初始每个位置的数字均为 1，现在做有限次操作，每次把某行或某列的所有数换成相反数. 称非负整数 N 是好的，如果最终数表中 -1 的个数恰为 N. 则好的 N 的个数为__________.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设 A 是正交矩阵（即 AA^T=A^TA=I），且 A 的每个矩阵元均为 \df{1}{2} 或 -\df{1}{2}. 则这样的 A 的个数为__________.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

已知 10 阶实方阵 A 满足 A^5=0，则 A 的秩的所有可能值之和为__________.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

若实数列 \{a_n\} 满足：(1) a_0\in\bN_+；(2) 对任意非负整数 i，a_{i+1}=2a_i+1 或 a_{i+1}=\df{a_i}{a_i+2}；(3) 存在正整数 k，使得 a_k=2014，则称数列 \{a_n\} 为"好的".求最小的正整数 k，使得存在一个好的实数列 \{a_n\}，满足 a_k=2014.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设非负整数 a_1,a_2,\cdots,a_{2022} 的和为 2022. 用 x 表示满足 a_i+a_{i+1}\geq3 的 i 的个数；用 y 表示满足 a_i\neq a_{i+1} 的 i 的个数，其中 i\in\{1,2,\cdots,2022\}，a_{2023}=a_1. 求 x+y 的最大可能值.（张端阳供题）

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设 \{a_i\}_{i=0}^{\infty} 是正整数数列，满足 a_0=1，且对任意正整数 n，a_n 是最小的正整数，使得 a_n>1 且\sigma(a_0a_1\cdots a_{n-1})\mid\sigma(a_0a_1\cdots a_n).问：2024^{2024} 有多少个因数在数列中出现？

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设 \left[\df{10^{2022}}{10^{100}-9}\right] 能被 10^n 整除，则非负整数 n 的最大可能值为__________.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设 a_n=\left(\df{1+q}{1-q}\right)\cdot\left(\df{1+q^2}{1-q^2}\right)^{\frac{1}{2}}\cdot\left(\df{1+q^4}{1-q^4}\right)^{\frac{1}{4}}\cdot\cdots\cdot\left(\df{1+q^{2^n}}{1-q^{2^n}}\right)^{\frac{1}{2^n}}，其中 q=\df{11}{13}. 记 a=\displaystyle\lim_{n\rightarrow+\infty}a_n，则 [a]=__________.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

记 I=\displaystyle\int_{\frac{\pi}{3}}^{\frac{\pi}{2}}\df{\dd x}{\sin x}，则 [100I]=__________.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设函数 \varphi_1(x),\varphi_2(x) 满足\varphi_i''(x)-\ln(x+1)\varphi_i'(x)+e^x\varphi_i(x)=0,i=1,2,x>-1.定义 w(x)=\varphi_1'(x)\varphi_2(x)-\varphi_1(x)\varphi_2'(x)，若 w(0)=1，则 [w(2)]=__________.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

已知欧拉常数\gamma=\displaystyle\lim_{n\rightarrow+\infty}\left(1+\df{1}{2}+\df{1}{3}+\cdots+\df{1}{n}-\ln n\right)约等于 0.5772156649. 记 I=\df{100}{\displaystyle\int_0^1\left(\df{1}{x}-\left[\df{1}{x}\right]\right)\dd x}，则 [I]=__________.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设 V 为关于 x 的次数小于或等于 10 的实多项式构成的集合，且 a 为实数. 定义 V 到 V 的线性映射 F 为 F(f)=(x^2f)''+f'-af. 记 I 为使得 F 不是满射的 a 的所有可能值之和，则 [I]=__________.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

对正整数 n，考虑含有 n 阶元的有限非交换群，记 a_n 为这些群的阶的最小可能值，则 a_1+a_2+a_3+a_4 的值为__________.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设 n 是正整数，3^n-1 没有 12 以上的质因子，则所有满足条件的 n 的和是__________.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

已知四面体 PABC，点 A_1 在 \triangle PBC 内，满足 \triangle A_1BP, \triangle A_1CP, \triangle A_1BC 的面积之比为 3:2:1. G 在线段 AA_1 上，直线 PG 交平面 ABC 于点 M，且 \df{AG}{GA_1}=\df{PG}{GM}，则四面体 PABC 与 A_1AMB 的体积之比为__________.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

若方格 P 的左右两侧分别有 a,b 个方格，上下两侧分别有 c,d 个方格，其中 a,b,c,d 是正整数，且满足 (a-b)(c-d)=0，则称由这些方格所组成的图形为一个"十字星".现有一张由 2014 个方格所组成的 38\times53 方格表，求该方格表中十字星的个数.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

已知 a_k 为 \log_2k, \log_3k, \cdots, \log_{2018}k 中整数项的个数，则 \ds{k=1}{2018}a_k=__________.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

若关于 x 的不等式 (x-2)e^x<ax+1 有且仅有三个不同的整数解，则整数 a 的最小值为__________.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

若正实数 x,y 满足 x(x+2y)=9，则 x^5y 的最大值为__________.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

若整数 a,b,c 满足：0\leq a\leq 10, 0\leq b\leq 10, 0\leq c\leq 10，10\leq a+b+c\leq 20，则满足条件的有序数组 (a,b,c) 共有__________.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

满足下述条件的最小正实数 k：对任意不小于 k 的 4 个互不相同的实数 a,b,c,d，都存在 a,b,c,d 的一个排列 p, q, r, s，使得方程 (x^2+px+q)(x^2+rx+s)=0 有 4 个互不相同的实根.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

n 支球队进行主客场双循环比赛（即每两支球队比赛两场，其中各有一场主场比赛）. 已知每支球队在一周（从周日到周六的七天）内可以进行多场客场比赛，但如果某周内某球队有主场比赛，那么在这一周内不能安排该球队的客场比赛. 若 4 周内能够完成全部比赛，求 n 的最大值.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

梦境国有 2016 座城市，星程航空公司希望在若干座城市对之间开通单向航线，使得每座城市恰有一班飞离的航线. 求最小的正整数 k，使得该公司无论怎样开通航线，总能将 2016 座城市分成 k 组，满足每组中任意一座城市不可能用不超过 28 次飞行到达这一组中的另一座城市.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

在数列 \{x_n\}_{n=1}^{+\infty} 中，x_1=1，且对任意正整数 n，x_{n+1}=x_n+[\sqrt{x_n}]，这里 [x] 表示不超过 x 的最大整数. 则满足 x_n\geq2019 的最小正整数 n 为__________.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设 m,n\in\bN，且 m,n\geq2，若 \df{2^3-1}{2^3+1}\cdot\df{3^3-1}{3^3+1}\cdot\cdots\cdot\df{n^3-1}{n^3+1}=\df{m^3-1}{m^3+2}, 则 m^2+n^2 的值为__________.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设正整数 a_1,a_2,\cdots,a_{31},b_1,b_2,\cdots,b_{31} 满足：(1) a_1<a_2<\cdots<a_{31}\leq2015, b_1<b_2<\cdots<b_{31}\leq 2015；(2) a_1+a_2+\cdots+a_{31}=b_1+b_2+\cdots+b_{31}.求 S=|a_1-b_1|+|a_2-b_2|+\cdots+|a_{31}-b_{31}| 的最大值.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

