Solve the following math problem step 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 $b_{n}=a_{n+1} a_{n+2}-a_{n}^{2}, n=1,2, \cdots$ 。
（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.

Evaluate
$$
\lim _{n \rightarrow \infty} \int_{0}^{1} \int_{0}^{1} \cdots \int_{0}^{1} \cos ^{2}\left\{\frac{\pi}{2 n}\left(x_{1}+x_{2}+\cdots x_{n}\right)\right\} d x_{1} d x_{2} \cdots d x_{n}
$$
#*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.

Let $C$ be a fixed unit circle in the Cartesian plane. For any convex polygon $P$ each of whose sides is tangent to $C$. let $N(P, h, k)$ be the number of points common to $P$ and the unit circle with center at ( $h, k$ ). Let $H(P)$ be the region of all points $(x, y)$ for which $N(P, x, y) \geq 1$ and $F(P)$ be the area of $H(P)$. Find the smallest number $u$ with
$$
\frac{1}{F(P)} \iint N(P, x, y) d x d y<u
$$
for all polygons $P$, where the double integral is taken over $H(P)$.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.

For each continuous function $f:[0,1]\to\mathbb{R}$, let $I(f)=$ $\int _{0}^{1}x^{2}f( x)$ $dx$ and $J( x) = \int _0^{1}x\left ( f( x) \right ) ^{2}$ $dx.$ Find the maximum value of $I(f)-J(f)$ over all such functions $f.$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.

两人轮流投掷股子，每人每次投掷两颗，第一个使两颗股子点数和大于 6 者为胜，否则轮由另一人投郑。先投郑人的获胜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.

已知函数 $f(x)=a x^{3}+b x^{2}+c x+d(a \neq 0)$, 当 $0 \leqslant x \leqslant 1$ 时, $\left|f^{\prime}(x)\right| \leqslant 1$, 试求 $a$ 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 $\triangle A B C$ 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\_\_\_\_\_\_.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.

从一个正方体的八个顶点中取出三个.则以这三个点为顶点构成直角三角形的概率是$\underline{\quad}$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.

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.

设 $a 、 b 、 c \in(0,1], \lambda$ 为实数,使得
$$
\frac{\sqrt{3}}{\sqrt{a+b+c}} \geqslant 1+\lambda(1-a)(1-b)(1-c).
$$
求入的最大值.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.

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.

设复数 $x, y, z$ 满足: $|x+2 y+3 z|=1$ 。
求 $|x|^{2}+|y|^{2}+|z|^{2}+\left|x^{2}+y^{2}+z^{2}\right|$ 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.

袋中装有 2 个红球、 3 个白球、 4 个黄球，从中任取 4 个球，则其中三种颜色的球都有的概率是 $\qquad$。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.

在$\triangle ABC$中，若$\tan\dfrac{A}{2}+\tan\dfrac{B}{2}=1$，则$\tan\dfrac{C}{2}$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.

设$\triangle ABC$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_.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.

Determine the greatest possible value of $\sum_{i=1}^{10}\cos(3x_i)$ for real numbers $x_1,x_2,\ldots,x_{10}$ satisfying $\sum_{i=1}^{10}\cos(x_i)=0.$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.

求所有的实数 $k$, 使得不等式
$$
a^{3}+b^{3}+c^{3}+d^{3}+1 \geqslant k(a+b+c+d)
$$
对任意 $a 、 b 、 c 、 d \in[-1,+\infty)$ 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.

实数 $a 、 b$ 使得方程
$x^{3}-a x^{2}+b x-a=0\quad①$
有三个正实根. 求 $\frac{2 a^{3}-3 a b+3 a}{b+1}$ 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.

已知向量 $\boldsymbol{a}=(x, 1), \boldsymbol{b}=(2, y), \boldsymbol{c}=$ $(1,1)$ ，且 $\boldsymbol{a}-\boldsymbol{b}$ 与 $\boldsymbol{c}$ 共线. 则 $|\boldsymbol{a}|+2|\boldsymbol{b}|$ 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.

在平面直角坐标系 $x O y$ 中, $Rt \triangle A B C$ 的三个顶点均在椭圆 $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ $(a>b>0)$ 上, 其中, $A(0, b)$ 为直角顶点.
(1) 若 $b=1$, 该三角形的面积的最大值为 $\frac{27}{8}$ ，求实数 $a$ 的值;
(2)求该三角形面积的最大值.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.

在梯形 $A B C D$ 中, $B C / / A D, A B=A D$, $\angle A B C=\frac{2 \pi}{3}, \angle B C D=\frac{\pi}{2}$. 将 $\triangle A B D$ 沿 $B D$翻折，点 $A$ 在面 $B C D$ 上的投影为点 $P$. 已知 $A B$ 与 $C D$ 所成角的余弦值为 $\frac{\sqrt{3}}{6}$ 。则 $B P$ 与 $C D$ 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.

使得 $\frac{x+3}{x^{2}-2 x+2}$ 为正整数的所有实数 $x$的和为 $\qquad$.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.

有五张分别写有 $1 、 2 、 3 、 4 、 5$ 数字的卡片，每次从中抽取一张，记下卡片上的数字，然后放回。这样抽取了四次，则抽到的最大数与最小数的差小于 4 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.

利用24小时制计时，可得到某一时刻的四个数字之和比时刻$20:21$的四个数字之和小的概率为_______。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$f(x)$的定义域为$\mathbf{R}$，且$f(x-1)$为偶函数，$f(x-2)$为奇函数。当$x\in[1,2]$时，$f(x)=ax^{2}+bx$。若$f(0)+f(3)=-1$，则$f\left(\frac{9}{2}\right)=\qquad.$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.

已知复数 $z$ 、w 满足:
$z+\frac{20 \mathrm{i}}{w}=5+\mathrm{i}, w+\frac{12 \mathrm{i}}{z}=-4+10 \mathrm{i}$ 。
则 $|z w|$ 的最小值是 $\qquad$.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.

已知边长为 $a$ 的正方形 $A B C D$ 的顶点 $A$ 在平面 $\beta$ 内，其余顶点均在平面 $\beta$ 的同侧，且 $B 、 D$ 两点到平面 $\beta$ 的距离分别为 $1 、 2$ 。若平面 $A B C D$ 与平面 $\beta$ 的夹角为 $30^{\circ}$, 则 $a=$
$\qquad$.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.

已知复数 $z_{1}, z_{2}$ 满足:
$$
\left|\overline{z_{1}}\right|=3,\left|\overline{z_{2}}\right|=4,\left|z_{1}+z_{2}\right|=6.
$$
则 $\frac{\overline{z_{1}}}{\overline{z_{2}}}+\frac{z_{1}}{z_{2}}=$ $\qquad$.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.

从 $1 \sim 9$ 这九个正整数中任选三个,则这三个数两两互素的概率是 $\qquad$。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.

设点 $A(2,0)$, 点 $B$ 在曲线 $y=\sqrt{1-x^{2}}$上, $\triangle A B C$ 是以 $A$ 为直角顶点的等腰直角三角形. 则 $|O C|$ 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.

从集合$\{1,2,\cdots,105\}$中任取一个元素$a$，使得$x^{2}+ax+6a=0$只有整数解的概率为_______。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.

一只小虫在正八面体的表面上爬行，每秒从某一个顶点等可能地爬往四个相邻顶点之一。则小虫在第8秒爬回初始位置的概率为_______。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.

在正四面体 $P-A B C$ 中, 点 $E 、 F$ 分别在棱 $P B 、 P C$ 上. 若 $P E \neq P F$ ，且 $A E=A F=2$ ， $E F=\sqrt{3}$, 则四面体 $P-A E F$ 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.

Evaluate$$\lim_{n\rightarrow\infty}\int_{0}^{1}\int_{0}^{1}\cdots\int_{0}^{1}\cos^{2}\left\{\frac{\pi}{2n}\left(x_{1}+x_{2}+\cdots x_{n}\right)\right\}dx_{1}dx_{2}\cdots dx_{n}.$$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.

设集合 $\left\{\left.\frac{5}{x}+y \right\rvert\, 1 \leqslant x \leqslant y \leqslant 5\right\}$ 中的最大元素与最小元素分别为 $M, N$, 则 $M N=$ $\qquad$.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.

双曲线 $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{16}=1$ 的左、右焦点分别为 $A, B, P$ 是双曲线上一点, 若 $\triangle P A B$ 的内切圆圆心为 $(3,1)$, 则 $\triangle P A B$ 外接圆的半径为 ?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.

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.

在三角形 $A B C$ 中, $B C=4, C A=5, A B=6$, 则 $\sin ^{6} \frac{A}{2}+\cos ^{6} \frac{A}{2}=$ $\qquad$.  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.

若平面上非零向量 $\vec{\alpha}, \vec{\beta}, \vec{\gamma}$ 满足 $\vec{\alpha} \perp \vec{\beta}, \vec{\beta} \cdot \vec{\gamma}=2|\vec{\alpha}|, \vec{\gamma} \cdot \vec{\alpha}=3|\vec{\beta}|$, 则 $|\vec{\gamma}|$ 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.

