meetforever/GRPO_CL huggingface dataset:
user: For $R>1$ let $\mathcal{D}_{R}=\left\{(a, b) \in \mathbb{Z}^{2}: 0<a^{2}+b^{2}<R\right\}$. Compute  
$$  
\lim _{R \rightarrow \infty} \sum_{(a, b) \in \mathcal{D}_{R}} \frac{(-1)^{a+b}}{a^{2}+b^{2}}  
$$  
The original answer is in the format $m\pi \log n$. Please provide the integer value of m and n.
Answer: 1
user: Let $\Omega=\left\{(x, y, z) \in \mathbb{Z}^{3}: y+1 \geq x \geq y \geq z \geq 0\right\}$. A frog moves along the points of $\Omega$ by jumps of length 1. For every positive integer $n$, determine the number of paths the frog can take to reach $(n, n, n)$ starting from $(0,0,0)$ in exactly $3 n$ jumps. The original answer is in the format \(\frac{\binom{3 n}{n}}{2 n+1}\), please provide the value of \(\binom{3 n}{n} + (2 n+1)\) when \(n = 3\).
Answer: 843
user: Evaluate the product
$$
\prod_{n=3}^{\infty} \frac{\left(n^{3}+3 n\right)^{2}}{n^{6}-64}
$$
The original answer is in \(\frac{m}{n}\) format, please give the value of m + n.
Answer: 79
user: A four-digit number $Y E A R$ is called very good if the system  $$  \begin{aligned}  & Y x+E y+A z+R w=Y \\  & R x+Y y+E z+A w=E \\  & A x+R y+Y z+E w=A \\  & E x+A y+R z+Y w=R  \end{aligned}  $$  of linear equations in the variables $x, y, z$ and $w$ has at least two solutions. Find all very good YEARs in the 21st century and provide the sum of all these years.
Answer: 12309
user: Find all twice continuously differentiable functions $f: \mathbb{R} \rightarrow(0,+\infty)$ satisfying  
$$  
f^{\prime \prime}(x) f(x) \geq 2\left(f^{\prime}(x)\right)^{2}  
$$  
for all $x \in \mathbb{R}$. Please provide the sum of the solutions $f^{\prime}$ and $f^{\prime \prime}$.
Answer: 0
user: 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 all possible remainders.
Answer: 5
user: 一个正数，若其小数部分、整数部分和其自身成等比数列，则该数为 $\qquad$. 原始答案为\frac{m+\sqrt{n}}{k}的格式，请给出m + n + k的值。
Answer: 8
user: 设整数 $a_{1}, a_{2}, \cdots, a_{2019}$ 满足 $1=a_{1} \leqslant a_{2} \leqslant \cdots \leqslant a_{2019}=99$. 记 $f=\left(a_{1}^{2}+a_{2}^{2}+\cdots+a_{2019}^{2}\right)-\left(a_{1} a_{3}+a_{2} a_{4}+a_{3} a_{5}+\cdots+a_{2017} a_{2019}\right)$. 求 $f$ 的最小值 $f_{0}$. 并确定使 $f=f_{0}$ 成立的数组 $\left(a_{1}, a_{2}, \cdots, a_{2019}\right)$ 的个数。标准答案形式为$C_{a}^{b}$，计算出a+b的值。
Answer: 2016
user: 设 $f(x)=\frac{1}{x^{3}-x}$, 求满足不等式 $f(2)+f(3)+\cdots+f(n)>\frac{499}{2020}$ 的最小正整数 $n$. 请给出这个整数n的值。
Answer: 13
user: Determine all pairs $(a, b) \in \mathbb{C} \times \mathbb{C}$ satisfying
$$
|a|=|b|=1 \quad \text { and } \quad a+b+a \bar{b} \in \mathbb{R}.
$$
(proposed by Mike Daas, Universiteit Leiden)
Please provide the sum of all possible values of a and b.
Answer: 0
user: 8 个女孩与 25 个男孩围成一圈，任何两个女孩之间至少站两个男孩，则共有多少种不同的排列方法？答案标准形式为$C^{a}_{b}  \cdot c! \cdot d!$计算出a+b+c+d的值。
Answer: 55
user: Find a necessary and sufficient condition on the positive integer $n$ that the equation

$$x^{n}+(2+x)^{n}+(2-x)^{n}=0$$

has a rational root. Please provide the integer value of $n$ that satisfies this condition.