一项赛事共有 100 位选手参加. 对于任意两位选手 x,y，他们之间恰比赛一次且分出胜负，用 x\rightarrow y 表示 x 战胜 y. 如果对任意两位选手 x,y，均能找到某个选手序列 u_1, u_2, \cdots, u_k (k\geq2)，使得 x=u_1 \rightarrow u_2 \rightarrow \cdots \rightarrow u_k=y，那么称该赛事结果是"友好"的.(1) 求证：对任意一个友好的赛事结果，存在正整数 m 满足如下条件：对任意两位选手 x,y，均能找到某个长度为 m 的选手序列 z_1,z_2,\cdots,z_m（这里 z_1, z_2, \cdots, z_m 可以有重复），使得 x=z_1\rightarrow z_2\rightarrow\cdots\rightarrow z_m=y.(2) 对任意一个友好的赛事结果 T，将符合 (1) 中条件的最小正整数 m 记为 m(T). 求 m(T) 的最小值.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

记 m=7997\times7999^3-7998\times7996^3，求 \sqrt[3]{m}.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

有 n 个人排成一行，自左至右按 1,2,\cdots,n 的顺序报数，凡序号为平方数者退出队伍；剩下的人自左至右再按 1,2,3,\cdots 的顺序重新报数，凡序号为平方数者退出队伍；如此继续. 在此过程中，每个人都将先后从队伍中退出.用 f(n) 表示最后一个退出队伍的人在最初报数时的序号，求 f(n) 的表达式，并给出 f(2019) 的值.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

前 100 个正整数中，能写成两个正整数平方差的数的总和为__________.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

若在 1,2,3,\cdots,18 的任意一个排列中，总能找到连续 6 个数之和不小于 m，则实数 m 的最大值为__________.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

若长方体的长、宽、高都是自然数，且所有棱长之和等于它的体积，则称此长方体为"完美长方体". "完美长方体''的体积的最大值为__________.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

若正数 x,y 满足 xy=4000，且 (\lg x-\lg2)(\lg y-\lg2)=\df{17}{36}，则 x,y 中较大数的整数部分的位数为__________.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

给定一个边长为 2021 的立方体. 有多少种不同的方式可以在立方体边界的某处添加一个单位小立方体，使得新得到的几何体可以分割为若干 1\times1\times k 的长方体，其中 k 是某个大于 1 的整数？

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

将正 2017 边形用其 2014 条在内部不相交的对角线划分成 2015 个三角形. 求其中等腰三角形个数的最大可能值.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

有 2022 个格排成一行，A,B 两人在玩游戏. 初始时，在每个左起奇数号格中填入 A、偶数号格中填入 B. 接下来从 A 开始，两人轮流进行以下操作：选择两个写有自己名字的格，这两个格不相邻，且它们之间的所有格都写有另一人的名字，然后将这些格都改为自己的名字. 当某人不能操作时游戏结束. 求最大的正整数 m，使得无论 B 如何操作，A 能保证游戏结束时总有至少 m 个格写有 A 的名字.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

某班有 32 名学生，班上有 10 个兴趣小组，每个兴趣小组恰好有 16 名该班学生参加. 对于任意两名学生，将他们两人中恰有一人参加的兴趣小组的个数的平方称为这两名学生的兴趣差. 设 S 是所有 \text{C}_{32}^2 个两人组的兴趣差之和，求 S 的最小可能值.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设方程 xyz=900 的全部正整数解为 (x_i,y_i,z_i) (1\leq i\leq n). 则 \ds{k=1}{n}(x_k+y_k+z_k)=\underline{\qquad\qquad}.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设集合 S=\{1,2,\cdots,2005\}. 若 S 中任意 n 个两两互质的数中都至少有一个质数，求 n 的最小值.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设 I=\displaystyle\int_{\frac{\pi}{3}}^{\frac{\pi}{2}}\frac{1}{\sin x}\ \dd x，则 [100I]=__________.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设地球是一个半径为 1 的球，点 A 在北纬 60 度、东经 30 度，点 B 在南纬 30 度、西经 90 度. 记 A 到 B 的球面距离为 \theta，则 \left[100\cos^2\theta\right]=__________.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

已知平面直角坐标系中的点集L=\{(x,y)\ |\ x,y\in\{1,2,\cdots,100\}\}.设 A 是由平面上若干个凸多边形组成的集合，满足：A 中每个凸多边形的所有顶点均属于 L，并且 L 中的每个点都恰好是 A 中一个凸多边形的顶点.求 A 中所有凸多边形的面积之和的最小可能值.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

称一个正整数是好的，如果它有三个不同的正因数的和等于 2022. 求最小的好数.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

求最大的正整数 m，使得存在集合 \{1,2,\cdots,100\} 的 m 个子集 A_1, A_2, \cdots, A_m，满足对任意 1\leq i<j\leq m，都有|A_i\cup A_j|-|A_i\cap A_j|=2.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

一个小镇上共有 2023 位居民，他们成立了一些互不相同的俱乐部 A_1, A_2, \cdots, A_n，每个俱乐部至少一人，每个人可以在任意个不同的俱乐部内. 已知对于每个俱乐部 A_i，存在一个不在其中的居民 b_i，考虑所有 b_i 未加入的俱乐部，它们要么为 A_i，要么和 A_i 没有公共成员. 求该小镇上俱乐部个数的最大值.（程国根）

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设 f(x)=\ds{i=1}{2013}\left[\df{x}{i!}\right]. 对整数 n，若关于 x 的方程 f(x)=n 有实数解，则称 n 为"好数". 求 \{1,3,5,\cdots,2013\} 中好数的个数.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

已知多项式 f(z)=z^3+az^2+bz+c(a,b,c\in\bZ) 的所有根的模均为 20 或 15，则这样的多项式的个数为__________.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设函数 f:\{1,2,\cdots,2019\}\rightarrow\{-1,1\}，满足对任意 1\leq k\leq 2019，存在 1\leq l\leq 2019，使得\ds{i:(l-i)(i-k)\geq0}{}f(i)\leq0.求 \ds{i=1}{2019}f(i) 的最大值.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设 S 是一个 35 元集合，\mathcal{F} 是一个由一些 S 到自身的映射组成的集合. 对正整数 k，称 \mathcal{F} 具有性质 P(k)，如果对任意 x,y\in S，都存在 \mathcal{F} 中的 k 个映射f_1, f_2, \cdots, f_k（可以相同），使得f_k(\cdots(f_2(f_1(x)))\cdots)=f_k(\cdots(f_2(f_1(y)))\cdots).求最小的正整数 m，使得所有具有性质 P(2019) 的集合 \mathcal{F} 都具有性质 P(m).

