sungyub/dapo-math-17k-verl huggingface dataset:
user: 设正四面体 $A B C D$ 的棱长为 2. 则与此正四面体的四个顶点距离相等的所有平面截这个四面体的外接球所得截面面积的总和为一个分数形式$\frac{m\pi}{n}$，请给出m + n的值。

Answer: 65
user: 已知 \( \sin \alpha+2 \cos \beta=2 \). 则\( y=\sin \left(\alpha+\frac{\pi}{4}\right)+2 \sin \left(\beta+\frac{\pi}{4}\right) \)的值域为 $\left[\sqrt{a}-\frac{\sqrt{b}}{c}, \sqrt{d}+\frac{\sqrt{e}}{f}\right]$.请提供a+b+c+d+e+f的值

Answer: 28
user: 已知\(f(x)=\log _{a} x(a>0, a \neq 1)\), \(g(x)=x^{2}-6 x+9\). 若存在唯一实数 \(x\), 使得 \(f(g(x))=g(f(x))=0\), 求实数 \(a\) 的值。标准答案格式为\sqrt[m]{n}，计算a所有取值的m+n的值的和

Answer: 12
user: 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。

Answer: 5
user: 已知双曲线$\frac{x^2}9-\frac{y^2}{16}=1$ 的左、右焦点分别为$F_1$、$F_2.$ 过$F_1$ 的直线与左支交于 $A$、$B$ 两点，且 $\frac{|AF_1|}{|AF_2|}=\frac12$ .则 $\triangle AF_1F_2$ 与$\triangle ABF_2$ 的内切圆面积之比为 $\qquad.$ 请给出比值两数之和。

Answer: 13
user: 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。

Answer: 1
user: 在 $\triangle A B C$ 中, $O$ 为 $\triangle A B C$ 的外心，且满足 $\overrightarrow{A O} \cdot \overrightarrow{A B}+\overrightarrow{B O} \cdot \overrightarrow{B C}=\overrightarrow{C O} \cdot \overrightarrow{C A}$. 计算$\angle B$ 的大小，答案形式为 \frac{m\pi}{n}，请给出 m + n 的值。

Answer: 3
user: 已知数列 $\left\{a_{n}\right\},\left\{b_{n}\right\}$ 满足:
$a_{1}=a_{2}=2019, b_{1}=b_{2}=2017$ ，
且对于一切正整数 $n \geqslant 2$ ，均有
$$
\begin{aligned}
& a_{n+1}=\left(a_{n-1}-1\right)\left(b_{n}+1\right) \\
& b_{n+1}=a_{n} b_{n-1}-1
\end{aligned}
$$
求 $\sum_{n=1}^{\infty} b_{n}\left(\frac{1}{a_{n+1}}-\frac{1}{a_{n+3}}\right)$ 的值。原始答案是 \frac{m}{n} 的形式，请给出 m + n 的值。

Answer: 8148685
user: 已知正四棱锥的五个顶点在同一个球面上. 若该正四棱锥的体积为 9 ，则球的表面积的最小值为。原始答案为\frac{m\pi}{n}的形式，请给出m + n的值。

Answer: 85
user: 已知 $\theta$ 为 $g(x)=x^{4}-7 x^{2}+1$ 的一个零点, $h(x)$ 为一个有理系数的二次函数, 且满足 $h\left(\theta^{2}-1\right)=0$. 请给出 $h(x)=0$ 的根的形式为 $\frac{m \pm n\sqrt{k}}{p}$，请计算 m + n + k +p的值。

Answer: 15
user: 已知抛物线 $x^{2}=4 y$ 上的两个动点 $A\left(x_{1}, y_{1}\right) 、 B\left(x_{2}, y_{2}\right)\left(y_{1}+y_{2}=2, y_{1} \neq y_{2}\right)$ 。若线段 $A B$ 的垂直平分线与 $y$ 轴交于点 $C$, 则 $\triangle A B C$ 面积的最大值为某个复杂分数形式，标准答案格式为$\frac{a\sqrt{b} }{c}$，则a+b+c的值为多少。

Answer: 31
user: 已知 $a 、 b \in[1,3], a+b=4$ 。求 $f(a, b)=\left|\sqrt{a+\frac{1}{b}}-\sqrt{b+\frac{1}{a}}\right|$ 的最大值。原始答案为m-\frac{n}{\sqrt{k}}的形式，请给出m + n + k的值。

Answer: 7
user: Find all positive integer pairs $(x, y)$ such that $x^{2}-y^{119}=1$. Please provide the sum of all possible $x$ and $y$ values from the solutions.If the equation has no solution, then the answer is 0.