一个不均匀的骰子，掷出 $1,2,3,4,5,6$ 点的概率依次成等差数列.独立地先后掷该骰子两次，所得的点数分别记为 $a, b$ .若事件 " $a+b=7$ " 发生的概率为 $\frac{1}{7}$ ，则事件 " $a=b$ "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.

在平面直角坐标系 $x O y$ 中, 设一条动直线 $l$ 与抛物线 $\Gamma: y^{2}=4 x$ 相切，且与双曲线 $\Omega: x^{2}-y^{2}=1$ 交于左、右两支各一点 $A 、 B$ 。求 $\triangle A O B$ 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.

对有限非空实数集 $X$, 记 $X$ 的元素个数为 $|X|$ ，均有
$f(X)=\frac{1}{|X|} \sum\limits_{a \in X} a$.
集合对 $(A, B)$ 满足
$A \cup B=\{1,2, \cdots, 100\}, A \cap B=\varnothing$,
且 $1 \leqslant|A| \leqslant 98$.
任取 $p \in B$ ，令
$A_{p}=A \cup\{p\}, B_{p}=\{x \mid x \in B, x \neq p\}$.
对所有满足上述条件的集合对 $(A, B)$ 与 $p \in$ $B$ ，求 $\left(f\left(A_{p}\right)-f(A)\right)\left(f\left(B_{p}\right)-f(B)\right)$ 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 $f(x)$ 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 $f(8)= ?$ 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 $f(x)$ 为定义在区间 $(0,+\infty)$ 上的单调函数. 对于任意的 $x>0$ ，均有
$f(x)>-\frac{4}{x}, f\left(f(x)+\frac{4}{x}\right)=3$.
则 $f(8)= ?$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.

对于四面体 $A B C D, A B \perp B C, C D \perp$ $B C, B C=2$ ，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 $A B C D$ 的体积的最大值为         .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.

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 $l_{1}$ 交 $E$ 于 $A, C$ 两点, $l_{2}$ 交 E 于 $B, D$ 两点, 则 $|A C|+2|B D|$ 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.

有一人进行投篮训练，投篮 5 次，失误一次扣 1 分，进一次得 1 分，连进 2次得 3 分，连进 3 次得 5 分. 若投篮的命中率为 $\frac{2}{5}$ ，则投篮 3 次恰好得 2 分的概
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.

$x_{1}, x_{2}, x_{3}, x_{4}$ 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.

Let $PQR$ be an equilateral triangle with sides of length three units. $U$, $V$, $W$, $X$, $Y$, and $Z$ divide the sides into lengths of one unit. Find the ratio of the area of the shaded quadrilateral $UWXY$ to the area of the triangle $PQR$.  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.

已知实数 $x_{1}, x_{2}, \cdots, x_{2018}$ 满足：对任意的 $1 \leqslant i<j \leqslant 2018$, 均有 $x_{i}+x_{j} \geqslant(-1)^{i+j}$. 求 $\sum\limits_{i=1}^{2018} i x_{i}$ 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.

设 $x 、 y 、 z \in(0,1)$ ，满足  $\sqrt{\frac{1-x}{y z}}+\sqrt{\frac{1-y}{z x}}+\sqrt{\frac{1-z}{x y}}=2$.  求 $x y z$ 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.

对区间 $[0,1]$ 上的实数 $x 、 y 、 z$, 记 $M$ 为 $x-x y-x z+y z 、 y-y x-y z+x z 、 z-z x-z y+x y$中的最小数. 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.

已知正数 $a 、 b 、 c$ 满足 $a^{2}+b^{2}+2 c^{2}=1$ .则 $\sqrt{2} a b+2 b c+7 a c$ 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.

一个盒中有 9 个正品和 3 个废品, 每次取一个产品, 取出后不再放回。则在取出 2个正品前已取出 3 个废品的概率为 $\qquad$.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.

在直三棱柱 $A B C-A_{1} B_{1} C_{1}$ 中, $A B=1, B C=C C_{1}=\sqrt{3}, \angle A B C=90^{\circ}$, 点 $P$ 是平面 $A B C$上一动点, 则 $A_{1} P+\frac{1}{2} P C$ 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.

设复数 $x, y, z$ 满足: $|x+2 y+3 z|=1$ 。
求 $|x|^{2}+|y|^{2}+|z|^{2}+\left|x^{2}+y^{2}+z^{2}\right|$ 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.

求最大的正实数 $\alpha$, 对于满足 $a+b+c \mid a^{2}+b^{2}+c^{2}$ 的不全相等的正整数 $a, b, c$, 都有方程
$$
(x+b)^{2}=(x+c)(x+a)
$$
在区间 $(0, \alpha)$ 内没有解.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.

将编号为 $1,2,3, \cdots, 9$ 的九个小球随机放置在圆周的九个等分点上，每个等分点上各放一个小球，设圆周上所有相邻两个球号码之差的绝对值之和为 $S$ ，求使 $S$ 达到最小值的放法的概率。（注：如果某种放法，经旋转或镜面反射后与另一种放法重合，则认为是相同的放法）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.

若函数 $f(x)=\frac{x}{\sqrt{1+x^{2}}}$ 且 $f^{(n)}(x)=\underbrace{f[f[f \cdots f(x)]]}_{n}$, 则 $f^{(99)}(1)=$ $\qquad$.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.

某情报站有 $A, B, C, D$ 四种互不相同的密码, 每周使用其中的一种密码, 且每周都是从上周未使用的三种密码中等可能地随机选用一种。设第1周使用 $A$ 种密码, 那么第 7 周也使用 $A$ 种密码的概率是 $\qquad$. (用最简分数表示)  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.

某情报站有 $A, B, C, D$ 四种互不相同的密码，每周使用其中的一种密码，且每周都是从上周未使用的三种密码中等可能地随机选用一种。设第1周使用 $A$种密码，那么第7周也使用 $A$ 种密码的概率是 $\qquad$ (用最简分数表示)。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 $1,2, \cdots, 20$ 中任取 5 个不同的数, 其中至少有两个是相邻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.

若从 $1,2,3,4,5,6,7,8,9$ 中随机地取出四个两两互异的数，分别记为 $a, b, c, d$ ，则 $a+b$ 与 $c+d$ 奇偶性相同的概率是 $\qquad$.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.

一个半径为 1 的小球在一个内壁棱长为 $4 \sqrt{6}$ 的正四面体容器内可向各个方向自由运动,则该小球永远不可能接触到的容器内壁的面积是 $\qquad$.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.

在 $\triangle A B C$ 中, $E, F$ 分别是 $A C, A B$ 的中点, $A B=\frac{2}{3} A C$. 若 $\frac{B E}{C F}<t$ 恒成立, 则 $t$ 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.

设等差数列 $\left\{a_{n}\right\}$ 的前 $n$ 项和为 $S_{n}$, 若 $S_{2010}-S_{1}=1$, 则 $S_{2011}=$ $\qquad$.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 $A, B, C, D$ 是空间四个不共面的点, 以 $\frac{1}{2}$ 的概率在每对点之前连一条边, 任意两对点之间是否连边是相互独立的, 则 $A, B$ 可用空间折线（一条或若干条边组成的）连接的概率为 $\qquad$。
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.

数列 $\left\{a_{n}\right\}$ 满足 $a_{1}=2, a_{n+1}=\frac{2(n+2)}{n+1} a_{n}\left(n \in \mathbb{N}^{*}\right)$, 则 $\frac{a_{2014}}{a_{1}+a_{2}+\cdots+a_{2013}}=$
$\qquad$.
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 $a_{1}=2, a_{n+1}=\frac{2(n+2)}{n+1} a_{n}\left(n \in \mathbf{N}^{*}\right)$ ，则 $\frac{a_{2014}}{a_{1}+a_{2}+\cdots+a_{2013}}$ $=$ $\qquad$ $\cdot$
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.

已知正四棱雉 $S-A B C D$ 的侧棱长为 $4, \angle A S B=30^{\circ}$, 在侧棱 $S B 、 S C 、 S D$ 分别取点 $E 、 F 、 G$. 则空间四边形 $A E F G$ 周长的最小值为 $\qquad$.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.

Today, Ivan the Confessor prefers continuous functions $f:[0,1] \rightarrow \mathbb{R}$ satisfying $f(x)+f(y) \geq|x-y|$ for all pairs $x, y \in[0,1]$. Find the minimum of $\int_{0}^{1} f$ over all preferred functions.
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.

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 $l_{1}$ 交 $E$ 于 $A, C$ 两点, $l_{2}$ 交 $E$ 于 $B, D$ 两点, 则 $|A C|+2|B D|$ 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.

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.

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 $\frac{1-p}{3}$的概率保持不变，有 $\frac{1-p}{3}$ 的概率分裂成两个，有 $\frac{1-p}{3}$ 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 $\frac{1}{2}$ ，则 $p$ 至多为 $\qquad$.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.

有一人进行投篮训练，投篮 5 次，失误一次扣 1 分，进一次得 1 分，连进 2次得 3 分，连进 3 次得 5 分. 若投篮的命中率为 $\frac{2}{5}$ ，则投篮 3 次恰好得 2 分的概率为 $\qquad$.  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.

从正九边形中任取三个顶点构成三角形, 则正九边形的中心在三角形内的概率为 $\qquad$.  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.

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.