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

在一次数学奥林匹克比赛中，学生获得的分数来自四部分：代数、几何、数论和组合. 任意两名学生在所有四部分中的分数互不相同. 若一组学生能在至少两部分的分数同时按递增的次序排列，则称该组学生是"好的". 求最小的正整数 N，使得任意 N 名学生中均存在 10 名学生是好的.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设 a_1,a_2,a_3,a_4,k 是两两不同的正整数，且不小于 80，满足 a_1^2+a_2^2+a_3^2+a_4^2-4k^2 是正整数. 求 (a_1^2+a_2^2+a_3^2+a_4^2-4k^2)\cdot k^2 的最小值.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

若一个有理数的十进制表示的小数部分有限，则称它是"短的". 对一个正整数 m，若正整数 t 满足：存在 c\in\{1,2,\cdots,2017\}，使得 \df{10^t-1}{cm} 是短的，且对任意 1\leq k<t，均有 \df{10^k-1}{cm} 不是短的，则称 t 为"m-型"的.用 S(m) 表示所有 m-型的数构成的集合，求 |S(m)| 的最大值.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

求最小的正整数 n，使得存在整数 a_1,a_2,\cdots,a_n，满足二次三项式x^2-2(a_1+a_2+\cdots+a_n)^2x+(a_1^4+a_2^4+\cdots+a_n^4+1)至少有一个整根.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

记 S=\displaystyle\lim_{n\rightarrow\infty}\int_{1}^{2023}(\cos(nx+n!))^4\ \dd x，求 [S].

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设 g(x)=\displaystyle\int_x^{\pi}\df{(\sin t)^5}{t}\ \dd t，记 S=\displaystyle\int_0^{\pi}g(x)\ \dd x，求 [100S].

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设方阵M=\begin{bmatrix}10&-9&0&0&0&0&0&-9\\ -9&10&-9&0&0&0&0&0\\ 0&-9&10&-9&0&0&0&0\\ 0&0&-9&10&-9&0&0&0\\ 0&0&0&-9&10&-9&0&0\\ 0&0&0&0&-9&10&-9&0\\ 0&0&0&0&0&-9&10&-9\\ -9&0&0&0&0&0&-9&10\end{bmatrix}的特征值从小到大为 \lambda_1\leq\lambda_2\leq\cdots\leq\lambda_8，求 [|\lambda_6|].

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

矩阵微分方程 \df{\dd }{\dd t}M(t)=B(t) 代表每个元素都满足微分方程 \df{\dd}{\dd t}M_{ij}(t)=B_{ij}(t). 考虑矩阵微分方程 \df{\dd}{\dd t}M(t)=AM(t)，其中矩阵 A 和初始矩阵 M(0) 分别为A=\begin{bmatrix}2&1&3\\ 1&3&2\\ 3&2&1\end{bmatrix},\quad M(0)=\begin{bmatrix} 3&2&1\\ 2&1&3\\ 1&3&2\end{bmatrix}.令 a=\ln|\det(M(2))|，求 [|a|].

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设 A 是 5\times5 的方阵，A 的第一行是 (1,1,1,1,1)，第二行是 (1,-1,1,-1,1). 已知 \text{tr}(AA^T)=28，求 [\det A] 的最大可能值.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设 n=5，L_n(x)=\df{1}{n!}\cdot\df{\dd^n}{\dd x^n}(x^2-1)^n. 记 S=\displaystyle\int_{-1}^1|L_n(x)|^2\ \dd x，求 [S].

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设方程(1+2+\cdots+k)-3(1+2+\cdots+t)=1的正整数解 (k,t) 按 k+t 从小到大排列为 (k_1,t_1), (k_2,t_2), \cdots. 求 k_6.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

已知函数 f:\{1,2,\cdots,10\}\rightarrow\{1,2,3,4,5\}，且对一切 k=1, 2, \cdots, 9，有 |f(k+1)-f(k)|\geq 3. 则符合条件的函数 f 的个数为__________.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

已知函数 f(x) 在 (0,+\infty) 上严格单调递减，对任意 x\in(0,+\infty)，均有 f(x)f\left(f(x)+\df{2}{x}\right)=\df{1}{3}. 记 g(x)=f(x)+4x^2，则函数 g(x) 的最小值是__________.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

A5. Alice and Bob play a game on a board consisting of one row of 2022 consecutive squares. They take turns placing tiles that cover two adjacent squares, with Alice going first. By rule, a tile must not cover a square that is already covered by another tile. The game ends when no tile can be placed according to this rule. Alice's goal is to maximize the number of uncovered squares when the game ends; Bob's goal is to minimize it. What is the greatest number of uncovered squares that Alice can ensure at the end of the game, no matter how Bob plays?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

若 x_1,x_2,\cdots,x_{100} 是 1,2,\cdots,100 的一个排列，则 S=|x_1-x_2|+|x_2-x_3|+\cdots+|x_{99}-x_{100}|+|x_{100}-x_1| 的最大值为__________.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

有 2018 名学生围成一圈. 称一个同学是优秀的，如果他两边站着的人和自己的性别都不同. 求优秀的男生数与优秀的女生数的平方差的最大值.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

正实数 x,y 满足如下条件：存在 a\in[0,x], b\in[0,y]，使得a^2+y^2=2,\quad b^2+x^2=1,\quad ax+by=1,则 x+y 的最大值为__________.原始的答案是k\sqrt{m}+n的形式，请给出k + m + n的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

已知 a_1,a_2,\cdots,a_{22}\in[1,2]，求\df{\ds{i=1}{22}a_ia_{i+1}}{\left(\ds{i=1}{22}a_i\right)^2}的最大值，其中 a_{23}=a_1.（王正供题）原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

将一枚骰子连续投掷五次，则事件"五次出现的点数既不全相同，也不两两互异，且从第二次起每一次的点数都不小于前一次的点数"的概率为__________.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

若等差数列 \{a_n\} 及正整数 m(m\geq3) 满足：a_1=1,a_m=2，且\df{1}{a_1a_2}+\df{1}{a_2a_3}+\cdots+\df{1}{a_{m-1}a_m}=3,则 a_1+a_2+\cdots+a_m 的值为__________.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设实数 x,y,z\in[0,1]，记M=\min\{x-xy-xz+yz,y-yx-yz+xz,z-zx-zy+xy\},求 M 的最大可能值.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

一个骰子连续掷两次，得到的点数依次为 a 和 b，则使得关于 x 的三次方程 x^3-(3a+1)x^2+(3a+2b)x-2b=0 有三个互不相等的实数根的概率 P=__________.（用数字作答）原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

若复数 z 满足 \left|\df{z^2+1}{z+i}\right|+\left|\df{z^2+4i-3}{z-i+2}\right|=4，则 |z-1| 的最小值为__________.原始的答案是k\sqrt{m}+n的形式，请给出k + m + n的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

已知 x 是一个锐角，那么 \df{8}{\sin x}+\df{1}{\cos x} 的最小值是__________.原始的答案是k\sqrt{m}+n的形式，请给出k + m + n的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

将一枚均匀的骰子独立投掷三次，所得的点数依次记为 x,y,z，则事件"\text{C}_7^x<\text{C}_7^y<\text{C}_7^z"发生的概率为__________.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

若平面上非零向量 \overrightarrow{\alpha},\overrightarrow{\beta},\overrightarrow{\gamma} 满足 \overrightarrow{\alpha}\bot\overrightarrow{\beta}，\overrightarrow{\beta}\cdot\overrightarrow{\gamma}=2|\overrightarrow{\alpha}|，\overrightarrow{\gamma}\cdot\overrightarrow{\alpha}=3|\overrightarrow{\beta}|，则 |\overrightarrow{\gamma}| 的最小值为__________.原始的答案是k\sqrt{m}+n的形式，请给出k + m + n的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