Answer: 0
user: 2020 个不同的小球放入 2020 个不同的盒子里，每个球落入各个盒子的可能性相同，不同小球之间落入盒子的可能性互不影响。则空盒子个数的期望为 \qquad 。原始答案为 \frac{m^{n}}{k^{q}} 格式，请给出 m + n +k+q的值。

Answer: 8078
user: $\frac{2020}{6063}$

Answer: 8083
user: 已知$x_{1},x_{2},\cdots,x_{50}\in\mathbf{R}$，且$\sum\limits_{i=1}^{50}\frac{x_{i}}{k+i}=\frac{2021}{2k+1}$对于任意$k\in\{1,2,\cdots,50\}$成立。求$\sum\limits_{i=1}^{50}\frac{x_{i}}{4042i+2021}$的值。原始答案为\(\frac{m}{n}\)格式，请给出m + n的值。

Answer: 12751
user: 已知关于$x$的不等式$$(2ax-\ln x)\left(x^{2}-(a+1)x+1\right)\geqslant0$$ 对于任意$x\in(0,+\infty)$恒成立。则实数$a$的取值范围是_______。标准答案格式为$\left[\frac{a}{be},c\right]$，计算a+b+c的值。

Answer: 4
user: 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。

Answer: 4
user: 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。

Answer: 2055
user: 在 $\triangle ABC$ 中，$BC=2\sqrt{2}$，$AC=\sqrt{6}-\sqrt{2}$，$\angle BAC=\angle ABC+30^{\circ}$。求 $S_{\triangle ABC}$ 的整数值，假设原始答案为 $a-\sqrtb$ 的形式，请给出 $a+b$ 的整数值。

Answer: 6
user: 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。

Answer: 8
user: 已知复数 $z$ 满足 $|z|=3$. 则 $\left|\frac{z^{2}+1}{z^{2}+\mathrm{i} z+z+\mathrm{i}}\right|$ 的最大值为原始答案的形式为\(\dfrac{m\sqrt{n}+\sqrt{k}}{q}\)，请给出m + n + k+q的值。

Answer: 95
user: 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。

Answer: 8
user: 设 $a_{1}, a_{2}, \cdots, a_{2021}$ 为 2021 个整数，满足 $1=a_{1} \leqslant a_{2} \leqslant \cdots \leqslant a_{2021}=100$ 。记 
$$ 
\begin{aligned} 
f= & \left(a_{1}^{2}+a_{2}^{2}+\cdots+a_{2021}^{2}\right)- \\
& \left(a_{1} a_{3}+a_{2} a_{4}+\cdots+a_{2019} a_{2021}\right). 
\end{aligned} 
$$ 
求 $f$ 的最大值 $f_{0}$, 并求使得 $f=f_{0}$ 成立的数组 $\left(a_{1}, a_{2}, \cdots, a_{2021}\right)$ 的个数。请给出 $f_0$ 和数组个数之和。

Answer: 19901
user: 当$m=3$ 时，则使得 $m^{2}+n$ 整除 $n^{2}+m$ 的最大正整数 $n$ 是 $\qquad$。

Answer: 5
user: 设 $[x]$ 表示不超过实数 $x$ 的最大整数。则函数 $f(x)=\frac{x+1}{[x]+1}$ 的值域是一个区间。请给出该区间两个端点的和。

Answer: 2
user: 在平面直角坐标系 $x O_{y}$ 中, 椭圆 $\Gamma_{1}:$ $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)$ 和抛物线 $\Gamma_{2}: y^{2}=\frac{1}{2} a x$交于 $A 、 B$ 两点， $P$ 为 $\Gamma_{1}$ 的右顶点。若 $O 、 A 、 P 、$ $B$ 四点共圆, 则 $\Gamma_{1}$ 的离心率的$\sqrt{6}$倍是 $\qquad$。

Answer: 2
user: 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 $\qquad$.