若 $x, y$ 为实数, 则 $|2 x+y|,|x-y|,|1+y|$ 这三个数中的最大数的最小值是$\qquad$.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.

Given real numbers $b_0,b_1,\ldots,b_{2019}$ with $b_{2019}\neq0$, let $z_1,z_2,\ldots,z_{2019}$ be the roots in the complex plane of the polynomial $$P(z)=\sum_{k=0}^{2019}b_kz^k.$$ Let $\mu=(|z_{1}|+\cdots+|z_{2019}|)/2019$ be the average of the distances from $z_1,z_2,\ldots,z_{2019}$ to the origin. Determine the largest constant $M$ such that $\mu\geq M$ for all choices of $b_0,b_1,\ldots,b_{2019}$ that satisfy $$1\leq b_0<b_1<b_2<\cdots<b_{2019}\leq2019.$$ The original answer is in the form $M=a^{b/ c}$, please provide the value of a+ b+c .

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

Suppose that $X_1,X_2,\ldots$ are real numbers between 0 and 1 that are chosen independently and uniformly at random. Let $S= \sum _{i= 1}^{k}X_{i}/ 2^{i}$, where $k$ is the least positive integer such that $X_k<X_{k+1}$, or $k=\infty$ if there is no such integer. Find the expected value of S.  The original answer is in $ae^{b / c}-d$ format, please provide the value of a+b+c+d.

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

Let $p$ be an odd prime number, and let ${\Bbb {F}}_p$ denote the field of integers modulo $p$. Let ${\Bbb {F}}_p[x]$ be the ring of polynomials over ${\Bbb {F}}_p$, and let $q(x)\in {\Bbb {F}}_p[x]$ be given by
$$q(x)=\sum\limits_{k=1}^{p-1}a_kx^k,$$
where
$$a_k=k^{(p-1)/2}\mod p.$$
Find the greatest nonnegative integer $n$ such that $(x-1)^n$ divides $q(x)$ in ${\Bbb {F}}_p[x]$. What is the answer when p = 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.

Find the smallest constant C such that for every real polynomial $P(x)$ of degree 3 that has a root in the interval $[0,1]$, $$\int_{0}^{1} \left\vert P(x)\right\vert\, {\rm d}x\leq C\max _{x \in[0,1]} \left\vert P(x) \right\vert$$. The original answer is in \frac{m}{n} format, please give 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.

The 30 edges of a regular icosahedron are distinguished by labeling them $1,2,…,30$. How many different ways are there to paint each edge red, white, or blue such that each of the 20 triangular faces of the icosahedron has two edges of the same color and a third edge of a different color? The standard answer format is $a^{b} c^{d}$. Please give the value of a + b + c + d.

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

For a positive integer $N$, let $f_N^{1}$ be the function defined by
$$f_N(x)=\sum_{n=0}^N\frac{N+1/2-n}{(N+1)(2n+1)}\sin((2n+1)x).$$
Determine the smallest constant $M$ such that $f_N(x)\leq M$
 for all $N$ and all real $x.$ The original answer is in $\frac{m\pi}{n}$ format, please give 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.

A dart, thrown at random, hits a square target. Assuming that any two parts of the target of equal area are equally likely to be hit, find the probability that the point hit is nearer to the center than to any edge. The original answer is in the form \frac{a\sqrt{b}- c}{d}, please provide the value of a + b + c + d.

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

Let $a$ and $b$ be positive numbers. Find the largest number $c$, in terms of $a$ and $b$, such that $$a^xb^{1-x}\leq a\frac{\sinh ux}{\sinh u}+b\frac{\sinh u(1-x)}{\sinh u}$$ for all $u$ with $0<|u| ≤c$ and for all $x,0<x<1$. (Note: $\sinh u= (e^u- e^{-u})/2.$) Please provide the sum of the bounds of $u$ when $|u| = 1 and a=b=1$.

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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.

Suppose that the plane is tiled with an infinite checkerboard of unit squares. If another unit square is dropped on the plane at random with position and orientation independent of the checkerboard tiling, what is the probability that it does not cover any of the corners of the squares of the checkerboard? The original answer is in the format $a - \frac{b}{\pi}$, please provide the value of $a + b $.

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

For all $n\geq1$, let $$a_n=\sum_{k=1}^{n-1}\frac{\sin\left(\frac{(2k-1)\pi}{2n}\right)}{\cos^2\left(\frac{(k-1)\pi}{2n}\right)\cos^2\left(\frac{k\pi}{2n}\right)}.$$ Determine $$\lim_{n\to\infty}\frac{a_n}{n^3}.$$ The original answer is in the format $\frac{m}{\pi^{n}}$. 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.

Let A be a $2n\times2n$ matrix, with entries chosen independently at random. Every entry is chosen to be 0 or 1, each with probability 1/2. Find the expected value of $\det(A-A^{t})$ (as a function of n), where $A^{t}$ is the transpose of $A.$ The original answer is in \frac{(an)!}{b^{n}n!} format, please give the value of a+b.

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

Let $\mathscr{F}$ be the set of functions $f(x,y)$ that are twice continuously differentiable for $x\geq1,y\geq1$ and that satisfy the following two equations (where subscripts denote partial derivatives): $$\begin{array}{c}xf_x+yf_y=xy\ln(xy),\\x^2f_{xx}+y^2f_{yy}=xy.\end{array}$$ For each $f\in\mathscr{F}$,let $$m(f)=\min_{s\geq1}\left(f(s+1,s+1)-f(s+1,s)-f(s,s+1)+f(s,s)\right).$$ Determine $m(f)$, and show that it is independent of the choice of $f.$ The original answer is in the format of $m \ln n - \frac{a}{b}$, please give the value of m + n +a +b.

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

A line in the plane of a triangle $T$ is called an equalizer if it divides $T$ into two regions having equal area and equal perimeter. Find positive integers $a > b > c$, with $a$ as small as possible, such that there exists a triangle with side lengths $a, b, c$ that has exactly two distinct equalizers. Please provide the value of $a + b + c$.

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

For $0\leq p\leq1/2$,let $X_1,X_2,\ldots$ be independent random variables such that  $$X_i=\begin{cases}1&with~ probability~p,\\-1&{with ~probability}~p,\\0&{with ~probability}~1-2p,\end{cases}$$  for all $i\geq1.$ Given a positive integer $n$ and integers $b,a_1,\ldots,a_n$, let $P(b,a_1,\ldots,a_n)$ denote the probability that $a_1X_1+\cdots+a_nX_n=b.$ For which values of $p$ is it the case that  $$P(0,a_1,\ldots,a_n)\geq P(b,a_1,\ldots,a_n)$$  for all positive integers $n$ and all integers $b,a_1,\ldots,a_n?$ The original answer is in the format $p \leq m / n$, 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.

For each positive integer $k$, let $A(k)$ be the number of odd divisors of $k$ in the interval $[1,\sqrt{2k}).$ Evaluate $$\sum_{k=1}^{\infty}(-1)^{k-1}\frac{A(k)}{k}.$$ The original answer is in the format $\frac{m\pi^j}{n}$, please provide the value of m + n + j.

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

Evaluate \(\int_{0}^{a} \int_{0}^{b} e^{\max \left(b^{2} x^{2}, a^{2} y^{2}\right)} d y d x\), where \(a\) and \(b\) are positive. If \(a = 1\) and \(b = 1\), find the value of \(m + n\) in the expression \(\frac{e^{a^mb^n}-j}{ab}\), where the answer is in the format \(m + n + j\).

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

Two real numbers x and y are chosen at random in the interval $(0,1)$ with respect to the uniform distribution. What is the probability that the closest integer to $x / y$ is even? The original answer is in the form $(m-\pi)/n$, 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.

有一个圆柱形水杯（厚度忽略不计），底面直径为 8, 高为 $8 \sqrt{3}$. 水杯中有适量水,缓慢倾斜水杯至水恰好溢出，此时，水杯的母线与水平面的夹角为 $\frac{\pi}{3}$. 继续缓慢倾斜水杯, 停止倒水时水杯的母线与水平面的夹角为 $\frac{\pi}{6}$. 则所倒出的水的体积的原始答案是 $\frac{m \sqrt{k} \pi}{n}$ 形式，请给出 m + n+k 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.

已知复数 $z_{1} 、 z_{2}$ 满足 $\left|z_{1}\right|=\left|z_{1}+2 z_{2}\right|$ ，且 $\overline{z_{1}} z_{2}=a(2-i)$ ( $a$ 为非零实数). 则 $\frac{z_{2}}{z_{1}}=\_$。已知原始答案为 $-\frac{a}{b}+\frac{c}{d} i$的形式，请给出 a+b+c+d 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.

已知方程 $4 x|x|-a x+1=0$ 恰有三个解. 请给出$a$的最小整数值。

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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.

某城镇有 $n(n \geqslant 3)$ 座村庄,某些村庄之间有道路相连，且每条道路仅连接两座村庄，任两座村庄之间至多有一条路，同时，路交叉处没有十字路口，但有桥或隧道。已知任意两村庄之间可通过道路到达，但由于某项自行车赛事关闭任意环路之后，前述结论不再成立。求道路条数的最大值，并请给出当n=2时的最大值。

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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.

