sungyub/dapo-math-17k-verl huggingface dataset:
user: 已知等差数列 $\left\{a_{n}\right\}$ 的公差 $d$ 不为 0, 等比数列 $\left\{b_{n}\right\}$ 的公比 $q$ 是小于 1 的正有理数. 若 $a_{1}=d, b_{1}=$ $d^{2}$, 且 $\frac{a_{1}^{2}+a_{2}^{2}+a_{3}^{2}}{b^{1}+b^{2}+b^{3}}$ 是正整数, 则 $q$ 等于 $\qquad$.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值

Answer: 3
user: 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。

Answer: 75
user: 已知函数 $f(x)=a x^{3}+b x^{2}+c x+d(a \neq 0)$, 当 $0 \leq x \leq 1$ 时, $\left|f^{\prime}(x)\right| \leq 1$,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

Answer: 11
user: 两人轮流投掷骰子，每人每次投掷两颗，第一个使两颗骰子点数和大于 6 者为胜，否则轮由另一人投郑。先投掷人的获胜概率是 $\qquad$.
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

Answer: 29
user: 把扑克牌中的 $A, 2, \cdots, J, Q, K$ 分别看作数字 $1,2, \cdots, 11,12,13$. 现将一幅扑克牌中的
黑桃、红桃各 13 张放在一起，从中随机取出 2 张牌，其花色相同且两个数的积是完全平方数的概率为 $\qquad$.
原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值

Answer: 67
user: 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

Answer: 15
user: 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

Answer: 4020
user: 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

Answer: 7
user: 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

Answer: 4028
user: 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

Answer: 4028
user: 已知 $\triangle A B C$ 的外心为 $O$, 且 $2 \overrightarrow{O A}+3 \overrightarrow{O B}+4 \overrightarrow{O C}=0$, 则 $\cos \angle B A C$ 的值是 $\qquad$. 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

Answer: 5
user: 实数 $x, y, z, w$ 满足 $x+y+z+w=1$ ，则 $M=x w+2 y w+3 x y+3 z w+4 x z+5 y z$ 的最大值是 $\qquad$.原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值

Answer: 5
user: 已知正四棱雉 $S-A B C D$ 的侧棱长为 $4, \angle A S B=30^{\circ}$, 过点 $A$ 作截面与侧棱 $S B, S C, S D$ 分别交于 $E, F, G$ ，则截面 $A E F G$ 周长的最小值是 $\qquad$.原始的答案是k\sqrt{m}+n的形式，请给出k + m + n的值。

Answer: 7
user: 已知 $\triangle A B C$ 的外心为 $O$, 且
$2 \overrightarrow{O A}+3 \overrightarrow{O B}+4 \overrightarrow{O C}=\mathbf{0}$.
则 $\cos \angle B A C=$原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值

Answer: 5
user: 设实数 $x 、 y 、 z 、 w$ 满足 $x+y+z+w=1$.则 $M=x w+2 y w+3 x y+3 z w+4 x z+5 y z$ 的最大值为 $\qquad$.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

Answer: 5
user: 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。

Answer: 7
user: 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.

Answer: 5
user: 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。

Answer: 16
user: 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。

Answer: 16
user: 将一枚骰子连续投掷五次，则事件 "五次出现的点数既不全相同，也不两两互异，且从第二次起每一次的点数都不小于前一次的点数"的概率为 $\qquad$. 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

Answer: 167
user: 已知 $a_{1}=1, a_{n+1}=\lambda a_{n}^{2}+2\left(n \in \mathbf{N}^{*}\right)$, 若数列 $\left\{a_{n}\right\}$ 有上界, 即存在常数 $M$, 使得 $a_{n} \leqslant M$ 对 $n \in \mathbf{N}^{*}$ 恒成立，则实数 $\lambda$ 的最大值为 $\qquad$. 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

Answer: 9
user: 已知 $a_{1}=1, a_{n+1}=\lambda a_{n}^{2}+2\left(n \in \mathbf{Z}_{+}\right)$.若数列 $\left\{a_{n}\right\}$ 有上界，即存在常数 $M>0$ ，使得 $a_{n} \leqslant M$ 对 $n \in \mathbf{Z}_{+}$恒成立，则实数 $\lambda$ 的最大值是 $\qquad$。 原始的答案是\frac{m}{n}的形式，其中m、n是互质的。请给出最终m + n的值