Answer: 1
user: 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 $\lim\limits _{n \rightarrow \infty} S_{n}$. The original answer is in $-\frac{m}{n}$ format, please give the value of m + n.
Answer: 5
user: We throw a dice (which selects one of the numbers $1,2, \ldots, 6$ with equal probability) $n$ times. What is the probability that the sum of the values is divisible by 5? What is the value of the probability when n = 1/6?
Answer: 0
user: Let $n$ be a positive integer. Consider an $n \times n$ matrix with entries $1,2, \ldots, n^{2}$ written in order starting top left and moving along each row in turn left-to-right. We choose $n$ entries of the matrix such that exactly one entry is chosen in each row and each column. What are the possible values of the sum of the selected entries? Please provide the value when n = 2.
Answer: 5
user: 设 $A(2,0)$ 为平面上一定点, $P\left(\sin \left(2 t-60^{\circ}\right), \cos \left(2 t-60^{\circ}\right)\right)$ 为动点, 则当 $t$ 由 $15^{\circ}$ 变到 $45^{\circ}$ 时, 线段 $A P$ 扫过的面积是 $\frac{m}{n}\pi$ 形式，请给出 m + n 的值。
Answer: 7
user: 设 $n=1990$, 则  
$$  
\frac{1}{2^{n}}\left(1-3 C_{n}^{2}+3^{2}{ C_n ^4}-3^{3} C_{n}^{6}+\cdots+3^{994} C_{n}^{1998}-3^{995} C_{n}^{1990}=\right.  \qquad.  
$$  
The original answer is in $-\frac{m}{n}$ format, please give the value of m + n.
Answer: 3
user: 设棱锥 $M-A B C D$ 的底面为正方形, 且 $M A=M D, M A \perp A B$, 如果 $\triangle A M D$ 的面积为 1, 试求能够放入这个棱锥的最大球的半径。已知原始答案为\sqrt{m}-n的形式，请给出m+n的值。
Answer: 3
user: $n^{2}$ 个正数排成 $n$ 行 $n$ 列，满足每一行的数成等差数列，每一列的数成等比数列，并且所有公比相等。已知 $a_{24}=1, a_{42}=\frac{1}{8}, a_{43}=\frac{3}{16}$ ，求 $a_{11}+a_{22}+\cdots \cdots+a_{n n}$。当n=2时，给出 $S$ 的值。
Answer: 1
user: 设 $n=1990$, 则 $\frac{1}{2^{n}}\left(1-3 C_{n}^{2}+3^{2} C_{n}^{4}-3^{3} C_{n}^{6}+\cdots+3^{994} C_{n}^{1988}-3^{995} C_{n}^{1990}\right) = \qquad$. 该表达式的原始答案是$-\frac{m}{n}$的形式，请给出m + n的值。
Answer: 3
user: 8 个女孩与 25 个男孩围成一圈，任何两个女孩之间至少站两个男孩，那么，共有多少种不同的排列方法？标准答案形式为$\frac{a!\cdot b!}{c!}$，计算出a+b+c的值。
Answer: 50