Answer: 2
user: 给定正整数 $n$, 非负实数 $a_{1}$, $a_{2}, \cdots, a_{n}$ 满足 $a_{1}+a_{2}+\cdots+a_{n}=4$ 。记
$$
S=54(a_{1}+a_{1} a_{2}+\cdots+a_{1} a_{2} \cdots a_{n})-13\sqrt{13}.
$$
求 $S$ 的最大值.

Answer: 305
user: 求所有的正整数 $n$ ，存在一种方案将集合 $\\{1,2, \\cdots, 2 n\\}$ 划分为元素个数为 2 且两两不交的集合 $A_{1}, A_{2}, \\cdots, A_{n}$ ，使得不存在 $x_{1}, x_{2}, \\cdots, x_{n}$ 同时满足:（1） $x_{i} \\in A_{i}(i=1,2, \\cdots, n)$;(2) $2 n \\mid\\left(x_{1}+x_{2}+\\cdots+x_{n}\\right)$. 请给出满足条件的n的最小值。

Answer: 2
user: 求正整数 $a, b$ 应满足的条件，使得能够将集合 $\{1,2, \cdots, a b\}$ 分成 $a$ 个不相交的 $b$ 元子集，满足每个子集的元素和都相等。请给出所有满足条件的(a, b)对中a + b的最小值。

Answer: 4
user: 已知数列 $\left\{a_{n}\right\}$ 满足:
$$
a_{1}=1, a_{2}=3,
$$
$a_{n}= \begin{cases}a_{\frac{n}{}}^{2}+2, & n \text { 是偶数; } \\ 2 a_{\frac{n+1}{2}}-1, & n \text { 是奇数. }\end{cases}$
当$n=6$，求数列 $\left\{a_{n}\right\}$ 的前 $2^{n}$ 项和 $S_{2^{n}}$.

Answer: 1394
user: 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 $f(\frac{3}{4})$。

Answer: 0
user: 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 $\qquad$。

Answer: 2
user: 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 $\qquad$。

Answer: 27
user: 已知向量 $\boldsymbol{a} 、 \boldsymbol{b} 、 \boldsymbol{c}$ 满足 $|\boldsymbol{a}|=|\boldsymbol{b}|=|\boldsymbol{c}|=4$, $\boldsymbol{a} \cdot \boldsymbol{b}=8$. 则 $|3 \boldsymbol{c}-\boldsymbol{a}|+\left|\frac{1}{4} \boldsymbol{c}-\boldsymbol{b}\right|$ 的最小值的$\sqrt{133}$倍是 $\qquad$.

Answer: 133
user: 给定正整数 $6$ 。求最大的实数 $M$ ，使得对于任意的实数 $x_{i}(1 \leqslant i \leqslant 2 n+1)$ ，若 $x_{1} \leqslant x_{2} \leqslant \cdots \leqslant x_{2 n+1}$, 则 $\sum_{i=1}^{2 n+1} x_{i}^{2} \geqslant M x_{n+1}^{2}$.

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

Answer: 225
user: 已知函数 \(f(x)=3 \lg (x+2)-\log _{2} x\),集合 \(M=\left\{n \in \mathbf{Z} \mid f\left(n^{3}-3 n^{2}-10 n+32\right) \geqslant \mathbf{0}\right\}\) 。求 \(M\) 中所有元素的和。

Answer: 6
user: 已知 $n$ 为给定整数且 $n \geqslant 2$ ，集合 $A$ 的元素均为正整数， $A$ 的最小元素为 1 ，最大元素为 $a$ ，且 $7 \times 3^{n}<a<3^{n+2}$ ，满足对于 $A$ 的任一元素 $x(x \neq 1)$ ，均存在 $s 、 t 、 p \in$ $A$ ，使得 $x=s+t+p$ 。求集合 $A$ 元素个数的最小值，并给出 n=2 时的答案。

Answer: 6
user: 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

Answer: 1
user: 已知$f(x)$为定义在$\mathbf{R}$上的奇函数，当$x\leqslant0$时，
$$f(x)=\left|x+\frac{1}{2}\sin\alpha\right|+\left|x+\frac{3}{2}\sin\alpha\right|-2\sin\alpha$$
其中，$\alpha\in(-\pi,\pi)$。若对任意实数$x$，均有$f(x)\geqslant f(x+2)$，则$\alpha$的最小值