Answer: 9
user: 将一枚骰子连续投掷五次，则事件 "五次出现的点数既不全相同也不两两互异，且从第二次起每一次的点数都不小于前一次的点数"的概率为 $\qquad$。 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

Answer: 167
user: 一只蚂蚁在正方体 $A B C D-E F G H$ 的顶点 $A$ 处，每次等概率地爬行到相邻三个顶点中的一个，那么六次爬行之后回到顶点 $A$ 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

Answer: 304
user: 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

Answer: 22
user: 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

Answer: 149
user: 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

Answer: 19
user: 设函数 $f(x)=a x^{2}+b x+c \quad(a \neq 0)$ 满足
$|f(0)| \leq 2, \quad|f(2)| \leq 2, \quad|f(-2)| \leq 2$,
求当 $x \in[-2,2]$ 时 $y=|f(x)|$ 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

Answer: 7
user: 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

Answer: 2019
user: 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

Answer: 3
user: 四面体 $A B C D$ 中, $A B \perp B C, C D \perp B C, B C=2$, 且异面直线 $A B$ 与 $C D$ 所成的角为 $60^{\circ}$ 。若四面体 $A B C D$ 的外接球半径为 $\sqrt{5}$, 则四面体 $A B C D$ 体积的最大值为 $\qquad$. 原始的答案是k\sqrt{m}+n的形式，请给出k + m + n的值。

Answer: 5
user: 求满足方程 $\cos x+\cos 2 x-\cos 3 x=1, x \in R$ 的所有 $\cos x$ 的平方和.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

Answer: 13
user: 正 2015 边形 $A_{1} A_{2} \cdots A_{2015}$ 内接于单位圆 $O$, 任取它的两个不同顶点 $A_{i}, A_{j}$,则 $\left|\overrightarrow{O A_{i}}+\overrightarrow{O A_{j}}\right| \geq 1$ 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

Answer: 1678
user: 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 original answer is in \frac{m}{n} format, please give the value of m + n.

Answer: 529
user: 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 .

Answer: 4037
user: For each real number $x$, let
$$f(x)=\sum_{n\in S_x}\frac{1}{2^n},$$
where $S_x$ is the set of positive integers $n$ for which $\lfloor nx\rfloor$ is even. What is the largest real number $L$ such that $f(x) ≥ L$ for all $x ∈ [0,1)?$ The original answer is in the format \frac{m}{n}, please provide the value of m + n.

Answer: 11
user: Let
$$I(R)=\iint_{x^2+y^2\leq R^2}\left(\frac{1+2x^2}{1+x^4+6x^2y^2+y^4}-\frac{1+y^2}{2+x^4+y^4}\right)dxdy.$$ Find
$$\lim_{R\to\infty}I(R),$$
or show that this limit does not exist. The original answer is in \frac{\sqrt{a}}{b}\pi\logc format, please provide the value of a+b+c.

Answer: 6
user: 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.

Answer: 8
user: 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?

Answer: 1
user: Let A be the set of all integers $n$ such that $1\leq n\leq2021$ and $\gcd(n,2021)=1.$ For every nonnegative integer $j$, let $$S(j)=\sum_{n\in A}n^j.$$ Determine the smallest positive integer $k$ such that all values of $j$ not divisible by $k$ make $S(j)$ a multiple of 2021. Calculate the sum of all possible values of k.

Answer: 88
user: 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.

Answer: 11
user: 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.

Answer: 35
user: 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.

Answer: 5
user: 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.

Answer: 14
user: Let $f$ be a function on $[0,\infty)$, differentiable and satisfying $$f'(x)=-3f(x)+6f(2x)$$ for $x>0.$ Assume that $|f(x)|\leqslant e^{-\sqrt{x}}$ for $x\geqslant0$ (so that $f(x)$ tends rapidly to 0 as $x$ increases). For $n$ a non-negative integer, define $$\mu_n=\int_0^\infty x^nf(\:x\:)\:dx$$ (sometimes called the nth moment of $f).$ a. Express $\mu_n$ in terms of $\mu_0.$ Given that $n = 1$, please provide the value of m + n where the expression is in the format \(\frac{m}{n}\).