如图 1, 在棱锥 $P-A B C D$ 中, 底面 $A B C D$ 为平行四边形， $P A \perp$ 平面 $A B C D$ ， $\angle A B C=\theta\left(\theta \in\left(0, \frac{\pi}{2}\right)\right). F$ 为 $P C$ 的中点，过 $F$ 且与平面 $P A B$ 平行的平面 $\alpha$ 分别与 $B C$ 、 $P D$ 交于点 $E 、 H$. 若 $A B=A C, P A=A D=4$, 则点 $F$ 到平面 $A E H$ 的距离为\frac{\sqrt{m}}{n}的形式，请给出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$a_{1},a_{2},	ext{...},a_{2021}$均为非负实数，满足$a_{1}+a_{2}+	ext{...}+a_{2021}=1$。定义：$S=	ext{...}rac{a_{i}}{i},T=	ext{...}rac{a_{i}a_{j}}{ij}$。求$S+2T$的最小可能值和最大可能值。原始答案为两个分数，请给出这些分数中所有分子和分母的和。

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

已知四边形 $A B C D$ 为圆柱轴截面，母线 $A B$ 长为 1, 底面直径 $B C$ 长为 $\sqrt{2}$, $E$ 为底面圆周上的动点， $P$ 、 $Q$ 分别为点 $B$ 在 $A E 、 A C$ 上的射影。则当三棱锥 $A-P B Q$ 的体积最大时， $C E$ 的原始答案为 \(\dfrac{\sqrt{m}}{n}\) 的形式，请给出 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 $x$、$y$、$z$ 满足 $x+y+z=1$, 设 $S=\frac{1-x}{1+x}+\frac{1-y}{1+y}+\frac{1-z}{1+z}$. 求 $S$ 的最小值和最大值。原始答案分别为\dfrac{m_1}{n_1}和\dfrac{m_2}{n_2}格式，请给出m_1+n_1+m_2+n_2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.

在正三棱柱 $A B C-A_{1} B_{1} C_{1}$ 中, $A B=$ $B C=C A=A A_{1}=2$, 棱 $A_{1} C_{1}$ 上的点 $D$ 满足 $B C_{1} / /$平面 $A B_{1} \dot{D}$ 。以 $D$ 为球心、 $\frac{\sqrt{7}}{2}$ 为半径的球面与侧面 $A A_{1} B_{1} B$ 的交线长的$\frac{3}{\pi}$倍是 $\qquad$.

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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.

已知椭圆 $\Gamma: \frac{x^{2}}{4}+\frac{y^{2}}{3}=1, F$ 为 $\Gamma$ 的右焦点。过 $F$ 作直线交 $\Gamma$ 于 $M 、 N$ 两点，直线 $l$ 过点 $F$ 且与直线 $M N$ 垂直, 并交直线 $x=4$ 于点 $P$, 直线 $O P$ 交 $\Gamma$ 于 $A 、 B$ 两点. 设直线 $M P 、 M N 、 M A 、 M B$ 的斜率依次为 $m 、 n 、 p$ 、 $q$ (四条直线的斜率均存在且均不为 0 ). 记 $m+n=s, p+q=t, m n=r, s t=f(r)$. 求 $f(\frac{3}{4})$。

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Solve the following math problem step 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 $P$ 在椭圆 $\frac{x^{2}}{2}+y^{2}=1$ 上, 记椭圆上顶点和右顶点分别为 $A$ 和 $B$. 则 $\triangle A B P$ 面积的最大值的2倍减去$\sqrt{2}$是 $\qquad$。

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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.

已知四面体 $A B C D$ 的外接球 $\Gamma$ 的半径为 3, 球心为 $O$, 其中 $A B$ 为球 $\Gamma$ 的直径, $C D=3, \angle A O C=\theta \in\left[\frac{\pi}{3}, \frac{5}{6} \pi\right]$, 四面体 $A B C D$的体积的最大值记为 $V(\theta)$. 则 $4\sqrt{3}V(\theta)$ 的最小值是 $\qquad$。

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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.

函数 $f(x)=8\sqrt{15}(7 \sin x+\sin 2 x)$ 的最大值为 $\qquad$

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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.

函数 $f(x)=\frac{\sin 2 x}{2+\sin x+\cos x}$ 的最小值减去1得到的结果的$\sqrt{2}$倍

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Solve the following math problem step 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 $C \in\left\{(x, y) \mid x^{2}+y^{2}=1, y \geq 0\right\}, A(-1,0),B(1,0)$, 延长 $A C$ 至 $D$ 使  $|C D|=3|B C|$, 那么点 $D$ 到 $E(4,5)$ 距离最大值和最小值之积减去$2\sqrt{170}-2\sqrt{85}$为 $\qquad$.

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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.

解方程组 $\left\{\begin{array}{l}x y=2 x+y-1 \\ y z=2 z+3 y-8. \\ z x=4 z+3 x-8\end{array}\right.$中x所有的可能值之和。

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Solve the following math problem step 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 $A, B$ 为抛物线 $y=1-x^{2}$ 上位于 $y$ 轴两侧的点, 求该抛物线过点 $A, B$ 的切线与 $x$ 轴围成的图形面积的最小值的$3\sqrt{3}$倍。

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Solve the following math problem step 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
$$
a_{0}=1, a_{n+1}=\sum_{\substack{i, j, k \geq 0 \\ i+j+k=n}} a_{i} a_{j} a_{k}
$$
对任意 $n \geq 0$ 成立. 求数列 $\left\{a_{n}\right\}$ 的通项. 原始答案为\frac{m}{n}格式，请给出m + n的值，当n = 2时。

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

在正三棱雉 $P-A B C$ 中, 底面是边长为 1 的正三角形. 若点 $P$ 到平面 $A B C$ 的距离为 $x$, 异面直线 $A B$ 与 $C P$ 的距离为 $y$, 则 $\lim _{x \rightarrow+\infty} 2\sqrt{3}y=$

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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.

Evaluate $29 \frac{27}{28} \times 27\frac{14}{15}$. The original answer is in the form of $a + \frac{m}{n}$, where a is the integer part. Please provide the value of a + 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.

In a language college, $72\%$ students can speak Chinese, $65\%$ students can speak English, and $10\%$ students can speak neither Chinese nor English. Find the percentage of students who can speak both Chinese and English. Please provide the sum of the original percentage and 100.

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

Replace the letters $a$, $b$, $c$ and $d$ in the following expression with the numbers $1$, $2$, $3$ and $4$, without repetition: $a+\cfrac{1}{b+\cfrac{1}{c+\cfrac{1}{d}}}$ Find the difference between the maximum value and the minimum value of the expression. The original answer is in \frac{m}{n} format, please give 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.

Find all pairs of primes $(p,q)$ for which $p-q$ and $pq-q$ are both perfect squares. Please provide the value of $p + q + pq$.

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

Find all triples of positive integers $(x,y,z)$ that satisfy the equation

$$\begin{align*} 2(x+y+z+2xyz)^2=(2xy+2yz+2zx+1)^2+2023 \end{align*}$$

Please provide the sum of all elements in one such triple.

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

Three distinct vertices are chosen at random from the vertices of a given regular polygon of $(2n+1)$ sides. If all such choices are equally likely, what is the probability that the center of the given polygon lies in the interior of the triangle determined by the three chosen random points? The original answer is in the format \frac{m}{k}, please provide the value of m + k when n = 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.

给定正整数 $n$. 求最小的实数 $\lambda$, 使得存在区间 $[0,1]$ 内的实数 $a_{1}, a_{2}$, $\cdots, a_{n}$, 对于满足 $0 \leqslant x_{1} \leqslant x_{2} \leqslant \cdots \leqslant x_{n} \leqslant 1$ 的任意实数 $x_{i}(i=1,2, \cdots, n)$ ，均有
## $\min _{1 \leqslant i \leqslant n}\left|x_{i}-a_{i}\right| \leqslant \lambda$.
请提供当n = 1时，\lambda的值以分数形式\frac{m}{n}，并给出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.

Determine the smallest number $M$ such that the inequality
$$\begin{vmatrix}ab(a^2-b^2)+bc(b^2-c^2)+ca(c^2-a^2)\end{vmatrix}\le M\bigl(a^2+b^2+c^2\bigr)^2$$
holds for all real numbers $a,b,c.$ The original answer is in the format \frac{m}{n}\sqrt{2}, please give 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.

Find all functions $f\colon(0,\infty)\to(0,\infty)$ such that $\frac{f(p)^2+f(q)^2}{f(r^2)+f(s^2)}=\frac{p^2+q^2}{r^2+s^2}$ for all $p,q, r, s > 0$ with $pq = rs$. There are multiple solutions for $f(x)$, represented by their respective function numbers. Please provide the sum of these numbers.

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

On some planet, there are $2^N$ countries ($N \ge 4)$. Each country has a flag $N$ units wide and one unit high composed of $N$ fields of size $1 \times 1$, each field being either yellow or blue. No two countries have the same flag. We say that a set of $N$ flags is diverse if these flags can be arranged into an $N \times N$ square so that all $N$ fields on its main diagonal will have the same color. Determine the smallest positive integer $M$ such that among any $M$ distinct flags, there exist $N$ flags forming a diverse set. Assume $N = 4$, provide the value of $M$.

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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.