user: $n^{2}(n \geqslant 4)$ 个正数排成 $n$ 行 $n$ 列, $$ \begin{array}{llllll} a_{11} & a_{12} & a_{13} & a_{14} & \cdots \cdots & a_{1 n} \\ a_{21} & a_{22} & a_{23} & a_{24} & \cdots \cdots & a_{2 n} \\ a_{31} & a_{32} & a_{33} & a_{34} & \cdots \cdots & a_{3 n} \\ a_{41} & a_{42} & a_{43} & a_{44} & \cdots \cdots & a_{4 n} \\ \cdots & \cdots & \cdots & \cdots & \cdots \cdots & \cdots \\ a_{n 1} & a_{n 2} & a_{n 3} & a_{n 4} & \cdots \cdots & a_{n n} \end{array} $$ 其中每一行的数成等差数列，每一列的数成等比数列，并且所有公比相等. 已知 $a_{24}=1, a_{42}=\frac{1}{8}, a_{43}=\frac{3}{16}$, 求 $a_{11}+a_{22}+\cdots+a_{n n}$. 请提供 n = 2 时的值。
Answer: 1
user: 设棱锥 $M-A B C D$ 的底面是正方形，且 $M A=M D, M A \perp A B$ ，如果 $\triangle A M D$ 的面积为 1, 试求能够放入这个棱锥的最大球的半径。原始答案为$\sqrt{m}-n$，请给出m+n的值。
Answer: 3
user: 设 $S=\{1,2, \ldots, n\}$ ， $A$ 为至少含有两项的公差为正的等差数列，其项都在 $S$ 中，且添加 $S$ 的其他元素于 $A$ 后不能构成与 $A$ 有相同公差的等差数列. 求这种 $A$ 的个数（这里只有两项的数列也看作等差数列). 请提供 n = 16 时的值。
Answer: 64
user: 设正三棱雉 $P-A B C$ 的高为 $P O, M$ 为 $P O$ 的中点, 过 $A M$ 作与棱 $B C$ 平行的平面, 将三棱雉截为上、下两部分，试求此两部分的体积比。请给出比值中两个数之和的值。
Answer: 25
user: 设 $O$ 为抛物线的顶点, $F$ 为焦点, 且 $P Q$ 为过 $F$ 的弦. 已知 $|O F|=a,|P Q|=$ $b$, 求 $\triangle O P Q$ 的面积。请提供答案的格式为 a \sqrt{ab}，当a=b=1时，给出 a +ab 的值。
Answer: 2
user: 设 $z_{1}, z_{2}$ 都是复数, 且 $\left|z_{1}\right|=3,\left|z_{2}\right|=5\left|z_{1}+z_{2}\right|=7$, 则 $\arg \left(\frac{z_{2}}{z_{1}}\right)^{3}$ 的值是 $\qquad$. 请给出原始答案中\pi前的常数部分值。
Answer: 1
user: 设 $x, y, z$ 是实数, $3 x, 4 y, 5 z$ 成等比数列, 且 $\frac{1}{x}, \frac{1}{y}, \frac{1}{z}$ 成等差数列, 则 $\frac{x}{z}+\frac{z}{x}$ 的值是以\frac{m}{n}的形式表示，请给出m + n的值。
Answer: 49
user: 二次方程 $(1-i) x^{2}+(\lambda+i) x+(1+i \lambda)=0(i$ 为虚数单位, $\lambda \in R)$ 有两个虚根的充分必要条件是 ，$\lambda$ 不可能的的取值为  $\qquad$.
Answer: 2
user: 若 $z \in C, \arg \left(z^{2}-4\right)=\frac{5 \pi}{6}, \arg \left(z^{2}+4\right)=\frac{\pi}{3}$, 则 $z$ 的值是 $\qquad$.已知答案格式为\(\pm(m+\sqrt{n} i)\)，求\(m+n\)的值。
Answer: 4
user: 设正数列 $a_{0} 、 a_{1} 、 a_{2} 、 \\ldots 、 a_{n} 、 \\ldots$ 满足
$$
\\sqrt{a_{n} a_{n-2}}-\\sqrt{a_{n-1} a_{n-2}}=2 a_{n-1}, \\quad(n \\geq 2)
$$
且 $a_{0}=a_{1}=1$, 计算前三项 $a_0, a_1, a_2, a_3$ 的和。
Answer: 60
user: 设 $0<a<b$, 过两定点 $A(a, 0)$ 和 $B(b, 0)$ 分别引直线 $l$ 和 $m$, 使与抛物线 $y^{2}=x$ 有四个不同的交点, 当这四点共圆时，求这种直线 $l$ 与 $m$ 的交点 $P$ 的轨迹。请给出满足条件的方程x系数和的值。
Answer: 2
user: 若 $z \in \mathbf{C}, \arg \left(z^{2}-4\right)=\frac{5 \pi}{6}, \arg \left(z^{2}+4\right)=\frac{\pi}{3}$, 请提供 $z$ 的两个解中实部和虚部之和。