从 1,2,\cdots,9 中任选 3 个数，它们两两互质的概率为__________.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设函数 f(x)=|x\log_{2023}x|，实数 a<b<c 满足 f(a)=f(b)=f(c). 若 a, b, c 构成等比数列，则其公比为__________.原始的答案是k\sqrt{m}+n的形式，请给出k + m + n的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

在平面直角坐标系中，点 A,B,C 在双曲线 xy=1 上，满足 \triangle ABC 为等腰直角三角形. 求 \triangle ABC 的面积的最小值.原始的答案是k\sqrt{m}+n的形式，请给出k + m + n的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

在平面直角坐标系 xOy 中，点集 K=\{(x,y)\ |\ x,y=-1,0,1\}. 在 K 中随机取出三个点，则这三个点两两之间的距离均不超过 2 的概率为________.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

正实数 x,y,z 满足 x+2y^2+4x^2y^2z^2=8，则 \log_4x+\log_2y+\log_8z 的最大值为__________.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

已知直三棱柱 ABC-A_1B_1C_1 的底面为直角三角形，\angle ACB=90^\circ，AC=6，BC=CC_1=\sqrt{2}. P 是 BC_1 上一动点，则 CP+PA_1 的最小值是__________.原始的答案是k\sqrt{m}+n的形式，请给出k + m + n的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

求最小常数 \lambda，使不等式xy(x^2+y^2)+yz(y^2+z^2)+zx(z^2+x^2)\leq \lambda(x+y+z)^4对所有非负实数 x,y,z 都成立.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

在正四面体 P-ABC 中，点 E,F 分别在棱 PB, PC 上. 若 PE\neq PF，且 AE=AF=2, EF=\sqrt{3}，则四面体 P-AEF 的体积为__________.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

在平面直角坐标系 xOy 中，A 为椭圆 \df{x^2}{4}+y^2=1 上一点，M 为线段 OA 上的动点. 过 M 作直线与椭圆交于 P, Q 两点，若 \overrightarrow{PM}=2\overrightarrow{MQ}，求四边形 OPAQ 面积的最大值.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设 x,y,z 是正实数，求2\sqrt{(x+y+z)\left(\df{1}{x}+\df{1}{y}+\df{1}{z}\right)}-\sqrt{\left(1+\df{x}{y}\right)\left(1+\df{y}{z}\right)}的最小值.原始的答案是k\sqrt{m}+n的形式，请给出k + m + n的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

已知非负实数 a_i,b_i(1\leq i\leq 2024) 满足 \ds{i=1}{2024}a_i=\ds{i=1}{2024}b_i=1. 求\ds{i=1}{2024}a_ib_{i+1}b_{i+2}b_{i+3}a_{i+4}的最大值，其中下标按模 2024 理解.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

将 1,2,3,4,5,6 随机排成一行，记为 a, b, c, d, e, f，则 abc+def 是偶数的概率为________.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设复数 z 满足 |z|=1，且使得关于 x 的方程 zx^2+2\bar{z}x+2=0 有实根. 则这样的 z 的和为________.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

将一枚硬币和一个骰子同时投掷，硬币出现正面记为 2、出现反面记为 1. 此数与骰子的点数之积记为 x（例如硬币出现正面，骰子点数为 3，则 x=2\times3=6），那么 x 的数学期望是__________.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

已知 x,y\in[0,+\infty)，且满足 x^3+y^3+6xy=8. 则 2x^2+y^2 的最小值为__________.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

已知 x_1,x_2,\cdots,x_{50} 是实数，且 \ds{i=1}{50}\df{x_i}{k+i}=\df{2021}{2k+1} 对任意 k\in\{1,2,\cdots,50\} 成立. 求 \ds{i=1}{50}\df{x_i}{4042i+2021} 的值.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

已知实数 x,y 满足 x<y,|x|<3,|y|<3，且关于 t 的方程 (x^3-y^3)t^4+(3x+y)t^2+\df{1}{x-y}=0 有实根. 则点 (x,y) 构成的区域的面积为__________.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

在平面直角坐标系 xOy 中，抛物线 C:y^2=4x 的焦点为 F，过 C 上一点 P（异于 O）作 C 的切线与 y 轴交于点 Q. 若点 Q 关于直线 PF 的对称点在 x 轴上，则 \triangle PQF 的面积为__________.原始的答案是k\sqrt{m}+n的形式，请给出k + m + n的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

盒子中有大小、形状完全相同的 3 个红球和 3 个白球. 现抛掷一枚质地均匀的骰子，掷出几点就从盒子中取出几个球. 则取出的球中红球个数大于白球个数的概率为_________.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

已知双曲线 C:\df{x^2}{a^2}-\df{y^2}{b^2}=1(a>0,b>0) 的离心率为 \df{\sqrt{17}}{3}，F 为右焦点，点 A,B 在右支上. 设 D 为 A 关于原点 O 的对称点，且 DF\bot AB. 若 \overrightarrow{AF}=\lambda\overrightarrow{FB}，则 \lambda=_________.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设实数 a,b,c,d 满足 a^2+b^2+c^2+d^2=1. 求 (a-b)(b-c)(c-d)(d-a) 的最小值，并求达到最小值时所有的 (a,b,c,d).原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

已知函数 f(x)=\cos^4x+\sin^4x+a\sin4x-b，且 f(x+\frac{\pi}{6}) 为奇函数. 若方程 f(x)+m=0 在 [0,\pi] 上有四个不同的实数解 x_1, x_2, x_3, x_4，则 f(\frac{x_1+x_2+x_3+x_4}{4}) 的平方值为__________.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

从面积为 1 的正 7 边形的 7 个顶点中随机选出 4 个不同的顶点，它们构成的四边形的面积大于 \df{1}{2} 的概率是__________.（王正供题）原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设复数 x,y,z 满足：|x+2y+3z|=1. 求|x|^2+|y|^2+|z|^2+|x^2+y^2+z^2|的最小值.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

点 M,N 分别在 \triangle ABC 的边 AB,AC 上，且\overrightarrow{AM}=x\overrightarrow{AB},\quad \overrightarrow{AN}=y\overrightarrow{AC}.D 为线段 BC 的中点，G 为线段 MN 与 AD 的交点. 若 \overrightarrow{AG}=\df{3}{5}\overrightarrow{AD}，则 \df{1}{x^2}+\df{4}{y^2} 的最小值为__________.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

若 \triangle ABC 的内角 A,B,C 满足 \sin A=\cos B=\tan C，求 \cos^3A+\cos^2A-\cos A 的值.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

直角三角形 DEF 的三个顶点分别在等边三角形 ABC 的边 AB, BC, CA 上，且 \angle DEF=90^\circ, \angle EDF=30^\circ. 求 \df{S_{\triangle DEF}}{S_{\triangle ABC}} 的最小值.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

求具有下述性质的最小正数 c：对任意整数 n\geq4，以及集合 A\subseteq\{1,2,\cdots,n\}，若 |A|>cn，则存在函数 f:A\rightarrow\{-1,1\}，满足\left|\ds{a\in A}{}f(a)\cdot a\right|\leq1.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

求最大的实数 k，使得对任意正实数 a,b，均有(a+b)(ab+1)(b+1)\geq kab^2.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