For any set $A = \{a_1, a_2, a_3, a_4\}$ of four distinct positive integers with sum $s_A = a_1 + a_2 + a_3 + a_4$, let $p_A$ denote the number of pairs $(i, j)$ with $1 \leq i < j \leq 4$ for which $a_i + a_j$ divides $s_A$. Determine the maximum value of $p_A$ among all sets of four distinct positive integers.

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

Determine all pairs $(f,g)$ of functions from the set of positive integers to itself that satisfy $f^{g(n)+1}(n)+g^{f(n)}(n)=f(n+1)-g(n+1)+1$ for every positive integer $n$. Here, $f^k(n)$ means $\underbrace{f(f(\cdots f}_k(n)\cdots)).$ Provide the value of $f(1) + g(1)$.

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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.

Let $n \geq 1$ be an odd integer. Determine all functions $f$ from the set of integers to itself such that for all integers $x$ and $y$, the difference $f(x) − f(y)$ divides $x^n − y^n$. The original answer involves parameters \varepsilon, d, and c. Provide the sum of all distinct possible values of \varepsilon + d + c.

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

For each positive integer $k$, let $t(k)$ be the largest odd divisor of $k.$ Determine all positive integers $a$ for which there exists a positive integer $n$ such that all the differences $\begin{aligned}t(n+a)-t(n), t(n+a+1)-t(n+1),\cdots,t(n+2a-1)-t(n+a-1)\end{aligned}$ are divisible by $4$. Please provide the sum of all possible values of $a$.

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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.

Find all the functions $f: \mathbb{Z} \to \mathbb{Z}$ such that $$f(a)^2+f(b)^2+f(c)^2=2f(a)f(b)+2f(b)f(c)+2f(c)f(a)$$ for all integers $a,b,c$ satisfying $a+b+c=0.$ Please provide the sum of all elements within each tuple of possible values for f(a), f(b), and f(c).

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

In a $999 × 999$ square table some cells are white and the remaining ones are red. Let $T$ be the number of triples $(C_1, C_2, C_3)$ of cells, the first two in the same row and the last two in the same column, with $C_1$ and $C_3$ white and $C_2$ red. Find the maximum value T can attain. The original answer is in \frac{m}{n} format, please give 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.

Call admissible a set $A$ of integers that has the following property: If $x,y\in A$ (possibly $x=y)$ then $x^2+kxy+y^2\in A$ for every integer $k.$ Determine all pairs $m,n$ of nonzero integers such that the only admissible set containing both $m$ and $n$ is the set of all integers. Please provide the sum of the pair $(m, n)$.

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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.

Find all triples $(x, y, z)$ of positive integers such that $x ≤ y ≤ z$ and $x^3(y^3 + z^3) = 2012(xyz + 2)$. Please provide the sum of the values of x, y, and z for all such triples.

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

Find all $n\in\mathbb{N}$ for which there exist nonnegative integers $a_1,a_2,\cdots,a_n$ such that $\frac{1}{2^{a_{1}}}+\frac{1}{2^{a_{2}}}+\cdots+\frac{1}{2^{a_{n}}}=\frac{1}{3^{a_{1}}}+\frac{2}{3^{a_{2}}}+\cdots+\frac{n}{3^{a_{n}}}=1$. Please provide the sum of all such possible values of $n$.

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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.

Find the least positive integer $n$ for which there exists a set $\{s_1,s_2,\ldots,s_n\}$ consisting of $n$ distinct positive integers such that
$\left(1-\dfrac{1}{s_1}\right)\left(1-\dfrac{1}{s_2}\right)\dots\left(1-\dfrac{1}{s_n}\right)=\dfrac{51}{2010}.$
Please provide the value of $n$.

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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.

Determine all functions $f:\mathbb{Z}\to\mathbb{Z}$ satisfying $f\left(f(m)+n\right)+f(m)=f(n)+f(3m)+2014\quad(1)$ for all integers $m$ and $n$. The original answer is in the form of a function $n\longmapsto2n+1007$. Please provide the value of the function when $n = 1000$. Directly give the resulting integer value.

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

Determine all functions $f:\mathbb{Z}\to\mathbb{Z}$ with the property that $$f\big(x-f(y)\big)=f\big(f(x)\big)-f(y)-1\quad(1)$$ holds for all $x,y\in\mathbb{Z}.$ Please provide several such functions.

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

Let $n$ be a fixed positive integer. Find the maximum possible value of $$\sum\limits_{1\leqslant r<s\leqslant2n}(s-r-n)x_rx_s,$$ where $-1\leqslant x_{i}\leqslant1$ for all $i=1,2,\cdots,2n.$ Please provide the value of the expression when $n$ is a specific integer, such as $n=5$. The answer should be given in integer form, which is the product of n and (n-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.

$x_{1}, x_{2}, x_{3}, x_{4}$ 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。原始答案是\frac{m}{n}格式，请给出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.

若一个 $k$ 项等差数列的公差是 4, 且首项的平方与除首项外的各项和小于 100 ，则 $k$ 的取值范围是哪些整数的集合？请提供该集合中所有元素的和。

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Solve the following math problem step 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 $x^{2}=4 y, M(2,2)$ ，过 M 点的直线交抛物线于 $A, B$ 两点，过 $A, B$ 两点作抛物线的切线交与点 P ，求 $	riangle A B P$ 的最小面积。请给出最小面积的数值。

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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.

Given the functions $$ f(x)=\frac{x}{\ln x}, $$ and $g(x)=k(x-1)$ where $k \in \mathbf{R}$.
(1) Prove that for any $k \in \mathbf{R}$, the line $y=g(x)$ is not a tangent to the curve $y=f(x)$;
(2) If there exists $x \in[\mathrm{e}, \mathrm{e}^{2}]$, such that $$ f(x) \leqslant g(x)+\frac{1}{2}, $$ find the range of values for $k$. Please translate the following text into English.
And give the value of \(2k\) when \(k\) takes the minimum value.

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

两个半径都是 1 的球 $O_{1}$ 和球 $O_{2}$ 相切，且均与直二面角 $\alpha-l-\beta$ 的两个半平面都相切，另有一个半径为 $r(r<1)$ 的小球 $O$与这个二面角的两个半平面也都相切, 同时,与球 $O_{1}$ 和球 $O_{2}$ 都外切。则 $r$ 的值为原始答案为 $n-\sqrt{k}$ 的形式，请给出n + k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.

设动圆圆心在抛物线 $y=\frac{1}{4} x^{2}$ 上, 半径等于该圆圆心的纵坐标. 求所有这样的圆上点的集合的边界条件。请给出边界条件中最大y值的答案。

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Solve the following math problem step 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 $a 、 b$ 都是正实数, $A=\frac{a+b}{2}, B=\frac{2}{\frac{1}{a}+\frac{1}{b}}$. 若 $A+B=a-b$, 则 $\frac{a}{b}=$ 原始答案的形式为 $a+b\sqrt{c}$，请给出a+b+c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.

现代社会对破译密码的要求越来越高。在密码学中，直接可以看到的内容为明码，对明码进行某种处理后得到的内容为密码。有一种密码将英文的 26 个字母 $a, b, \cdots, z$ （不论大小写）依次对应 $1,2, \cdots, 26$ 这 26 个自然数，见表1.\n| $a$ | $b$ | $c$ | $d$ | $e$ | $f$ | $g$ | $h$ | $i$ | $j$ | $k$ | $l$ | $m$ |\n| :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |\n| 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 |\n| $n$ | $o$ | $p$ | $q$ | $r$ | $s$ | $t$ | $u$ | $v$ | $w$ | $x$ | $y$ | $z$ |\n| 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | 22 | 23 | 24 | 25 | 26 |\n给出明码对应的序号 $x$ 和密码对应的序号 $y$ 的变换公式：\n$$\ny= \begin{cases}\frac{x+1}{2}, & x \text { 为奇数,且 } 1 \leqslant x \leqslant 26 ; \\ \frac{x}{2}+13, & x \text { 为偶数,且 } 1 \leqslant x \leqslant 26.\end{cases}\n$$\n利用它可将明码转换成密码，如 $5 \rightarrow$ $\frac{5+1}{2}=3$, 即 $e$ 变成 $c ; 8 \rightarrow \frac{8}{2}+13=17$, 即 $h$ 变成 $q$ 。\n按上述公式，若将某明码译成的密码是\n$s h x c$ ，请提供原始明码中每个字母对应数字的和。

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Solve the following math problem step 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 $P$ 为直线 $y=x-2$ 上的动点，过 $P$ 作抛物线 $y=\frac{1}{2} x^{2}$ 的切线，切点分别为 $A 、 B$.
(1) 证明: 直线 $A B$ 过定点;
(2) 求 $\triangle P A B$ 面积 $S$ 的最小值，以及取得最小值时点 $P$ 的坐标。请提供点P的坐标之和。

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

设不经过坐标原点 $O$ 的直线 $l$ 与圆 $x^{2}+y^{2}=1$ 交于不同的两点 $P 、 Q$. 若直线 $P Q$ 的斜率是直线 $O P$ 和 $O Q$ 斜率的等比中项,求 $	riangle P O Q$ 面积 $S$ 的取值范围。请给出区间两端点之和的整数值。

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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: $a_{0}=\sqrt{6}, a_{n+1}=\left[a_{n}\right]+\frac{1}{\left\{a_{n}\right\}}$ (其中 $\left[a_{n}\right]$ 和 $\left\{a_{n}\right\}$ 分别表示实数 $a_{n}$ 的整数部分与小数部分), 求 $a_{2020}$ 的整数部分

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Solve the following math problem step 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 $x, y$ 满足: $\frac{1}{x+3 y}+\frac{1}{2 x+y}=1$, 则 $x+y$ 的最小值是。原始答案是 \frac{m+2\sqrt{n}}{k} 的格式，请给出 m + n + k 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 $\triangle A B C$ 的外接圆圆心为 $O$, 且 $3 \overrightarrow{O A}+4 \overrightarrow{O B}+5 \overrightarrow{O C}=\mathbf{0}$, 则 $\angle C$ 的大小是？请提供 $\angle C$ 的度数的2倍值。

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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.