Answer: 5
user: 已知 \( \frac{1-4 \sin \alpha}{\tan \alpha}=\sqrt{3}(\alpha \in\left(0, \frac{\pi}{2}\right)) \). 原始答案为 \( \frac{m\pi}{n} \) 的形式，请给出 m + n 的值.

Answer: 19
user: 已知 $\left(1+x+x^{2}+x^{3}\right)^{400}$
$$
\begin{aligned}
& =\sum_{k=0}^{300} c_{4 k} x^{4 k}+\sum_{k=0}^{299} c_{4 k+1} x^{4 k+1}+ \\
& \sum_{k=0}^{299} c_{4 k+2} x^{4 k+2}+\sum_{k=0}^{299} c_{4 k+3} x^{4 k+3}. \\
\end{aligned}
$$
则 $\sum\limits_{k=0}^{300} c_{4 k}=$The standard form of the answer is $a^{b}$. What is the value of a + b?

Answer: 403
user: 集合 $M$ 是集合 $A=\{1,2, \cdots, 100\}$ 的子集，且 $M$ 中至少含有一个偶数或者立方数。则这种子集 $M$ 的个数是 $\qquad$。答案的标准形式$a^{b}(c^{d}-e)$，则a+b+c+d+e的值为多少。

Answer: 105
user: 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 $r + s\pi$, please give the value of $r + s$.

Answer: 4
user: Let $p$ be an odd prime and let $\mathbf{Z}_p$ denote (the field of) the integers modulo $p$. How many elements are in the set $$ \left\{x^2: x \in \mathbf{Z}_p\right\} \cap\left\{y^2+1: y \in \mathbf{Z}_p\right\} ? $$ Please provide the number of elements in the intersection when p = 13.

Answer: 4
user: 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 $|u| = 1 and a=b=1$.

Answer: 2
user: Find all $\mathbf{v}\in\mathscr{P}$ such that the set $\mathscr{P}\setminus\{\mathbf{v}\}$ obtained by omitting vector $\mathbf{v}$ from $\mathscr{P}$ can be partitioned into two sets of equal size and equal sum. The original answer is in ”vectors $(m, b)$ where $n \leq$ $b \leq j$ and $b$ is even. “format, Please provide the sum of $m+n+j$。

Answer: 101
user: Denote by $\mathbb{Z}^2$ the set of all points $(x,y)$ in the plane with integer coordinates. For each integer $n\geq0$, let $P_n$ be the subset of $\mathbb{Z}^2$ consisting of the point (0,0) together with all points $(x,y)$ such that $x^2+y^2=2^k$ for some integer $k\leq n.$ Determine, as a function of $n$, the number of four-point subsets of $P_n$ whose elements are the vertices of a square. Please provide the value of the function when $n = 10$. The answer should be an integer.

Answer: 51
user: 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 $.

Answer: 8
user: Consider an $m$-by-$n$ grid of unit squares, indexed by $(i,j)$ with $1\leq i\leq m$ and $1\leq j\leq n.$ There are $(m-1)(n-1)$ coins, which are initially placed in the squares $(i,j)$ with $1\leq i\leq m-1$ and $1\leq j\leq n-1.$ If a coin occupies the square $(i,j)$ with $i\leq m-1$ and $j\leq n-1$ and the squares $(i+1,j),(i,j+1)$, and $(i+1,j+1)$ are unoccupied, then a legal move is to slide the coin from $(i,j)$ to $(i+1,j+1).$ How many distinct configurations of coins can be reached starting from the initial configuration by a (possibly empty) sequence of legal moves? The original answer is in the format \binom{a}{b}, please provide the value of a+b when $m=1$ and $n=1$.

Answer: 0
user: Define a positive integer $n$ to be squarish if either $n$ is itself a perfect square or the distance from $n$ to the nearest perfect square is a perfect square. For example, 2016 is squarish, because the nearest perfect square to 2016 is $45^2=2025$ and 2025-2016=9 is a perfect square. (Of the positive integers between l and 10, only 6 and 7 are not squarish.) For a positive integer $N$, let $S(N)$ be the number of squarish integers between l and $N$, inclusive. Find positive constants $\alpha$ and $\beta$ such that $$\lim_{N\to\infty}\frac{S(N)}{N^{\alpha}}=\beta,$$ or show that no such constants exist. Please provide the sum of $12\alpha v$.