将一个正方形分割成有限个白色和红色矩形，其边均与正方形的边平行. 在每个白色矩形内，写下它的宽与高的比值；在每个红色矩形内，写下它的高与宽的比值. 我们计算出所有矩形内写的数的总和 x. 如果两种颜色矩形的总面积相同，求 x 的最小可能值.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设 y^2=4x 的三条切线的斜率为 1,2,3，则三条切线围成的三角形的外接圆直径为__________.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

若平面向量 \overrightarrow{a}=(2^m,-1) 与 \overrightarrow{b}=(2^m-1,2^{m+1}) 垂直，其中 m 为实数，则 \overrightarrow{a} 的模为________.原始的答案是k\sqrt{m}+n的形式，请给出k + m + n的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设 x\in[-\frac{\pi}{6},\frac{\pi}{6}]，则函数 f(x)=\sin^2x-2\sin x 的最大值为__________.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设实数 a_1,a_2,a_3,a_4,a_5 满足，对 k=1,2,3,4,5，\df{a_1}{k^2+1}+\df{a_2}{k^2+2}+\df{a_3}{k^2+3}+\df{a_4}{k^2+4}+\df{a_5}{k^2+5}=\df{1}{k^2}.求 \df{a_1}{37}+\df{a_2}{38}+\df{a_3}{39}+\df{a_4}{40}+\df{a_5}{41} 的值.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设非负实数 x_1,x_2,\cdots,x_{2018} 满足 \ds{1\leq i<j\leq 2018}{}x_ix_j=1. 求 \ds{i=1}{2018}\df{1}{s-x_i} 的最小值，其中 s=\ds{i=1}{2018}x_i.（赵斌、王坤供题）原始的答案是k\sqrt{m}+n的形式，请给出k + m + n的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设函数 f(x)=A\sin(\omega x+\varphi) (A\neq0,\omega>0,0<\varphi<\df{\pi}{2}). 若 f(\df{5\pi}{6})+f(0)=0，则 \omega 的最小值为__________.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

已知正实数 a,b 满足 a^b=(8a)^{9b}，则 \log_a(4a^2) 的值为__________.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

某校三个年级举办乒乓球比赛，每个年级选派 4 名选手参加比赛. 组委会随机将这 12 名选手分成 6 组，每组 2 人，则在上述分组方式中每组的 2 人均来自不同年级的概率为__________.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设实数 a,b,c\in(0,2]，且 b\geq 3a 或 a+b\leq\df{4}{3}，则 \max\{b-a,c-b,4-2c\} 的最小值为__________.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

在椭圆 \Gamma 中，A 为长轴的一个端点，B 为短轴的一个端点，F_1,F_2 为两个焦点. 若 \overrightarrow{AF_1}\cdot\overrightarrow{AF_2}+\overrightarrow{BF_1}\cdot\overrightarrow{BF_2}=0，求 \tan\angle ABF_1\cdot\tan\angle ABF_2 的值.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

求最大的实数 c，使得对平面上任意两个单位向量 \overrightarrow{v_1}, \overrightarrow{v_2}，均存在单位向量 \overrightarrow{v}，满足|\overrightarrow{v}\cdot\overrightarrow{v_1}|\cdot|\overrightarrow{v}\cdot\overrightarrow{v_2}|\geq c.（胡珏伟供题）原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设集合 S=\{1,2,3,4,5\}. 若 S 的子集 A 满足：若 x\in A，则 6-x\in A，则称子集 A 具有性质 p. 现从 S 的所有非空子集中，等可能地取出一个，则所取出的非空子集具有性质 p 的概率为__________.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设非负实数 a,b,c,x,y,z 满足 a+b+c=x+y+z=1. 求 (a-x^2)(b-y^2)(c-z^2) 的最大值.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设正实数 u,v,w 满足 u\sqrt{vw}+v\sqrt{wu}+w\sqrt{uv}\geq1. 求 u+v+w 的最小值.原始的答案是k\sqrt{m}+n的形式，请给出k + m + n的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

函数 y=\df{1}{3}\cos2x+\sin x-\cos x 在区间 \left[-\df{\pi}{2},\df{\pi}{2}\right] 上的最大值为__________.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

在平面直角坐标系中，甲从原点出发，到点 (4,4)；乙从点 (2,0) 出发，到点 (6,4). 已知两人同时出发，且每次向上或向右走一步. 则甲、乙的行走路线没有公共点的概率为__________.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

求最小的实数 C，使得对任意正实数 a_1, a_2, a_3, a_4, a_5（允许相同），总可以选择不同的下标 i,j,k,l，满足 \left|\df{a_i}{a_j}-\df{a_k}{a_l}\right|\leq C.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

求最大的实数 a，使得对任意正整数 n 及任意实数 0=x_0<x_1<\cdots<x_n，均有\ds{i=1}{n}\df{1}{x_i-x_{i-1}}\geq a\ds{i=1}{n}\df{i+1}{x_i}.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设非负实数 a,b,c,d 满足 a+b+c+d=4. 求\df{a}{b^3+4}+\df{b}{c^3+4}+\df{c}{d^3+4}+\df{d}{a^3+4}的最小值.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

若无穷等比数列 \{a_n\} 的各项和为 1，各项的绝对值之和为 2，则首项 a_1 的值为__________.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

在 1,2,\cdots,10 中随机选出三个不同的数，它们两两互素的概率为__________.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设 a,b 是从集合 \{1,2,3,4,5\} 中随机选取的数，则直线 y=ax+b 与圆 x^2+y^2=2 有公共点的概率是__________.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设 n 是正整数，A_1,A_2,\cdots,A_n 是 n 个不同的非空有限集，满足对任意 1\leq i,j,k\leq n，若 A_i\cap A_j=\emptyset，则存在 1\leq l\leq n，使得(A_i\cup A_k)\setminus A_j=A_l.求最大的实数 c，使得对任意正整数 n 和任意满足上述性质的集合 A_1, A_2, \cdots, A_n，存在 I\subseteq\{1,2,\cdots,n\}，使得 (1) |I|\geq cn，(2) 对任意 i,j\in I，A_i\cap A_j\neq\emptyset.（王一川供题）原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

已知 f(x) 是定义在 \bR 上的函数，且对任意实数 x，均有 2f(x)+f(x^2-1)=1，则 f(\sqrt{2}) 的值为__________.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

数列 \{a_n\} 满足 a_1=1,\df{a_{n+1}-a_n}{a_n}=\df{a_{n+2}-a_{n+1}}{a_{n+2}} (n\in\bN^\ast). 若 a_1a_2+a_2a_3+\cdots+a_6a_7=3，则 a_{2024}=__________.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

一珍稀物种出现在地球，对每个珍稀生物，每天有如下事件发生：有 p (0\leq p\leq 1) 的概率消失，有 \df{1-p}{3} 的概率保持不变，有 \df{1-p}{3} 的概率分裂成两个，有 \df{1-p}{3} 的概率分裂成三个. 对所有新产生的生物每天也会发生上述事件. 假设开始只有一个这样的珍稀生物，若希望最终这种生物灭绝的概率不超过 \df{1}{2}，则 p 至多为__________.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

在矩形 $A B C D$ 中, $A B=2, A D=1$, 边 $D C$ 上（包含点 $D 、 C$ ）的动点 $P$ 与 $C B$ 延长线上 (包含点 $B$ ) 的动点 $Q$ 满足 $|\overrightarrow{D P}|=|\overrightarrow{B Q}|$, 则向量 $\overrightarrow{P A}$ 与向量 $\overrightarrow{P Q}$ 的数量积 $\overrightarrow{P A} \cdot \overrightarrow{P Q}$ 的最小值为 $\qquad$.
原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