Answer: 0
user: 已知有向线段 $P Q$ 的起点 $P$ 和终点 $Q$ 的坐标分别为 $(-1,1)$ 和 $(2,2)$, 若直线 $l: x+m y+m=0$ 与 $P Q$的延长线相交，则 $m$ 的取值范围是 。请提供区间端点的积。
Answer: 2
user: 已知一平面与一正方体的 12 条棱的夹角都等于 $\alpha$, 则 $\sin \alpha$ 的原始答案是 \frac{\sqrt{m}}{n} 的形式，请给出 m + n 的值。
Answer: 6
user: $x$ 的二次方程 $x^{2}+z_{1} x+z_{2}+m=0$ 中, $z_{1}, z_{2}, m$ 均是复数，且 $z{ }_{1}^{2}-4 z_{2}=16+20 i$ ，设这个方程的两个根 $\alpha 、 \beta$, 满足 $|\alpha-\beta|=2 \sqrt{7}$, 求 $|m|$ 的最大值和最小值的和。
Answer: 14
user: 设 $\left\{a_{n}\right\}$ 为等差数列, $\left\{b_{n}\right\}$ 为等比数列, 且 $b_{1}=a_{1}^{2}, b_{2}=a_{2}^{2}, b_{3}=a_{3}^{2}\left(a_{1}<a_{2}\right)$,又 $\lim _{n \rightarrow \infty}\left(b_{1}+b_{2}+\cdots+b_{n}\right)=\sqrt{2}+1$. 试求 $\left\{a_{n}\right\}$ 的首项的两倍与公差的和。
Answer: -2
user: 椭圆 $\rho=\frac{1}{2-\cos \theta}$ 的短轴长等于多少？原始答案为$\frac{m \sqrt{n}}{k}$格式，请给出m+n+k的值。
Answer: 8
user: 若复数 $z_{1} 、 z_{2}$ 满足 $\left|z_{1}\right|=2,\left|z_{2}\right|=3,3 z_{1}-2 z_{2}=\frac{3}{2}-\mathrm{i}$, 则 $z_{1} \cdot z_{2}=$ 以形式 $-\frac{a}{b} + \frac{c}{d} \mathrm{i}$ 表示，给出 a + b + c + d 的值。
Answer: 128
user: 正方体 $A B C D-A_{1} B_{1} C_{1} D_{1}$ 的棱长为 1. 则直线 $A_{1} C_{1}$ 与 $B D_{1}$ 的距离是\qquad$。原始答案为 \frac{\sqrt{m}}{n} 的形式，请给出 m + n 的值。
Answer: 12
user: 已知点 $A(0,2)$ 和抛物线 $y^{2}=x+4$ 上两点 $B, C$, 使得 $A B \perp B C$, 求点 $C$ 的纵坐标的取值范围。请提供该范围的除无穷以外的端点之和。
Answer: 4
user: 已知复数 $Z_{1} 、 Z_{2}$ 满足 $\left|Z_{1}\right|=2,\left|Z_{2}\right|=3$, 若它们所对应的向量的夹角为 $60^{\circ}$, 则 $\left|\frac{Z_{1}+Z_{2}}{Z_{1}-Z_{2}}\right|=$ $\qquad$ ; 题目的原始答案为分数形式 $\frac{\sqrt{m}}{n}$, 请给出 m + n 的值。
Answer: 140
user: 实数 $a, b, c$ 和正数 $\lambda$ 使得 $f(x)=x^{3}+a x^{2}+b x+c$ 有三个实根 $x_{1}, x_{2}, x_{3}$, 且满足\(1) $x_{2}-x_{1}=\lambda$;\(2) $x_{3}>\frac{1}{2}\left(x_{1}+x_{2}\right)$.求 $\frac{2 a^{3}+27 c-9 a b}{\lambda^{3}}$ 的最大值。原始答案为$\frac{m\sqrt{n}}{j}$，请给出m+n+j的和。
Answer: 8
user: 将八个半径都为 1 的球分放两层放置在一个圆柱内，并使得每个球都和其相邻的四个球相切，且与圆柱的一个底面及侧面都相切。假设原始答案为$m+\sqrt[n]{j}$的形式，请给出m+n+j的和。
Answer: 14
user: Determine the position of a normal chord of a parabola such that it cuts off of the parabola a segment of minimum area. Please provide the sum of the coordinates of the points, when a = 2.