Answer: 25
user: 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.

Answer: 11
user: Determine the maximum value of the sum
$$S=\sum_{n=1}^{\infty}\frac{n}{2^{n}}\left(a_{1}a_{2}\cdots a_{n}\right)^{1/n}$$
over all sequences $a_{1},a_{2},a_{3},\cdots$ of nonnegative real numbers satisfying
$$\sum_{k=1}^{\infty}a_{k}=1.$$ The original answer is in \frac{m}{n} format, please give the value of m + n.

Answer: 5
user: Let $\times$ represent the cross product in $\mathbb{R}^3.$ For what positive integers $n$ does there exist a set $S\subset\mathbb{R}^3$ with exactly $n$ elements such that
$$S=\{v\times w:v,w\in S\}?$$ Please provide the sum of all possible values of $n$.

Answer: 8
user: Let $x_0=1$, and let $\delta$ be some constant satisfying $0<\delta<1.$ Iteratively, for $n=0,1,2,\ldots$, a point $x_{n+1}$ is chosen uniformly from the interval $[0,x_n].$ Let Z be the smallest value of $n$ for which $x_n<\delta.$ Find the expected value of Z, as a function of $\delta$. Please provide the value when $\delta = 10$. The answer should be an integer.

Answer: 0
user: 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.

Answer: 6
user: Evaluate the sum
$\sum_{k=0}^{\infty}\left(3\cdot\frac{\ln(4k+2)}{4k+2}-\frac{\ln(4k+3)}{4k+3}-\frac{\ln(4k+4)}{4k+4}-\frac{\ln(4k+5)}{4k+5}\right)$
$=3\cdot\frac{\ln2}{2}-\frac{\ln3}{3}-\frac{\ln4}{4}-\frac{\ln5}{5}+3\cdot\frac{\ln6}{6}-\frac{\ln7}{7}-\frac{\ln8}{8}-\frac{\ln9}{9}+3\cdot\frac{\ln10}{10}-\cdots.$
(As usual, $lnx$ denotes the natural logarithm of x.)
 The original answer is in $(\log m)^{n}$format, please give the value of m + n.

Answer: 4
user: 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.

Answer: 7
user: Let $n$ be a positive integer, and let $V_n$ be the set of integer $(2n+1)$-tuples $\mathbf{v}=(s_0,s_1,\cdots,s_{2n-1},s_{2n})$ for which $s_{0}=s_{2n}=0$ and $|s_j-s_{j-1}|=1$ for $j=1,2,\cdots,2n.$ Define $$q(\mathbf{v})=1+\sum_{j=1}^{2n-1}3^{s_{j}},$$ and let $M(n)$ be the average of $\frac{1}{q(\mathbf{v})}$ over all $\mathbf{v}\in V_n.$ Evaluate $M(2020). The original answer is in \frac{m}{n} format, please give the value of m + n.

Answer: 4041
user: 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 $a + b + c$.

Answer: 24
user: Find all functions $f$ from the interval $(1,\infty)$ to $(1,\infty)$ with the following property: if $x,y\in(1,\infty)$ and $x^2\leq y\leq x^{3}$,then$(f(x))^2\leq f(y)\leq(f(x))^3.$ Please provide the smallest value of c when assuming $f(x)=x^c$ and c is a specific constant value such that the function satisfies the given property.

Answer: 1
user: 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.

Answer: 5
user: Determine which positive integers n have the following property: For all integers m that are relatively prime to n, there exists a permutation $\pi:\{1,2,\ldots,n\}\to\{1,2,\ldots,n\}$ such that $\pi(\pi(k))\equiv mk(\bmod n)$ for all $k\in\{1,2,\ldots,n\}$. Provide the sum of all such positive integers n.

Answer: 3
user: 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.