Answer: 0
user: 三棱雉$P-ABC$中，三个侧面与底面所成角相等，三个侧面的面积分别为$3、4、5$，且底面面积为6。已知三棱雉$P-ABC$的外接球的表面积的原始答案形式为 $\frac{m\pi}{n}$，请给出 m + n 的值。

Answer: 82
user: 已知实数$x、y$满足：
$x>\max\{-3,y\},(x+3)\left(x^{2}-y^{2}\right)=8$。
则$3x+y-4\sqrt6$的最小值为_______。

Answer: -6
user: 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的256倍为_______。

Answer: 43
user: 已知复数 $z_{1} 、 z_{2} 、 z_{3}$ 的辐角分别为 $\alpha 、 \beta 、 \gamma$, 且满足: $$ \left|z_{1}\right|=1,\left|z_{2}\right|+\left|z_{3}\right|=2, z_{1}+z_{2}+z_{3}=0 $$ 求 $\cos (\alpha-\beta)+2 \cos (\beta-\gamma)+3 \cos (\gamma-\alpha)$的取值范围。请给出该范围的起点和终点的和。

Answer: -4
user: 设 $\alpha$ 为复数， $\bar{\alpha}$ 表示 $\alpha$ 的共轭复数。已知 $|\alpha-\bar{\alpha}|=2 \sqrt{3}$ 且 $\frac{\alpha}{\bar{\alpha}^{2}}$ 为纯虚数，则 $\sqrt{3}|\alpha|$ 的所有可能值的和为 $\qquad$.

Answer: 3
user: 已知正数 $a, b, c$, 满足 $(a+b-c)\left(\frac{1}{a}+\right.$ $\left.\frac{1}{b}-\frac{1}{c}\right)=3$, 求 $f=\left(a^{4}+b^{4}+c^{4}\right)\left(\frac{1}{a^{4}}+\frac{1}{b^{4}}+\frac{1}{c^{4}}\right)-240\sqrt{3}$ 的最小值.

Answer: 417
user: 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 $\qquad$.

Answer: 0
user: 函数 $f(x)=\min \left\{\sin x, \cos x,-\frac{1}{\pi} x+1\right\}$ 在 $[0, \pi]$ 上最大值的$\sqrt{2}$倍是 $\qquad$.

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

Answer: 5
user: 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。

Answer: 5
user: 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。

Answer: 8
user: 求 $f(x)=711(|x-1|+|2 x-1|+\cdots+|2011 x-1|)$ 的最小值.

Answer: 592043
user: 设整数 $n \geq 2$. 某天 $n$ 个人组成了一个社团，每人向政府交 1 元注册费。接下来每天都发生如下事情：每一个前一天组建的社团都分裂成两个团体，其中含至少两人的团体组建成一个新社团，每个新社团的成员向政府交 1 元注册费，只有一名成员的团体不再组建新社团。上述过程直到不能继续进行下去为止。当$n=6$时，求政府总共收到的注册费的最大值。

Answer: 20
user: 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。

Answer: 3
user: 已知实数 $x, y$ 满足 $\left\{\begin{array}{l}\sin x+\sin y=\frac{1}{3} \\ \cos x-\cos y=\frac{1}{5}\end{array}\right.$, 求 $17\sin (x-y)$.

Answer: -15
user: 函数 $f(x)=\frac{\sqrt{2}}{2}(\cos x-\sin x) \sin \left(x+\frac{\pi}{4}\right)-2 a \sin x+b(a>0)$ 的最大值为 1, 最小值为 -4, 求 $b$ 的值.

Answer: -1
user: 已知椭圆 $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ 与圆 $x^{2}+y^{2}=b^{2}$, 过椭圆上一点 $M$ 作圆的两切线, 切点分别为 $P, Q$, 直线 $P Q$ 与 $x, y$轴分别交于点 $E, F$, 当$a=2,b=2$时，求 $S_{\triangle E O F}$ 的最小值.

Answer: 4
user: 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 $\lim _{x \rightarrow+\infty} 2\sqrt{3}y=$

Answer: 3
user: 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.