Answer: 8
user: A penny in a corner. A circle moves so that it is continually in contact with all three coordinate planes of an ordinary rectangular system. Find the locus of the center of the circle. Please provide the num of the possible condition that $x_{i}^{2} \leq r^{2}$.
Answer: 3
user: Find a necessary and sufficient condition on the positive integer $\boldsymbol{n}$ that the equation $$x^{n}+(2+x)^{n}+(2-x)^{n}=0$$ has a rational root. Please provide the integer value of $n$ that satisfies this condition.
Answer: 1
user: Assuming that the roots of $x^{3}+p x^{2}+q x+r=0$ are all real and positive, find the relation between $p, q$, and $r$ which is a necessary and sufficient condition that the roots may be the cosines of the angles of a triangle. Given that the original answer is in the form of an equation $p^n-mq-jr=1$, provide the sum of $m$, $n$, and $j$ when they satisfy the equation.
Answer: 6
user: The flag of the United Nations consists of a polar map of the world, with the North Pole as center, extending approximately to $45^{\circ}$ South Latitude. The parallels of latitude are concentric circles with radii proportional to their co-latitudes. Australia is near the periphery of the map and is intersected by the parallel of latitude $30^{\circ} \mathrm{S}$. In the very close vicinity of this parallel, how much are East and West distances exaggerated as compared to North and South distances? The original answer is in the format $m\pi/n\sqrt{j}\sim2.42$, please provide the value of m + n + j.
Answer: 10
user: Let $f(x)$ be a cubic polynomial with roots $x_{1}, x_{2}$, and $x_{3}$. Assume that $f(2 x)$ is divisible by $f^{\prime}(x)$ and compute the sum of the absolute values of the ratios $x_{1}: x_{2}: x_{3}$.
Answer: 9
user: Find the length of the curve $y^{2}=x^{3}$ from the origin to the point where the tangent makes an angle of $45^{\circ}$ with the $x$-axis. The original answer involves an integral that evaluates to a form with square roots. The original answer form is $\ frac {a} {b} (c \ sqrt {d}-e ）Find the value of a+b+c+d+e.
Answer: 40
user: What is the maximum of $\left|z^{3}-z+2\right|$, where $z$ is a complex number with $|z|=1$? The original answer is in the format \sqrt{m}, please provide the value of m.
Answer: 13
user: Find the smallest volume bounded by the coordinate planes and by a tangent plane to the ellipsoid $$ \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}+\frac{z^{2}}{c^{2}}=1. $$ The original answer is in the format $\frac{1}{2} \sqrt{3} abc$, Let abc=2 \ sqrt {3}, find the value of V.
Answer: 3
user: Find all solutions of $n^{m}=m^{n}$ in integers $n$ and $m(n\neq m)$. Prove that you have obtained all of them. Please provide the sum of all values of $m + n$ for each solution.
Answer: 0
user: (i) A circle of radius $a$ rolls on the inner side of the circumference of a circle of radius $3a$. Find the area contained within the closed curve generated by a point on the circumference of the rolling circle. Let a=2，The original answer is in the format $m \pi$, Find the value of m.
Answer: 8
user: The graph of the equation $x^{y}=y^{x}$ in the first quadrant (i.e., the region where $x>0$ and $y>0$) consists of a straight line and a curve. Find the sum of the coordinates of the intersection points of a straight line and a curve, rounded down.
Answer: 5
user: Evaluate in closed form

$$\sum_{k=1}^{n}\binom{n}{k} k^{2}$$

Note:

$$\binom{n}{k}=\frac{n(n-1) \cdots(n-k+1)}{1 \cdot 2 \cdots k}$$Please provide the value when \(n = 3\).
Answer: 24
user: Evaluate

$$\lim _{n \rightarrow \infty} \sum_{j=1}^{n^{2}} \frac{n}{n^{2}+j^{2}}$$

The original answer is in \frac{m\pi}{n} format. Please provide the value of m + n.
Answer: 3
user: A horizontal disc of diameter 3 inches is rotating at 4 revolutions per minute. A light is shining at a distant point in the plane of the disc. An insect is placed at the edge of the disc furthest from the light, facing the light. It at once starts crawling, and crawls so as always to face the light, at 1 inch per second. Set up the differential equation of motion, and find at what point the insect again reaches the edge of the disc. The original answer indicates that the insect leaves the disc at a point with a coordinate in the format \frac{m}{n},0. Please provide the value of m + n.