Answer: 19
user: Let $n$ be a positive integer. For $i$ and $j$ in $\{1,2,\ldots,n\},$ let $s(i,j)$ be the number of pairs $(a,b)$ of nonnegative integers satisfying $ai+bj=n.$ Let S be the n-by-n matrix whose $(i,j)$ entry is $s(i,j).$ For example, when $n=5$, we have $S=\begin{bmatrix}6&3&2&2&2\\3&0&1&0&1\\2&1&0&0&1\\2&0&0&0&1\\2&1&1&1&2\end{bmatrix}.$ Compute the determinant of $S$ for $n=5$. The original answer is in the format $(-1)^{\lceil n / 2\rceil-1} 2\left\lceil\frac{n}{2}\right\rceil$, please give the value of the determinant for $n=2$.

Answer: 2
user: A $2 \times 3$ rectangle has vertices at $(0,0),(2,0),(0,3)$, and $(2,3)$. It rotates $90^{\circ}$ clockwise about the point $(2,0)$. It then rotates $90^{\circ}$ clockwise about the point $(5,0)$, then $90^{\circ}$ clockwise about the point $(7,0)$, and finally, $90^{\circ}$ clockwise about the point $(10,0)$. (The side originally on the $x$-axis is now back on the $x$-axis.) Find the area of the region above the $x$-axis and below the curve traced out by the point whose initial position is $(1,1)$. The original answer is in the form of \(\frac{m\pi}{n} + k\), please provide the value of \(m + n + k\).

Answer: 15
user: 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 \(m + n + j\).

Answer: 5
user: 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 give the value of a + b + c + d.

Answer: 14
user: Let f be a function on $[0, \infty)$, differentiable and satisfying 
$$ 
f^{\prime}(x)=-3 f(x)+6 f(2 x) 
$$ 
for $x>0$. Assume that $|f(x)| \leqslant e^{-\sqrt{x}}$ for $x \geqslant 0$ (so that $f(x)$ tends rapidly to 0 as x increases). For $n$ a non-negative integer, define 
$$ 
\mu_{n}=\int_{0}^{\infty} x^{n} f(x) d x 
$$ 
(sometimes called the $n$th moment of f ). 

Please express $\mu_{n}$ in terms of $\mu_{0}$ and give the value of $\mu_{2}$ when $\mu_{0} = 45$. Directly provide the integer value of $\mu_{2}$.

Answer: 32
user: For any pair $(x, y)$ of real numbers, a sequence $\left(a_{n}(x, y)\right)_{n \geq 0}$ is defined as follows: $$\begin{aligned} a_{0}(x, y) & =x \\ a_{n+1}(x, y) & =\frac{\left(a_{n}(x, y)\right)^{2}+y^{2}}{2}, \quad \text { for all } n \geq 0. \end{aligned}$$ Find the area of the region $\left\{(x, y) \mid\left(a_{n}(x, y)\right)_{n \geq 0}\right.$ converges $\}$. The original answer is in the format $m+n\pi$, please provide the value of m+n.

Answer: 5
user: 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.

Answer: 9
user: 设双曲线 $C: \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1, A_{1} 、 A_{2}$ 为双曲线 $C$ 的左、右顶点, $F$ 为 $C$ 的焦点. 以坐标原点 $O$ 为圆心、 $|O F|$ 为半径的圆与 $C$ 的渐近线的一个交点为 $B$. 若 $\angle A_{1} B A_{2}=60^{\circ}$, 则 $C$ 的离心率的原始答案为$\frac{\sqrt{m}}{n}$，请给出m + n的值。

Answer: 24
user: 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。

Answer: 134
user: 已知数列 $\left\{(-1)^{n} \frac{3 n+2}{n(n+1) \cdot 2^{n+1}}\right\}$ 的前 $n$ 项和为 $S_{n}$. 计算 $S_{18}$ 的值。标准答案格式为$\frac{a}{b^{c} \times d}-\frac{e}{f}$，计算a+b+c+d+e+f的值。

Answer: 44
user: 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。

Answer: 16
user: 已知椭圆 $\frac{x^{2}}{5}+\frac{y^{2}}{4}=1$ 的右焦点为 $F$,上顶点为 $A, P$ 为椭圆上位于第一象限内的动点. 则 $\triangle A P F$ 面积的最大值为$\qquad$。答案的标准格式为$\sqrt{a}-b$，则a+b的值为多少？

Answer: 7