Answer: 1358
user: During a rebuilding project by contractors 'A', 'B' and 'C', there was a shortage of tractors. The contractors lent each other tractors as needed. At first, 'A' lent 'B' and 'C' as many tractors as they each already had. A few months later, 'B' lent 'A' and 'C' as many as they each already had. Still later, 'C' lent 'A' and 'B' as many as they each already had. By then each contractor had $24$ tractors. Please provide the total number of tractors the contractors originally had.

Answer: 204
user: Find the two $10$-digit numbers which become nine times as large if the order of the digits is reversed. Please provide the sum of these two numbers.

Answer: 1287128711
user: 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.

Answer: 147
user: 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.

Answer: 67
user: Skipper’s doghouse has a regular hexagonal base that measures one metre on each side. Skipper is tethered to a $2$-metre rope which is fixed to a vertex. What is the area of the region outside the doghouse that Skipper can reach? Calculate an approximate answer by using $\pi=3.14$ or $\pi=22/7$. The original answer is in the format $n\pi$, please give the value of n.

Answer: 3
user: Find all pairs of primes $(p,q)$ for which $p-q$ and $pq-q$ are both perfect squares. Please provide the value of $p + q + pq$.

Answer: 11
user: 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.

Answer: 6
user: Consider an $n$-by-$n$ board of unit squares for some odd positive integer $n$. We say that a collection $C$ of identical dominoes is a maximal grid-aligned configuration on the board if $C$ consists of $(n^2-1)/2$ dominoes, where each domino covers exactly two neighboring squares and the dominoes don’t overlap: $C$ then covers all but one square on the board. We are allowed to slide (but not rotate) a domino on the board to cover the uncovered square, resulting in a new maximal grid-aligned configuration with another square uncovered. Let $k(C)$ be the number of distinct maximal grid-aligned configurations obtainable from $C$ by repeatedly sliding dominoes. Find the maximum value of $k(C)$ as a function of $n$. Please provide the value of k(C) when n = 11 . The answer should be in the format \frac{m}{n}, directly give the value of m + n .

Answer: 11
user: A random number selector can only select one of the nine integers $1, 2, ..., 9$, and it makes these selections with equal probability. Determine the probability that after $n$ selections ($n>1$), the product of the $n$ numbers selected will be divisible by $10$. Please calculate the probability when n = 2 and give the result in the form of \frac{m}{n}, and then provide the value of m + n.

Answer: 89
user: 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.

Answer: 4
user: On a given circle, six points $A$, $B$, $C$, $D$, $E$, and $F$ are chosen at random, independently and uniformly with respect to arc length. Determine the probability that the two triangles $ABC$ and $DEF$ are disjoint, i.e., have no common points. The original answer is in \frac{m}{n} format, please give the value of m + n.

Answer: 3
user: 在$4n\times4n\left(n\in\mathbf{N}_{+}\right)$实数表中，每个数的绝对值均不超过1，所有数的和为0。已知数表中的每行各数之和的绝对值、每列各数之和的绝对值均不小于$c$。求$c$的最大可能值。请给出$c$的值，当$n=2$时。

Answer: 0
user: 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
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。

Answer: 3
user: 求最大的实数 $c$ ，满足：对任意正整数 $n>1$ ，均存在 $\{1,2, \cdots, n\}$ 的子集 $A$ ，使得 $|A| \geqslant_{c n}$ ，且对任意的 $x 、 y_{、} z \in A$ （允许相同），均有 $n$ 不是 $x+y-z$ 的约数。请给出m + n的值，其中原始答案为\frac{m}{n}。

Answer: 4
user: 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.

Answer: 11
user: Determine all pairs $(x,y)$ of integers satisfying the equation $1+2^x+2^{2x+1}=y^2$. Please provide the sum of all individual elements from each solution pair.

Answer: 0
user: Let $n>1$ be an integer. In the space, consider the set $S=\{(x,y,z)\mid x,y,z\in\{0,1,\cdots,n\}, x+y+z>0\}.$ Find the smallest number of planes that jointly contain all $(n+1)^3-1$ points of $S$ but none of them passes through the origin. Provide the value of this expression when n = 2.

Answer: 6
user: Find all pairs $(k,n)$ of positive integers for which $7^k-3^n$ divides $k^4+n^2$. Please provide the sum of all values of $k$ and $n$ in the pairs.

Answer: 4
user: 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.