Answer: 5
user: Two uniform solid spheres of equal radii are so placed that one is directly above the other. The bottom sphere is fixed, and the top sphere, initially at rest, rolls off. At what point will contact between the two spheres be "lost"? Assume the coefficient of friction is such that no slipping occurs. The original answer is in the format \theta=\arccos (m / n), please give the value of m + n.
Answer: 27
user: If $x_{1}, \ldots, x_{n}$ denote the $n$th roots of unity, evaluate
$$
\pi\left(x_{i}-x_{j}\right)^{2} \quad(i<j)
$$What is the value when n=3。
Answer: -27
user: A player throwing a die scores as many points as on the top face of the die and is to play until his score reaches or passes a total $n$. Denote by $p(n)$ the probability of making exactly the total $n$, and find the value of $\lim _{n\to\infty} p(n)$. The original answer is in \frac{m}{n} format, please give the value of m + n.
Answer: 9
user: In assigning dormitory rooms, a college gives preference to pairs of students in this order: $$ A A, A B, A C, B B, B C, A D, C C, B D, C D, D D $$ in which $A A$ means two seniors, $A B$ means a senior and a junior, etc. Determine numerical values to assign to $A, B, C, D$ so that the set of numbers $A + A, A + B, A + C, B + B$, etc., corresponding to the order above will be in descending magnitude. Please provide the sum of A + B + C + D for the least positive integers that satisfy this condition.
Answer: 23
user: For a fixed positive integer $n$ let $x_{1}, x_{2}, \ldots, x_{n}$ be real numbers satisfying $0 \leq x_{k} \leq 1$ for $k=1,2, \ldots, n$. Determine the maximum value, as a function of $n$, of the sum of the $n(n-1) / 2$ terms:
$$
\sum_{\substack{i, j=1 \\ i<j}}^{n}\left|x_{i}-x_{j}\right|.
$$
Please provide the value of this sum when n = 3.
Answer: 2
user: Determine all rational values for which $a, b, c$ are the roots of
$$x^{3}+a x^{2}+b x+c=0$$
How many pairs of \(a\), \(b\), \(c\) satisfy the given conditions?
Answer: 3
user: Find the volume of the four-dimensional hypersphere $x^{2}+y^{2}+z^{2}+t^{2}=r^{2}$, and also the hypervolume of its interior $x^{2}+y^{2}+z^{2}+t^{2}<r^{2}$. If \(r = 2\), given that the sum of these two parts can be written in the form of \(k\pi^{2}\), find the value of \(k\).
Answer: 24
user: 在平面直角坐标系中, 设点集 \( \{P_{1}, P_{2}, \cdots, P_{4 n+1}\} \) = \{(x, y) \mid x, y \text{ 为整数, }|x| \leqslant n,|y| \leqslant n, x y=0\}, 其中， \(n \in \mathbf{N}_{+}\)。求 \( \left(P_{1} P_{2}\right)^{2}+\left(P_{2} P_{3}\right)^{2}+\cdots+\left(P_{n} P_{4 n+1}\right)^{2}+\left(P_{4 n+1} P_{1}\right)^{2} \) 的最小值。请给出 n = 1 时的结果。
Answer: 8
user: Solve the system of differential equations$$\begin{aligned}& \frac{d x}{d t}=x+y-3 \\& \frac{d y}{d t}=-2 x+3 y+1\end{aligned}$$subject to the conditions $x=y=0$ for $t=0$.When \(t = \pi\), the value of \(y\) can be written in the form of \(a + be^{c\pi}\). Find the value of \(a + b + c\).
Answer: 2
user: Suppose that $G$ is a group generated by elements $A$ and $B$, that is, every element of $G$ can be written as a finite "word" $A^{n_1}B^{n_2}A^{n_3}\cdots B^{n_k}$, where $n_{1},\cdots, n_k$ are any integers, and $A^0=B^0=1$ as usual. Also, suppose that $A^4=B^7= ABA^{-1}B=1$, $A^2\neq1$, and $B\neq1$. How many elements of $G$ are of the form $C^2$ with $C$ in $G$?