正四面体的 4 个表面上分别写有数字 $1,2,3,4$, 将 4 个这样的密度均匀的正四面体同时投掷于桌面上，与桌面接触的 4 个面上的 4 个数的和能被 4 整除的概率是 。请给出答案为\(\frac{m}{n}\)形式时，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 $x, y$ 满足: $\frac{1}{x+3 y}+\frac{1}{2 x+y}=1$, 则 $x+y$ 的最小值是 \( \frac{3+2\sqrt{2}}{5} \) 的形式。请给出 \( m + n + k \) 的值，其中原始答案为 \( \frac{m+n\sqrt{2}}{k} \)。

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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.

在三棱雉 $D-A B C$ 中, $\angle B A D=30^{\circ}$, $\angle A C B=\angle A B D=90^{\circ}, C A=C B$. 若点 $C$ 在平面 $A B D$ 上的射影恰在 $A D$ 上, 则二面角 $C$ $A B-D$ 的平面角的正弦值大小为 $\frac{\sqrt{m}}{n}$ 的形式，请给出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
$$
(1+\sqrt{3})^{n}=a_{n}+b_{n} \sqrt{3},
$$
其中, $a_{n}, b_{n}$ 为整数. 则 $\lim _{n \rightarrow+\infty} \frac{a_{n}}{b_{n}}=$ $\qquad$.
请提供形式为$\sqrt{m}$时m + 1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 $f(x)=\frac{x^{3}+3 x}{3 x^{2}+1}$, 数列 $\left\{x_{n}\right\}$ 满足 $x_{1}=2, x_{n+1}=f\left(x_{n}\right)\left(n \in \mathbf{N}_{+}\right)$. 记 $b_{n}=\log _{3}\left(\frac{x_{n+1}-1}{x_{n+1}+1}\right)\left(n \in \mathbf{N}_{+}\right)$. （1）证明: $\left\{b_{n}\right\}$ 为等比数列, 并求数列 $\left\{b_{n}\right\}$的通项公式； (2) 记 $c_{n}=-n b_{n}\left(n \in \mathbf{N}_{+}\right)$, 求数列 $\left\{c_{n}\right\}$的前 $n$ 项和公式 $T_{n}$ 。请提供 $T_{n}$ 的值当 $n = 1$ 时。答案应为整数形式。

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Solve the following math problem step 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 $x 、 y 、 z$ 为正实数. 求
$$
\left(x+\frac{1}{y}+\sqrt{2}\right)\left(y+\frac{1}{z}+\sqrt{2}\right)\left(z+\frac{1}{x}+\sqrt{2}\right)
$$
的最小值。请给出整数部分和非整数部分系数的和。

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

Find all $r>0$ such that whenever $f: \mathbb{R}^{2} \rightarrow \mathbb{R}$ is a differentiable function such that $|\operatorname{grad} f(0,0)|=1$ and $|\operatorname{grad} f(u)-\operatorname{grad} f(v)| \leq|u-v|$ for all $u, v \in \mathbb{R}^{2}$, then the maximum of $f$ on the disk $\left\{u \in \mathbb{R}^{2}:|u| \leq r\right\}$ is attained at exactly one point. $\left(\operatorname{grad} f(u)=\left(\partial_{1} f(u), \partial_{2} f(u)\right)\right.$ is the gradient vector of $f$ at the point $u$. For a vector $\left.u=(a, b),|u|=\sqrt{a^{2}+b^{2}}.\right)$ The original answer is in \frac{m}{n} format, please give 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.

Find the number of positive integers $x$ satisfying the following two conditions: 1. $x<10^{2006}$; 2. $x^{2}-x$ is divisible by $10^{2006}$. Please provide the sum of all such positive integers.

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

Let $p$ be a prime number and let $k$ be a positive integer. Suppose that the numbers $a_{i}=i^{k}+i$ for $i=0,1, \\ldots, p-1$ form a complete residue system modulo $p$. What is the set of possible remainders of $a_{2}$ upon division by $p$? Please provide the sum of the possible remainders.

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

For $n=1,2, \ldots$ let
$$
S_{n}=-\log \left(\sqrt[n^{2}]{1^{1} \cdot 2^{2} \cdot \ldots \cdot n^{n}}\right)+\log (\sqrt{n})
$$
where $\log$ denotes the natural logarithm. Find $\lim _{n \rightarrow \infty} S_{n}$.
 If the value can be expressed in the form $\frac{p}{q}$ for relatively prime positive integers $p$ and $q$, find $p+q$.

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

Let for $j=0, \ldots, n, a_{j}=a_{0}+j d$, where $a_{0}, d$ are fixed real numbers. Put
$$
A=\left(\begin{array}{ccccc}
a_{0} & a_{1} & a_{2} & \ldots & a_{n} \\
a_{1} & a_{0} & a_{1} & \ldots & a_{n-1} \\
a_{2} & a_{1} & a_{0} & \ldots & a_{n-2} \\
\ldots & \ldots & \ldots \ldots \ldots & \ldots & \ldots \\
a_{n} & a_{n-1} & a_{n-2} & \ldots & a_{0}
\end{array}\right)
$$
Calculate $\operatorname{det}(A)$, where $\operatorname{det}(A)$ denotes the determinant of $A$. Please provide the value of $\operatorname{det}(A)$ when $a_0 = 0$, $a_n = 1,n=4$, and $d = 0$

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

Let $A_{n}=\{1,2, \ldots, n\}$, where $n \geq 3$. Let $\mathcal{F}$ be the family of all non-constant functions $f: A_{n} \rightarrow A_{n}$ satisfying the following conditions:\n(1) $f(k) \leq f(k+1)$ for $k=1,2, \ldots, n-1$,\n(2) $f(k)=f(f(k+1))$ for $k=1,2, \ldots, n-1$.\nFind the number of functions in $\mathcal{F}$. for $n=3$

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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.

已知函数 $f(x)=a \sin x-\frac{1}{2} \cos 2 x+a-\frac{3}{a}+\frac{1}{2}, a \in \mathbf{R}, a \neq 0$.\n（1）若对任意 $x \in \mathbf{R}$, 都有 $f(x) \leqslant 0$, 求 $a$ 的取值范围;\n（2）若 $a \geqslant 2$ ，且存在 $x \in \mathbf{R}$ ，使得 $f(x) \leqslant 0$ ，已知$a$的范围是一个区间，区间两个端点的和为分数$\frac{p}{q}$的形式，请求出$p+q$

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Solve the following math problem step 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 $x>0, y>0$, 且满足 $\left\{\begin{array}{l}\cos ^{2}(\pi x)+2 \sin (\pi y)=1 \\ \sin (\pi x)+\sin (\pi y)=0 \\ x^{2}-y^{2}=12\end{array}\right.$。请给出 x + y 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 $x_{1}, x_{2}, x_{3}$ 是非负实数，满足 $x_{1}+x_{2}+x_{3}=1$ ，求 $(x_{1}+3 x_{2}+5 x_{3})(x_{1}+\frac{x_{2}}{3}+\frac{x_{3}}{5})$ 的最小值和最大值。最大值的原始答案为\frac{m}{n}的格式，请给出m + n的值，并加上最小值。

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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.

长为 $\sqrt{2}$, 宽为 1 的矩形, 以它的一条对角线所在的直线为轴旋转一周, 求得到的旋转体体积。原始答案为\frac{m \sqrt{p} \pi}{n}的格式，请给出m + n+p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 $\log _{a} \sqrt{2}<1$, 请给出$a$ 的取值范围。原始答案形式为$(a,b) \cup(\sqrt{c},+\infty)$,计算a+b+c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.

给定整数 $n \geq 2$. 设非负实数 $a_{1}, a_{2}, \cdots, a_{n}$ 满足
$$
a_{1} \geq a_{2} \geq \cdots \geq a_{n}, \quad a_{1}+a_{2}+\cdots+a_{n}=n.
$$
求 $a_{1}+a_{1} a_{2}+a_{1} a_{2} a_{3}+\cdots+a_{1} a_{2} \cdots a_{n}$ 的最小值。请给出当n=2时的答案。

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Solve the following math problem step 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 $a_{1}$ 的值为原始答案的格式为 \( \frac{m}{n} \)，请给出 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 $f(x)$ 满足: 对任意非零实数 $x$, 均有 $f(x)=f(1) \cdot x+\frac{f(2)}{x}-1$, 则 $f(x)$ 在 $(0,+\infty)$ 上的最小值为 $\qquad$.答案格式为$\sqrt{a}-b$,求a+b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, $A B=1, A C=2, B-C=\frac{2 \pi}{3}$, 则 $\triangle A B C$ 的面积为 $\qquad$.答案格式为\frac{a \sqrt{b}}{c},求a+b+c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.