Answer: 0
user: Let $n$ and $k$ be fixed positive integers of the same parity, $k \ge n$. We are given $2n$ lamps numbered $1$ through $2n$; each of them can be on or off. At the beginning all lamps are off. We consider sequences of $k$ steps. At each step one of the lamps is switched (from off to on or from on to off). Let $N$ be the number of $k$-step sequences ending in the state: lamps $1,....,n$ on, lamps $n+1,...,2n$ off. Let $M$ be the number of $k$-step sequences leading to the same state and not touching lamps $n+1,....2n$ at all. Find the ratio $N/M$ when $k = n + 1$. Please provide the integer value of the ratio.

Answer: 1
user: Find all positive integers n such that there exists a sequence of positive integers $a_{1},a_{2},\\ldots,a_{n}$ satisfying $a_{k+1}=\frac{a_k^2+1}{a_{k-1}+1}-1$ for every k with $2\leq k\leq n-1.$ When n = 4, a sequence exists with specific values for $a_1$, $a_2$, $a_3$, and $a_4$. Please provide the sum of these values.

Answer: 3
user: Determine all functions f : $\mathbb{R}\to\mathbb{R}$ such that the equality $f([x]y)=f(x)[f(y)].\quad(1)$ holds for all $x,y\in\mathbb{R}$. Here, by $[x]$ we denote the greatest integer not exceeding $x$. Please provide the sum of all possible integer values of the constant C.

Answer: 1
user: Denote by $\mathbb{Q}^{+}$ the set of all positive rational numbers. Determine all functions $f:\mathbb{Q}^{+}\to\mathbb{Q}^{+}$ which satisfy the following equation for all $x,y\in\mathbb{Q}^{+}:$ $f\big(f(x)^2y\big)=x^3f(xy).\quad(1)$ The original answer is in \frac{m}{n} format for a specific x, please provide the value of m + n when x = 1.

Answer: 2
user: 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 $M$.

Answer: 17
user: Find all pairs $( m, n)$ of nonnegative integers for which $$m^2+2\cdot3^n=m\big(2^{n+1}-1\big).\quad(1)$$ Please provide the sum of all values of $m + n$ for each pair $(m, n)$.

Answer: 21
user: 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.

Answer: 4
user: Determine all sequences $(x_1,x_2,\cdots,x_{2011})$ of positive integers such that for every positive integer $n$ there is an integer $a$ with $$x_1^n+2x_2^n+\cdots+2011x_{2011}^n=a^{n+1}+1.$$ Please provide the value of $x_1 + x_2 + \cdots + x_{2011}$ for the sequence that satisfies the condition.

Answer: 2011
user: 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 $f(1) + g(1)$.

Answer: 1
user: 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.

Answer: 0
user: 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 $a$.

Answer: 9
user: 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).

Answer: 0
user: 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.

Answer: 3980000000000
user: 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 $(m, n)$.

Answer: 1
user: 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.

Answer: 505
user: 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 $n$.

Answer: 37
user: 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 $n$.

Answer: 39
user: In the plane, $2013$ red points and $2014$ blue points are marked so that no three of the marked points are collinear. One needs to draw $k$ lines not passing through the marked points and dividing the plane into several regions. The goal is to do it in such a way that no region contains points of both colors. Find the minimal value of $k$ such that the goal is attainable for every possible configuration of $4027$ points. Please provide the value of k.

Answer: 2013
user: $A \pm1-sequence$ is a sequence of 2022 numbers $a_{1},\ldots,a_{2022}$, each equal to either +1 or -1. Determine the largest $C$ so that, for any $\pm1$ sequence, there exists an integer $k$ and indices $1\leqslant t_{1}<\ldots<t_{k}\leqslant2022$ so that $t_{i+1}-t_{i}\leqslant2$ for all $i$, and $$\left|\sum_{i=1}^ka_{t_i}\right|\geqslant C.$$ Please provide the value of C.

Answer: 506
user: Solve in integers the equation $xy+3x-5y=-3.$ Please provide the value of the number of solutions multiplied by 2.

Answer: 24
user: Find integers $x, y, z, u$ such that $$x^2 + y^2 + z^2 + u^2 = 2xyzu.$$ Provide the sum of all integers $x, y, z, u$ that satisfy the equation.

Answer: 0