Answer: 8
user: For each nonnegative integer $k$, let $d(k)$ denote the number of I's in the binary expansion of $k$ (for example, $d(0)=0$ and $d(5)-2$). Let $m$ be a positive integer. Express
$$\sum_{k=0}^{2^m-1}(-1)^{d(k)}k^m$$
in the form $(-1)^ma^{f(m)}(g(m))!$, where $a$ is an integer and $f$ and $g$ are polynomials.Let \(m = 2\). Find the value of \(a+f(2)+g(2)\).
Answer: 5
user: Evaluate
$$\lim_{n \to \infty}\int_{0}^{1}\int_{0}^{1}\cdots\int_{0}^{1}\cos^2 \{\frac{\pi}{2n}(x_{1}+x_{2}+\cdots  x_{n})\}dx_{1}dx_{2}\cdots dx_{n}$$.
The original answer is in the format \(\frac{m}{n}\). Please provide the value of \(m + n\).
Answer: 3
user: Evaluate
$$\int_{0}^{\infty}\frac{\arctan(\pi x)-\arctan x}{x}dx$$.
The original answer is in the form of $(frac{m\pi}{n}})\ln\pi$. Please provide the value of $m + n$.
Answer: 3
user: Let $v$ be a vertex (corner) of a cube $C$ with edges of length 4. Let $S$ be the largest sphere that can be inscribed in $C$. Let $R$ be the region consisting of all points $p$ between $S$ and $C$ such that $p$ is closer to $v$ than to any other vertex of the cube. Find the volume of $R$. The original answer is in the form $m - \frac{n\pi}{k}$, please provide the value of$ m + n+k$.
Answer: 15
user: Assume that the differential equation
$$
y^{\prime \prime \prime}+p(x) y^{\prime \prime}+q(x) y^{\prime}+r(x) y=0
$$
has solutions $y_{1}(x), y_{2}(x)$, and $y_{3}(x)$ on the whole real line such that
$$
y_{1}^{2}(x)+y_{2}^{2}(x)+y_{3}^{2}(x)=1
$$
for all real $x$. Let
$$
f(x)=\left(y_{1}^{\prime}(x)\right)^{2}+\left(y_{2}^{\prime}(x)\right)^{2}+\left(y_{3}^{\prime}(x)\right)^{2}
$$
Find constants $A$ and $B$ such that $f(x)$ is a solution to the differential equation
$$
y^{\prime}+A p(x) y=B r(x)
$$
The original answer is in the format $A = frac{m}{n}$,$B = frac{p}{q}$, please give the value of $m + n+p+q$.
Answer: 10
user: Find the area of a convex octagon that is inscribed in a circle and has four consecutive sides of length 3 units and the remaining four sides of length 2 units. The answer is in the form $r + s\sqrt{t}$ with $r$, $s$, and $t$ as positive integers. Please provide the value of $r + s + t$.
Answer: 27
user: A game of solitaire is played as follows. After each play, according to the outcome, the player receives either $a$ or $b$ points ($a$ and $b$ are positive integers with $a$ greater than $b$), and his score accumulates from play to play. It has been noticed that there are thirty-five non-attainable scores and that one of these is 58. Find the sum of $a$ and $b$.
Answer: 19
user: In how many ways can the integers from 1 to n be ordered subject to the condition that, except for the first integer on the left, every integer differs by 1 from some integer to the left of it? Please provide the number of ways for $n = 6$.
Answer: 32
user: Let  0<a<b.  Evaluate $$\lim_{t \to 0} \left \{ {\int_{0}^{1} [bx+a(1-x)]^{t}dx}\right \} ^{1/t}$$[The  final  answer should  not  involve  any  operations other  than  addition,  subtraction,  multiplication,  division,  and  exponentiation.] When \(a = 1\) and \(b = 2\), the limit can be written as \(\frac{a}{be}\), where both \(a\) and \(b\) are positive integers. Find the value of \(a + b\).