已知复数列 $\left\{z_{n}\right\}$ 满足:  $$  z_{1}=\frac{\sqrt{3}}{2}, z_{n+1}=\overline{z_{n}}\left(1+z_{n} \mathrm{i}\right)(n=1,2, \cdots),  $$  其中 $\mathrm{i}$ 为虚数单位. 求 $z_{2021}$ 的值。原始答案格式为$\frac{\sqrt a}{b}+(\frac{c}{d}+\frac{e}{f^{g^{h}}})\mathrm i$,求a+b+c+d+e+f+g+h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 $A, B$ 为平面直角坐标系 $x O y$ 中的曲线 $x y=1(x, y>0)$ 上的两点, 向量 $\vec{m}=(1,|O A|)$ ，则数量积 $\vec{m} \cdot \overrightarrow{O B}$ 的最小值是多少。原始答案是 $a \sqrt[4]{b}$ 的形式，请给出$a + b$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, $\cos A, \sin A, \tan B$ 依次构成以 $\frac{3}{4}$ 为公比的等比数列,则 $\cot C$ 的原始答案是以$-\frac{m}{n}$的形式给出，请给出$m + n$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，椭圆 $\Gamma: \frac{x^{2}}{4}+\frac{y^{2}}{3}=1$ ，点 $P$ 在椭圆 $\Gamma$ 内部，且在直线 $y=x$ 上移动。点 $K, L$ 在 $\Gamma$ 上，满足 $\overrightarrow{P K}, \overrightarrow{P L}$ 的方向分别为 $x$ 轴正方向与 $y$ 轴正方向，点 $Q$ 使得 $P K Q L$ 为矩形。是否存在平面上两点 $A, B$ ，使得当矩形 $P K Q L$ 变化时， $|Q A|-|Q B|$ 为非零常数？若存在，求出线段 $A B$ 的所有可能的长度；若不存在，请说明理由。请给出原始答案格式为$\frac{m\sqrt{3}}{n}$时，$m + n$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 $a, b, c$ 满足 $a+b+c=1$, 则 $a^{2}(b-c)+b^{2}(c-a)+c^{2}(a-b)$ 的最大值为 $\frac{m+\sqrt{n}}{p}$的格式，请给出 $m + n + p$ 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 $(\cdots((2 * 3) * 4) \cdots) * 21$结果的格式为$\frac{m}{n}$，给出$m + n$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.

若非负数 $a, b, c$ 满足 $a+b+c=1$ ，则 $a^{2}(b-c)+b^{2}(c-a)+c^{2}(a-b)$ 的最大值是 $\frac{\sqrt{m}}{n}$ 的格式，请给出 $m + n$ 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.

已知矩形 $A B C D$ 中, $A B=2, B C=1$, 折叠使点 $A, C$ 重合, 折痕为 $M N$, 打开平面 $A D M N$, 使二面角 $A-M N-C$ 为 $\frac{\pi}{3}$, 求直线 $M N$ 与直线 $A C$ 的距离，原始答案为 $\frac{\sqrt{m}}{n}$ 的格式，请给出 $m + n$ 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.

已知在 $\triangle A B C$ 中, $D$ 是边 $B C$ 的中点, 且 $\angle D A C=15^{\circ}$, 求 $\angle A B C$ 的最大值。原始答案为 $\frac{m\pi}{n}$ 的形式，请给出 $m + n$ 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.

已知正方体 $A B C D-A^{\prime} B^{\prime} C^{\prime} D^{\prime}$ 的棱长为 1, 点 $M, N$ 分别为线段 $A B^{\prime}, A C$ 上的动点, 点 $T$在屏幕 $B C C^{\prime} B^{\prime}$ 内，求 $|M T|+|N T|$ 的最小值。原始答案为 $\frac{a\sqrt{b}}{c}$ 格式，请给出 $a + b+c$ 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 $\mathrm{I}_{\mathrm{n}}=\left(1+\frac{\mathrm{i}}{2}\right)\left(1+\frac{\mathrm{i}}{4}\right)\left(1+\frac{\mathrm{i}}{16}\right) \cdots\left(1+\frac{\mathrm{i}}{2^{2 \mathrm{n}}}\right)$ ，求 $\left|\mathrm{I}_{\mathrm{n}}\right|$的表达式，给出当$n=1$时该式整数部分的值。

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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.

求椭圆 $C: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ 的内接六边形面积的最大值。已知原始答案的格式为$\frac{m\sqrt{3}}{n}$，请给出$m + n$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 $\left\{a_{n}\right\}$ ，且 $S=n a+n_{n}(n-1)$，点 $\left(a_{n}, \frac{S_{n}}{n}\right)$ 所在的直线方程的斜率的形式为$\frac{n}{m}$，求$m+n$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 $\operatorname{Re}(z) \geq 1$, 求 $\left|\frac{1}{z}+1+\mathrm{i}\right|_{\text {min }}$ 的值。原始答案为 $\frac{\sqrt{m}-n}{k}$ 的格式，请给出 $m + n + k$ 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 $x^{2}=4 y, M(2,2)$, 过 $M$ 点的直线交抛物线于 $A, B$ 两点, 过 $A, B$ 两点作抛物线的切线交与点 $P$, 求 $S_{\triangle A B P}$ 的最小值。请给出该最小值。

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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 $a x^{3}+b x^{2}+x+1=0(a<0)$ 恰有两个零点, 求 $a+b$ 的取值范围. 请给出区间右端点的整数部分。

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Solve the following math problem step 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, $\sin A=\frac{\sqrt{2}}{2}$,  $\cos B+\sqrt{2} \cos C$ 的取值范围的形式为$(a,b] \cup(c, \sqrt{d}]$，求$a+b+c+d$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 $\overrightarrow{A F_{1}} \cdot \overrightarrow{A F_{2}}+\overrightarrow{B F_{1}} \cdot \overrightarrow{B F_{2}}=0$, 求 $\tan \angle A B F_{1} \cdot \tan \angle A B F_{2}$ 的值。原始答案为 \frac{m}{n} 的格式，请给出 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 $a, b \in \mathbf{R}$ 。若关于 $z$ 的方程\[\left(z^{2}+a z+b\right)\left(z^{2}+a z+2 b\right)=0\]有 4 个互不相等的复数根 $z_{1}, z_{2}, z_{3}, z_{4}$, 且它们在复平面上对应的点恰是一个边长为 1 的正方形的四个顶点，求 $\left|z_{1}\right|+\left|z_{2}\right|+\left|z_{3}\right|+\left|z_{4}\right|$ 的值。原始答案为\sqrt{m} + n\sqrt{p}，请给出m + n + p + q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.

平面向量 $\boldsymbol{a}, \boldsymbol{b}$ 夹角为 $\frac{\pi}{3}$, 若 $|\boldsymbol{a}|,|\boldsymbol{b}|,|\boldsymbol{a}+\boldsymbol{b}|$ 依次成等差数列, 求 $|\boldsymbol{a}|:|\boldsymbol{b}|:|\boldsymbol{a}+\boldsymbol{b}|$. 请给出比例的和。

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Solve the following math problem step 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 $f(x)$ 的定义域为 $\mathbf{R}$, 且当 $x \geq 0$ 时, $f(x)=|x-2|+a$ (其中 $a$ 为实数). 若 $f(x)$ 为奇函数, 则不等式 $f(x) \geq 1$ 的解集为$[a,b] \cup[c,+\infty)$，求$a+b+c$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.

设正整数数列 $\left\{a_{n}\right\}$ 同时具有以下两个性质：\n（i）对任意正整数 $k$ ，均有 $a_{2 k-1}+a_{2 k}=2^{k}$ ；\n(ii) 对任意正整数 $m$, 均存在正整数 $l \leq m$, 使得 $a_{m+1}=\sum_{i=l}^{m} a_{i}$。 $a_{2}+a_{4}+a_{6}+\cdots+a_{2022}$ 的最大值为$\frac{2^{m}-n}{t}$，求$m + n+t$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 $\triangle A B C$ 满足 $\frac{\angle A}{3}=\frac{\angle B}{4}=\frac{\angle C}{5}$, 则 $\frac{\overrightarrow{A B} \cdot \overrightarrow{A C}}{|B C|^{2}}$ 的值为$\frac{a+\sqrt{b}}{c}$，求$a+b+c$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.