袋子 $A$ 中装有 2 张 10 元纸币和 3 张 1 元纸币，袋子 $B$ 中装有 4 张 5 元纸币和 3 张 1元纸币。现随机从两个袋子中各取出两张纸币，则 $A$ 中剩下的纸币面值之和大于 $B$ 中剩下的纸币面值之和的概率为 $\qquad$.
原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

在平面直角坐标系 $x O y$ 中，点集 $K=\{(x, y) \mid x, y=-1,0,1\}$ 。在 $K$ 中随机取出三个点，则这三点中存在两点之间距离为 $\sqrt{5}$ 的概率为 $\qquad$.
原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设数列 $\left\{a_{n}\right\}$ 是等差数列，数列 $\left\{b_{n}\right\}$ 满足 $b_{n}=a_{n+1} a_{n+2}-a_{n}^{2}, n=1,2, \cdots$ 。
（1）证明：数列 $\left\{b_{n}\right\}$ 也是等差数列；
（2）设数列 $\left\{a_{n}\right\},\left\{b_{n}\right\}$ 的公差均是 $d \neq 0$ ，并且存在正整数 $s, t$ ，使得 $a_{s}+b_{t}$ 是整数，求 $\left|a_{1}\right|$ 的最小值。
原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

在平面直角坐标系 $x O y$, 点集 $K=\{(x, y) \mid x, y \in-1,0,1\}$ 。在 $K$ 中随机取出三个点,则这三个点两两之间的距离均不超过 2 的概率是 $\qquad$.
原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

将 $1,2,3,4,5,6$ 随机排成一行, 记为 $a, b, c, d, e, f$, 则 $a b c+def$ 是偶数的概率为 $\qquad$.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

将 $1,2,3,4,5,6$ 随机排成一行, 记为 $a, b, c, d, e, f$, 则 $a b c+d e f$ 是奇数的概率为 $\qquad$.
原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

在 $\triangle A B C$ 中, $\sin A=\frac{3}{5}, \cos B=\frac{5}{13}$, 那么 $\cos C$ 的值等于 $\qquad$.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

在平面直角坐标系中, 点 $A, B, C$ 在双曲线 $x y=1$ 上, 满足 $\triangle A B C$ 为等腰直角三角形。求 $\triangle A B C$ 的面积的最小值.
原始的答案是k\sqrt{m}+n的形式，请给出k + m + n的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

在凸四边形 $A B C D$ 中, $\overrightarrow{B C}=2 \overrightarrow{A D}$. 点 $P$ 是四边形 $A B C D$ 所在平面上一点, 满足
$$
\overrightarrow{P A}+2020 \overrightarrow{P B}+\overrightarrow{P C}+2020 \overrightarrow{P D}=\overrightarrow{0}
$$
设 $s, t$ 分别为四边形 $A B C D$ 与 $\triangle P A B$ 的面积, 则 $\frac{t}{s}=$ $\qquad$.
原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

$x_{1}, x_{2}, x_{3}, x_{4}$ 为互不相等的正实数, $x_{i 1}, x_{i 2}, x_{i 3}, x_{i 4}$ 为 $x_{1}, x_{2}, x_{3}, x_{4}$ 的任意顺序排列, $X=\max \left\{\min \left\{x_{i 1}, x_{i 2}\right\}, \min \left\{x_{i 3}, x_{i 4}\right\}, Y=\min \left\{\max \left\{x_{i 1}, x_{i 2}\right\}, \max \left\{x_{i 3}, x_{i 4}\right\}\right.\right.$, 求 $x$ 大于 $y$ 的概率。  原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

在凸四边形 $A B C D$ 中, $\overrightarrow{B C}=2 \overrightarrow{A D}$. 点 $P$ 是四边形所在平面上一点, 满足 $\overrightarrow{P A}+2020 \overrightarrow{P B}+\overrightarrow{P C}+2020 \overrightarrow{P D}=\mathbf{0}$. 设 $s, t$ 分别为四边形 $A B C D$ 与 $\triangle P A B$的面积，则 $\frac{t}{s}=$ $\qquad$.
原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

数列 $\left\{a_{n}\right\}$ 满足 $a_{1}=1, a_{n}=\frac{1}{4 a_{n-1}}+\frac{1}{n}(n \geq 2)$, 则 $a_{100}$ 的值为 $\qquad$.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

在平面直角坐标系 $x O y$ 中， $\Gamma_{1}$ 是以 $(2,1)$ 为圆心的单位圆， $\Gamma_{2}$ 是以 $(10,11)$为圆心的单位圆。过原点 $O$ 作一条直线 $l$, 使得 $l$ 与 $\Gamma_{1}, \Gamma_{2}$ 各有两个交点，将 $\Gamma_{1}, \Gamma_{2}$共分成四段圆弧，且这四段圆弧中有两段等长. 所有满足条件的直线 $l$ 的斜率之和为 $\qquad$。原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

等差数列 $\left\{a_{n}\right\}$ 满足 $a_{2021}=a_{20}+a_{21}=1$, 则 $a_{1}$ 的值为 $\qquad$.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设 $m$ 为实数, 复数 $z_{1}=1+2 \mathrm{i}, z_{2}=m+3 \mathrm{i}$ (这里 i 为虚数单位), 若 $z_{1} \cdot \overline{z_{2}}$ 为纯虚数，则 $\left|z_{1}+z_{2}\right|$ 的值为 $\qquad$.原始的答案是k\sqrt{m}+n的形式，请给出k + m + n的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设 $\alpha, \beta \in \mathbf{R}$, 若 $\tan (\alpha+\beta)=2, \tan (\alpha+2 \beta)=3$, 则 $\tan \alpha$ 的值为 $\qquad$. 原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

在某次数学竞赛小组交流活动中，四名男生与三名女生按随机次序围坐一圈，则三名女生两两不相邻的概率为 $\qquad$.
原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

在 $1,2, \cdots, 10$ 中随机选出三个不同的数, 它们两两互素的概率为 $\qquad$.
原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

已知函数 $y=f(x)$ 的图像既关于点 $(1,1)$ 中心对称, 又关于直线 $x+y=0$轴对称。若 $x \in(0,1)$ 时, $f(x)=\log _{2}(x+1)$, 则 $f\left(\log _{2} 10\right)$ 的值为 $\qquad$.
原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

在平面直角坐标系中, 双曲线 $\Gamma: x^{2}-y^{2}=1$ 的右顶点为 $A$ 。将圆心在 $y$ 轴上，且与 $\Gamma$ 的两支各恰有一个公共点的圆称为 "好圆"。若两个好圆外切于点 $P$ ，圆心距为 $d$ ，求 $\frac{d}{|P A|}$ 的所有可能的值。原始的答案是k\sqrt{m}+n的形式，请给出k + m + n的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

一只青蛙在正方形 $A B C D$ 的四个顶点间跳跃, 每次跳跃总是等可能地跳至与当前所在顶点相邻的两个顶点之一，且各次跳跃是独立的. 若青蛙第一次跳跃前位于顶点 $A$ ，则它第 6 次跳跃后恰好仍位于顶点 $A$ 的概率为
原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