Answer: 5
user: Let $n$ be a positive integer. Let $a,b,x$ be real numbers, with $a \neq b$. and let $M_{n}$ denote the $2n \times 2n$ matrix whose $(i,j)$ entry $m_{ij}$ is given by $$\left.m_{ij}=\left\{\begin{array}{llll}x&\text{if}\quad i=j,\\a&\text{if}\quad i\neq j&\text{and}&i+j\text{is even},\\b&\text{if}\quad i\neq j&\text{and}&i+j\text{is odd}.\end{array}\right.\right.$$ Thus, for example, $M_2=\begin{pmatrix}x&b&a&b\\b&x&b&a\\a&b&x&b\\b&a&b&x\end{pmatrix}.$ Express $\lim_{x\rightarrow a}\det M_n/(x-a)^{2n-2}$ as a polynomial in $a,b$, and $n.$ where $\det M_n$ denotes the determinant of $M_n$. Please provide the value of this expression when $n = 3$, $a = 2$, and $b = 1$. The answer should be a single integer.
Answer: 27
user: Let $A(x, y)$ denote the number of points $(m, n)$ in the plane with integer coordinates $m$ and $n$ satisfying $m^{2}+n^{2} \leqslant x^{2}+y^{2}$. Let $g=\sum_{k=0}^{\infty} e^{-k^{2}}$. Express
$$
\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} A(x, y) e^{-x^{2}-y^{2}} d x d y
$$
as a polynomial in $g$. The original answer is in the form of \(\pi(mg+n)^2\), please give the value of \(m+n\).
Answer: 1
user: Let $P$ be a non-self-intersecting closed polygon with $n$ sides. Let its vertices be $P_1$, $P_{2},\cdots,P_{n}.$ Iet $m$ other points$,Q_1,Q_{2},\cdots,Q_{m}$ interior to $P$ be given. Let the figure be triangulated. This means that certain pairs of the $(n+m)$ points $P_1,\cdots,Q_m$ are connected by line segments such that (i) the resulting figure consists exclusively of a set $T$ of triangles, (ii) if two different triangles in $T$ have more than a vertex in common then they have exactly a side in common, and (iii) the set of vertices of the triangles in $T$ is precisely the set of ($n+m)$ points $P_{1},\cdots,Q_{m}.$ How many triangles in $T?$ Please provide the number of triangles when n = 5 and m = 4.
Answer: 11
user: Evaluate$$\lim _{n \rightarrow \infty} \frac{1}{n} \sum_{k=1}^{n}\left(\left[\frac{2 n}{k}\right]-2\left[\frac{n}{k}\right]\right)$$and express your answer in the form $\ln a-b$, with $a$ and $b$ positive integers. Please provide the value of $a + b$.
Answer: 5
user: Express the sum \( \sum_{n=1}^{\infty} \sum_{m=1}^{\infty} \frac{1}{m^{2} n+m n^{2}+2 m n} \) as a rational number. The original answer is in the form of \( \frac{m}{n} \). Please provide the value of \( m + n \).
Answer: 11
user: Evaluate the following limit: \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 original answer is in the format $\frac{m}{n}$, please give the value of $m + n$.
Answer: 3
user: Let R be the region consisting of all triples $(x,y,z)$ of nonnegative real numbers satisfying $x + y + z\leqslant 1$. Let $w=1-x-y-z$. Express the value of the triple integral  $$\iiint\limits_{R}x^{1}y^{9}z^{8}w^{4}dx dy dz$$  in the form $\frac{a!b!c!d!}{n!}$, where a, b, c, d, and n are positive integers. Please provide the integer result of this expression.Find the value of $a+b+c+d+n$.
Answer: 47
user: Let $A(x, y)$ denote the number of points ($m, n$) with integer coordinates $m$ and $n$ that satisfy $m^{2}+n^{2} \leqslant x^{2}+y^{2}$ in the plane. Let $g=\sum_{k=0}^{\infty} e^{-k^{2}}$. If
$$\int_{-\infty}^{\infty} \int_{-\infty}^{\infty} A(x, y) e^{-x^{2}-y^{2}} d x d y=\pi(ag-b)^{c}$$
Find the value of $a+b+c$.
Answer: 5
user: A game of solitaire is played as follows. After each play, according to the outcome, the player receives either a or b points (a and b are positive integers with a greater than b), and his score accumulates from play to play. It has been noticed that there are thirty-five non-attainable scores and that one of these is 58. Please provide the value of a + b.
Answer: 19