对于和为 1 的九个非负实数 $a_{1}, a_{2}, \cdots, a_{9}$, 令
$$
\begin{aligned}
& S=\min \left\{a_{1}, a_{2}\right\}+2 \min \left\{a_{2}, a_{3}\right\}+\cdots+8 \min \left\{a_{8}, a_{9}\right\}+9 \min \left\{a_{9}, a_{1}\right\}, \\
& T=\max \left\{a_{1}, a_{2}\right\}+2 \max \left\{a_{2}, a_{3}\right\}+\cdots+8 \max \left\{a_{8}, a_{9}\right\}+9 \max \left\{a_{9}, a_{1}\right\},
\end{aligned}
$$
这里, $\min \{x, y\}$ 表示 $x, y$ 中的较小者, $\max \{x, y\}$ 表示 $x, y$ 中的较大者.
记 $S$ 的最大可能值为 $S_{0}$ 。 当 $S=S_{0}$ 时,  $T$ 最大值与最小值之和为$\frac{b}{a}$，求$a+b$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.

若四棱雉 $P-A B C D$ 的棱 $A B, B C$ 的长均为 $\sqrt{2}$, 其他各条棱长均为 1, 则该四棱雉的体积为$\frac{\sqrt{m}}{n}$的形式，请给出$m + n$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: $\frac{z-3 i}{z+i}$ 为负实数 ( i 为虚数单位), $\frac{z-3}{z+1}$ 为纯虚数, 则 $z$ 的值为$\sqrt{m} \mathrm{i}$ 的形式，请给出 $m$ 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 $f(x)=\frac{x^{2}+x+16}{x}(2 \leq x \leq a)$, 其中实数 $a>2$. 若 $f(x)$ 的值域为 $[9,11]$, 请给出 $a$ 的取值范围端点的和。

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

设正数 $a_{1}, a_{2}, a_{3}, b_{1}, b_{2}, b_{3}$ 满足: $a_{1}, a_{2}, a_{3}$ 成公差为 $b_{1}$ 的等差数列, $b_{1}, b_{2}, b_{3}$ 成公比为 $a_{1}$ 的等比数列, 且 $a_{3}=b_{3}$. 求 $a_{3}$ 的最小值, 并确定当 $a_{3}$ 取到最小值时 $a_{2} b_{2}$ 的值。原始答案为$\frac{m\sqrt{n}}{k}$格式，请给出$m + n + k$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, $F_{1}, F_{2}$ 是双曲线 $\Gamma: \frac{x^{2}}{3}-y^{2}=1$ 的两个焦点, $\Gamma$ 上一点 $P$ 满足 $\overrightarrow{P F_{1}} \cdot \overrightarrow{P F_{2}}=1$. 求点 $P$ 到 $\Gamma$ 的两条渐近线的距离之和。原始答案为 $\frac{m \sqrt{n}}{k}$ 格式，请给出 $m + n + k$ 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.

若正四棱雉 $P-A B C D$ 的各条棱长均相等, $M$ 为棱 $A B$ 的中点, 则异面直线 $B P$ 与 $C M$ 所成的角的余弦值为 $\frac{\sqrt{m}}{n}$ 的形式，请给出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 $a=1+10^{-4}$ 。在 $2023 \times 2023$ 的方格表的每个小方格中填入区间 $[1, a]$ 中的一个实数。设第 $i$ 行的总和为 $x_{i}$ ，第 $i$ 列的总和为 $y_{i}$ ， $1 \leq i \leq 2023$ 。 用$a$表示$\frac{y_{1} y_{2} \cdots y_{2023}}{x_{1} x_{2} \cdots x_{2023}}$ 的最大值为$\frac{(m a+n)^{p}}{p^{p} a^{m}}$，求$m+n+p$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.

方程 $\sin x=\cos 2 x$ 的最小的 $20$ 个正实数解之和为$m\pi$, 请给出$m$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 $m, n$ 是给定的整数， $m \geq n \geq 3$ 。求具有下述性质的最小正整数 $k$ ：若将 $1,2, \cdots, k$ 中的每个数任意染为红色或者蓝色，则或者存在 $m$ 个红色的数 $x_{1}, x_{2}, \cdots, x_{m}$ （允许相同），满足 $x_{1}+x_{2}+\cdots+x_{m-1}<x_{m}$ ，或者存在 $n$ 个蓝色的数 $y_{1}, y_{2}, \cdots, y_{n}$ （允许相同），满足 $y_{1}+y_{2}+\cdots+y_{n-1}<y_{n}$ 。请给出 $m=5,n=4$时$k$ 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.

求出所有满足下面要求的不小于 -1 的实数 $t$ ：对任意 $a \in[-2, t]$, 总存在 $b, c \in[-2, t]$, 使得 $a b+c=1$. 请给出满足条件的最小整数。

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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.

设复数 $z, w$ 满足 $z+w=2$, 求 $S=\left|z^{2}-2 w\right|+\left|w^{2}-2 z\right|$的最小可能值。原始答案为$a\sqrt{b}-c$的格式，请给出$a+b+c$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 $\cos C$ 的值. 该答案的原始形式为 \(\frac{\sqrt{m}}{n}\)，请给出 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 $a_{1}=\frac{\pi}{6}, a_{n+1}=\arctan \left(\sec a_{n}\right)\left(n \in \mathbb{N}^{*}\right)$ 。求正整数 $m$ ，使得 $$ \sin a_{1} \cdot \sin a_{2} \cdots \sin a_{m}=\frac{1}{100} $$ 请给出正整数 $m$ 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 $\Omega$ 中 $F_{1}, F_{2}$ 为焦点, $A$ 为长轴的一个端点, $B$ 为短轴的一个端点, 若 $\angle F_{1} B F_{2}=\angle F_{1} A B$ ，则 $\Omega$ 的离心率为。原始答案为 \frac{\sqrt{m} - n}{k} 的格式，请给出 m + n + k 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.

方程 $\cos \frac{X}{4}=\cos x$ 的通解是 $\qquad$ ，在 $(0,24 \pi)$ 内不相同的解有多少个？请给出不相同解的个数。

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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.

方程 $\cos \frac{x}{4}=\cos x$ 的通解是 $\qquad$, 在 $(0,24 \pi)$ 内不相同的解有多少个？请给出在 $(0,24 \pi)$ 内不相同的解的个数加1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， $A B$ 是单位圆的直径，在 $A B$ 上任取一点 $D$ ，作 $D C \perp A B$, 交圆周于 $C$ 。若点 $D$ 的坐标为 $(x, 0)$,则当 $x \in$ $\qquad$时，线段 $A D 、 B D 、 C D$ 可以构成锐角三角形。请提供区间的两个端点之和。

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Solve the following math problem step 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, 在正方体 $A B C D-A_{1} B_{1} C_{1} D_{1}$ 中, $E$ 是 $B C$ 中点, $F$ 在 $A A_{1}$ 上, 且 $A_{1} F: F A=1: 2$. 求平面 $B_{1} E F$与底面 $A_{1} B_{1} C_{1} D_{1}$ 所成的二面角。原始答案为$\mathrm{arctan}\frac{\sqrt{m}}{n}$，请给出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，在正方体 $A B C D-A_{1} B_{1} C_{1} D_{1}$ 中， $E$ 是 $B C$ 的中点， $F$ 在 $A A_{1}$ 上，且 $A_{1} F: F A=1: 2$, 求平面 $B_{1} E F$ 与底面 $A_{1} B_{1} C_{1} D_{1}$ 所成的二面角。已知原始答案为\( \arctan \frac{\sqrt{m}}{n} \)的格式，请给出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.

长为 $\sqrt{2}$ ，宽为 1 的矩形，以它的一条对角线所在的直线为轴旋转一周，求得到的旋转体的体积。原始答案为 \frac{m}{n} \sqrt{k \pi} 的形式，请给出 m + n + k 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.

$(\sqrt{x}+2)^{2 n+1}$ 的展开式中, $x$ 的整数次幂的各项系数之和为 $\qquad$.请给出
n=2时的结果

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

已知复数数列 $\left\{z_{n}\right\}$ 满足 $z_{1}=1, z_{n+1}=\overline{z_{n}}+1+n \mathrm{i}$ ，其中 $n=1,2, \cdots$ ，其中 i 是虚数单位, $\overline{z_{n}}$ 表示 $z_{n}$ 的共轭复数, 则 $z_{2015}$ 的值中实数部分为 $\qquad$。

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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 $a_{1}, a_{2}, a_{3}, a_{4}$ 是 4 个有理数, 使得
$$
\left\{a_{i} a_{j} \mid 1 \leqslant i<j \leqslant 4\right\}=\left\{-24,-2,-\frac{3}{2},-\frac{1}{8}, 1,3\right\}
$$
求 $a_{1}+a_{2}+a_{3}+a_{4}$ 的所有可能值之和。

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Solve the following math problem step 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 $\lambda$的八倍， 使得不等式
$$
\sum_{k=1}^{n} x_{k}^{3}\left(x_{k}-x_{k-1}\right) \geq \frac{1}{4}+\frac{\lambda}{n}
$$
对任意正整数 n 以及任意实数 $0=x_{0} \leq x_{1} \leq x_{2} \leq \cdots \leq x_{n}=1$ 均成立.

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

Evaluate$$\lim_{n\rightarrow\infty}\frac{1}{n}\sum_{k=1}^{n}\left(\left[\frac{2n}{k}\right]-2\left[\frac{n}{k}\right]-\ln4\right)$$
and express your answer in the form $\log a-b$, with $a$ and $b$ positive integers.
Here $[x]$ is defined to be the integer such that $[x]\leqq x<[x]+1$ and $\log x$ is the logarithm of $x$ to base $e$.

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