在 $\triangle A B C$ 中, 已知
$\cos C=\frac{\sin A+\cos A}{2}=\frac{\sin B+\cos B}{2}$, 求 $\sin C$ 的值
原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

在矩形 $A B C D$ 中, $A B=2, A D=1$, 在 $D C$ 边上（包含点 $D 、 C$ ）的动点 $P$ 与 $C B$延长线上（包含 $B$ ）的动点 $Q$ 满足 $|\overrightarrow{D P}|=|\overrightarrow{B Q}|$ ，则向量 $\overrightarrow{P A}$ 与向量 $\overrightarrow{P Q}$ 的数量积 $\overrightarrow{P A} \cdot \overrightarrow{P Q}$ 的最小值为 $\qquad$.
原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

$x_{1}, x_{2}, x_{3}, x_{4}$ 为互不相等的正实数, $x_{i 1}, x_{i 2}, x_{i 3}, x_{i 4}$ 为 $x_{1}, x_{2}, x_{3}, x_{4}$ 的任意顺序排列, $X=\max \left\{\min \left\{x_{i 1}, x_{i 2}\right\}, \min \left\{x_{i 3}, x_{i 4}\right\}, Y=\min \left\{\max \left\{x_{i 1}, x_{i 2}\right\}, \max \left\{x_{i 3}, x_{i 4}\right\}\right.\right.$, 求 x 大于 y 的概率.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

定义 $x * y=\frac{x+y}{1+x y}$, 则 $(\cdots((2 * 3) * 4) \cdots) * 21=\qquad$.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

如图 1，抛物线 $y^{2}=2 p x(p>0)$ 的焦点为 $F$, 准线为 $l$, 过点 $F$ 的直线与抛物线交于 $A 、 B$ 两点, 且 $|A B|=3 p$. 设点 $A 、 B$ 在 $l$ 上的射影分别为 $A^{\prime}$ 、 $B^{\prime}$ ，向四边形 $A A^{\prime} B^{\prime} B$ 内任投
![](https://cdn.mathpix.com/cropped/2024_10_21_d55e3db49793c5b8fe33g-1.jpg?height=446&width=414&top_left_y=1693&top_left_x=1300)
图 1
一点 $M$. 则点 $M$落在 $\triangle F A^{\prime} B^{\prime}$ 内的概率为 $\qquad$.
原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设 $\triangle A B C$ 的内角 $\angle A 、 \angle B 、 \angle C$ 的对边分别为 $a 、 b 、 c$, 且 $\angle A-\angle C=\frac{\pi}{2}, a 、 b 、 c$ 成等差数列. 则 $\cos B$ 的值为 $\qquad$.
原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

有 20 张卡片上分别写有数字 1,2, $\cdots, 20$, 将它们放入一个盒子内。有 4 个人从中不放回地各抽取一张卡片，抽到两个较小数字的两人在同一组，抽到两个较大数字的两人在同一组。现其中有两人抽到 $5 、 14$, 则此两人在同一组的概率等于 $\qquad$ （用最简分数作答).
原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

在正三棱锥 $P-A B C$ 中, 已知侧棱与底面所成的角为 $45^{\circ}$ 。则相邻两侧面所成角的余弦值为 $\qquad$.
#*原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

在一个圆上随机取三点, 则以此三点为顶点的三角形是锐角三角形的概率为
$\qquad$.
#*原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

已知正四面体的四个表面上分别写有数字 $1 、 2 、 3 、 4$, 将四个这样的密度均匀的正四面体同时投郑于桌面上，与桌面接触的四个面上的四个数的和能被 4 整除的概率为
$\qquad$.
原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

已知 $f(x)$ 是定义在 $\mathbf{R}$ 上的函数, 且对任意实数 $x$, 均有 $2 f(x)+f\left(x^{2}-1\right)=1$, 则 $f(\sqrt{ } 2)$ 的值为 $\qquad$。
原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

已知抛物线 $y^{2}=2 p x(p>0), A, B$ 是抛物线上不同于顶点 $O$ 的两个动点. 记 $\angle A O B=\theta\left(\theta \neq 90^{\circ}\right)$. 若 $S_{\triangle A O B}=m \tan \theta$, 试求当 $m$ 取得最小值时 $\tan \theta$ 的最大值。
原始的答案是k\sqrt{m}+n的形式，请给出k + m + n的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

如图，设椭圆 $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)$ 的左、右焦点分别为 $F_{1} 、 F_{2}$ ，过点 $F_{2}$ 的直线交栯圆于 $A\left(x_{1}, y_{1}\right), B\left(x_{2}, y_{2}\right)$ 两点. 若 $\triangle A F_{1} B$ 内切圆的面积为 $\pi$ ，且 $\left|y_{1}-y_{2}\right|=4$ ，则椭圆的离心率为___原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设 $a, b, c$ 均大于 1, 满足 $\left\{\begin{array}{l}\lg a+\log _{b} c=3, \\ \lg b+\log _{a} c=4.\end{array}\right.$ 求 $\lg a \cdot \lg c$ 的最大值.
原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

求具有下述性质的最小正数 $c$ ：对任意整数 $n \geq 4$,以及集合 $A \subseteq\{1,2, \cdots, n\}$, 若 $|A|>c n$, 则存在函数 $f: A \rightarrow\{1,-1\}$, 满足
$$
\left|\sum_{a \in A} f(a) \cdot a\right| \leq 1
$$
原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设 $x \neq y$, 且两数列 $x, a_{1}, a_{2}, a_{3}, y$ 和 $b_{1}, x, b_{2}, b_{3}, y, b_{4}$ 均为等差数列, 那么 $\frac{b_{4}-b_{3}}{a_{2}-a_{1}}=$ $\qquad$.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

一颗质地均匀的正方体骰子，六个面上分别标有点数 $1,2,3,4,5,6$. 随机地抛掷该骰子三次 (各次抛掷结果相互独立), 所得的点数依次为 $a_{1}, a_{2}, a_{3}$, 则事件 " $\left|a_{1}-a_{2}\right|+\left|a_{2}-a_{3}\right|+\left|a_{3}-a_{1}\right|=6$ " 发生的概率为 $\qquad$。原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

求具有下述性质的最小正数 $c$ ：对任意整数 $n \geq 4$,以及集合 $A \subseteq\{1,2, \cdots, n\}$, 若 $|A|>c n$, 则存在函数 $f: A \rightarrow\{1,-1\}$, 满足
$$
\left|\sum_{a \in A} f(a) \cdot a\right| \leq 1.
$$原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

若等差数列 $\left\{a_{n}\right\}$ 及正整数 $m(m \geq 3)$ 满足: $a_{1}=1, a_{m}=2$, 且
$$
\frac{1}{a_{1} a_{2}}+\frac{1}{a_{2} a_{3}}+\cdots+\frac{1}{a_{m-1} a_{m}}=3,
$$
则 $a_{1}+a_{2}+\cdots+a_{m}$ 的值为 $\qquad$.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

已知数列 $\left\{a_{n}\right\}$ 的各项均为非负实数, 且满足：对任意整数 $n \geq 2$, 均有 $a_{n+1}=a_{n}-a_{n-1}+n$. 若 $a_{2} a_{2022}=1$, 求 $a_{1}$ 的最大可能值.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

若 $\triangle A B C$ 的内角 $A, B, C$ 满足 $\sin A=\cos B=\tan C$, 求 $\cos ^{3} A+\cos ^{2} A-\cos A$ 的值.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

在 $\triangle A B C$ 中, 已知
$\cos C=\frac{\sin A+\cos A}{2}=\frac{\sin B+\cos B}{2}$, 求 $\sin C$ 的值原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

将 $1,2,3,4,5,6$ 随机排成一行, 记为 $a, b, c, d, e, f$, 则 $a b c+\operatorname{def}$ 是奇数的概率为 $\qquad$.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

在 $1,2,3, \cdots, 10$ 中随机选出一个数 $a$ ，在 $-1,-2,-3, \cdots,-10$ 中随机选出一个数 $b$ ，则 $a^{2}+b$ 被 3 整除的概率为 $\qquad$.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

在 $\triangle A B C$ 中, $A B=6, B C=4$, 边 $A C$ 上的中线长为 $\sqrt{10}$, 则 $\sin ^{6} \frac{A}{2}+\cos ^{6} \frac{A}{2}$ 的值为 $\qquad$ $\cdot$原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

在凸四边形 $A B C D$ 中, $\overrightarrow{B C}=2 \overrightarrow{A D}$. 点 $P$ 是四边形 $A B C D$ 所在平面上一点, 满足
$$
\overrightarrow{P A}+2020 \overrightarrow{P B}+\overrightarrow{P C}+2020 \overrightarrow{P D}=\overrightarrow{0}.
$$
设 $s, t$ 分别为四边形 $A B C D$ 与 $\triangle P A B$ 的面积, 则 $\frac{t}{s}=$ $\qquad$.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

数列 $\left\{a_{n}\right\}$ 满足 $a_{n+1}=(-1)^{n} n-a_{n}, n=1,2,3, \cdots$, 且 $a_{10}=a_{1}$, 则 $a_{n} a_{n+1}$ 的最大值是 $\qquad$.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

在 $\triangle A B C$ 中, $A B=6, B C=4$, 边 $A C$ 上的中线长为 $\sqrt{10}$, 则 $\sin ^{6} \frac{A}{2}+\cos ^{6} \frac{A}{2}$的值为 $\qquad$.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

在凸四边形 $A B C D$ 中, $\overrightarrow{B C}=2 \overrightarrow{A D}$. 点 $P$ 是四边形所在平面上一点, 满足 $\overrightarrow{P A}+2020 \overrightarrow{P B}+\overrightarrow{P C}+2020 \overrightarrow{P D}=\mathbf{0}$. 设 $s, t$ 分别为四边形 $A B C D$ 与 $\triangle P A B$的面积，则 $\frac{t}{s}=$ $\qquad$.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

将正九边形的每个顶点等概率地涂上红、蓝两种颜色之一, 则存在三个同色的顶点构成锐角三角形的概率为 $\qquad$.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

正实数 $u, v, w$ 均不等于 1, 若 $\log _{u}(v w)+\log _{v} w=5, \log _{v} u+\log _{w} v=3$, 则 $\log _{w} u$ 的值为 $\qquad$.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

在平面直角坐标系 $x O y$ 中, 点集 $K=\{(x, y) \mid x, y=-1,0,1\}$. 在 $K$ 中随机取出三个点, 则这三个点两两之间的距离不超过 2 的概率为 $\qquad$.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设抛物线 $C: y^{2}=2 x$ 的准线与 $x$ 轴交于点 $A$, 过点 $B(-1,0)$ 作一直线 $l$ 与抛物线 $C$相切于点 $K$, 过点 $A$ 作 $l$ 的平行线, 与抛物线 $C$ 交于点 $M, N$, 则 $\triangle K M N$ 的面积为 $\qquad$.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

在 $\triangle A B C$ 中, $\sin ^{2} A+\sin ^{2} C=2018 \sin ^{2} B$, 则 $\frac{(\tan A+\tan C) \tan ^{2} B}{\tan A+\tan B+\tan C}=\qquad$.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设 $f(x)$ 是定义在 $(0,+\infty)$ 上的单调函数, 对任意 $x>0$ 有 $f(x)>-\frac{4}{x}, f\left(f(x)+\frac{4}{x}\right)=3$,则 $f(8)=$ $\qquad$.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

四面体 $A B C D$ 中, $A B \perp B C, C D \perp B C, B C=2$, 且异面直线 $A B$ 与 $C D$ 所成的角为 $60^{\circ}$ 。若四面体 $A B C D$ 的外接球半径为 $\sqrt{5}$, 则四面体 $A B C D$ 体积的最大值为 $\qquad$.原始的答案是k\sqrt{m}+n的形式，请给出k + m + n的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

过抛物线 $E: y^{2}=2 x$ 的焦点 $F$ 作两条斜率之积为 $-\frac{1}{2}$ 的直线 $l_{1} 、 l_{2}$, 其中, $l_{1}$ 与抛物线 $E$ 交于 $A 、 C$ 两点, $l_{2}$ 与抛物线 $E$ 交于 $B 、 D$两点.则 $|A C|+2|B D|$ 的最小值为 $\qquad$.原始的答案是k\sqrt{m}+n的形式，请给出k + m + n的值。

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

棱长为 $\sqrt{2}$ 的正四面体 $A B C D$ 中, 已知 $\overrightarrow{A P}=\frac{1}{2} \overrightarrow{A B}, \overrightarrow{A Q}=\frac{1}{3} \overrightarrow{A C}, \overrightarrow{A R}=\frac{1}{4} \overrightarrow{A D}$, 若点 $K$ 为 $\triangle B C D$ 的重心, 则四面体 $K P Q R$ 的体积为 $\qquad$.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

已知点 $P(x, y)$ 满足 $|x|+|y| \leqslant 2$. 则到 $x$ 轴的距离 $d \leqslant 1$ 的点 $P$ 的概率为 $\qquad$.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

设正三棱锥 $P-A B C$ 的高为 $P O, M$ 为 $P O$ 的中点, 过 $A M$ 作与棱 $B C$平行的平面，将三棱锥截为上、下两部分，试求此两部分的体积比.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

$\cos ^{2} 10^{\circ}+\cos ^{2} 50^{\circ}-\sin 40^{\circ} \sin 80^{\circ}=\qquad$.The answer is in the form \frac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

在 $\triangle A B C$ 中, 已知三个角 $A, B, C$ 成等差数列, 假设它们所对的边分别为 $a, b, c$, 并且 $c-a$ 等于 $A C$ 边上的高 $h$, 则 $\sin \frac{C-A}{2}=\qquad$.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

实数 $x 、 y$ 满足 $4 x^{2}-5 x y+4 y^{2}=5$ ，设 $S=x^{2}+y^{2}$ ，则 $\frac{1}{S_{\max }}+\frac{1}{S_{\min }}$ 的值为$\qquad$.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

给定曲线族 $2(2 \sin \theta-\cos \theta+3) x^{2}-(8 \sin \theta+\cos \theta+1) y=0, \theta$ 为参数，求该曲线在直线 $y=2 x$ 上所截得的弦长的最大值。原始的答案是k\sqrt{m}+n的形式，请给出k + m + n的值。

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