meetforever/GRPO_CL huggingface dataset:
user: 若直线 \(x=\frac{\pi}{4}\) 被曲线 \(C:(x-\arcsin a)(x-\arccos a)+(y-\arcsin a)(y+\arccos a)=0\) 所截的弦长为 \(d\)，当 \(a\) 变化时 \(d\) 的最小值的形式为\(\frac{k\pi}{m}\)，请给出k+m的值。
Answer: 3
user: 若 $M=\left\{(x, y)| | \tan \pi y \mid+\sin ^{2} \pi x=0\right\}, N=\left\{(x, y) \mid x^{2}+y^{2} \leqslant 2\right\}$ ，求 $M \cap N$ 的元素个数。
Answer: 9
user: 若直线 \(x=\frac{\pi}{4}\) 被曲线 \(C:(x-\arcsin a)(x-\arccos a)+(y-\arcsin a)(y+\arccos a)=0\) 所截得的弦长为 \(d\)，当 \(a\) 变化时，求 \(d\) 的最小值。答案的形式为 \(\frac{k\pi}{m}\)，请给出 \(k + m\) 的值。
Answer: 3
user: 设复平面上单位圆内接正 20 边形的 20 个顶点所对应的复数依次为 $Z_{1}, Z_{2}, \cdots, Z_{20}$, 求复数 $Z_1^{1995}, Z_2^{1995}, \cdots, Z_{20}^{1995}$ 所对应的不同的点的个数。
Answer: 4
user: 设复平面上单位圆内接正 20 边形的 20 个顶点所对应的复数依次为 $z_{1}, z_{2}, \cdots, z_{20}$.求复数 $z_{1}^{1995}, z_{2}^{1995}, \cdots, z_{20}^{1995}$ 所对应的不同的点的个数。
Answer: 4
user: 已知方程 $|x-2 n|=k \sqrt{x}(n \in \mathbf{N})$ 在区间 $(2 n-1,2 n+1]$ 上有两个不相等的实根，求 $k$ 的取值范围。答案的形式为 $0<k \leqslant \frac{1}{\sqrt{a n+b}}$，请给出$a+b$的值。
Answer: 3
user: 由曲线 $x^{2}=4 y, x^{2}=-4 y, x=4, x=-4$ 围成的图形绕 $y$ 轴旋转一周所得的旋转体的体积为 $V_{1}$; 满足 $x^{2}+y^{2} \leq 16, x^{2}+(y-2)^{2} \geqslant 4, x^{2}+(y+2)^{2} \geqslant 4$ 的点绕 $y$ 轴旋转一周所得的旋转体的体积为 $V_{2}$。设$V_{1}=aV_{2}+b$，请给出a+b的值。
Answer: 1
user: 已知两个实数集合 $A=\left\{a_{1}, a_{2}, \cdots, a_{100}\right\}$ 与 $B=\left\{b_{1}, b_{2}, \cdots, b_{50}\right\}$, 若从 $A$ 到 $B$ 的映射 $f$ 使得 $B$ 中每个元素都有原象，且 $f\left(a_{1}\right) \leqslant f\left(a_{2}\right) \leqslant \cdots \leqslant f\left(a_{100}\right)$ ，则这样的映射共有多少种？原始答案的形式为$C_{x}^{y}$，请给出x+y的值。
Answer: 148
user: 由曲线 $x^{2}=4 y, x^{2}=-4 y, x=4, x=-4$ 围成的图形绕 $y$ 轴旋转一周所得旋转体的体积为 $V_{1}$ ；满足 $x^{2}+y^{2} \leqslant 16, x^{2}+(y-2)^{2} \geqslant 4, x^{2}+(y+2)^{2} \geqslant 4$ 的点组成的图形绕 $y$ 轴旋转一周所得旋转体的体积为 $V_{2}$ ，求 $V_{1}$ 和 $V_{2}$ 的比值。
Answer: 1
user: 删去正整数数列 $1,2,3, \cdots \cdots$ 中的所有完全平方数, 得到一个新数列。求这个数列的第 2003 项。
Answer: 2048
user: Two obvious approximations to the length of the perimeter of the ellipse with semi-axes $a$ and $b$ are $\pi(a+b)$ and $2 \pi(a b)^{1/2}$. Which one comes nearer the truth when the ratio $b / a$ is very close to 1? Write your answer in the form of $\pi(x+y)$, where $x$ and $y$ are integers. Find the value of $x + y$.
Answer: 2
user: 过抛物线 \(y^{2}=8(x+2)\) 的焦点 \(F\) 作倾斜角为 \(60^{\circ}\) 的直线，若此直线与抛物线交于 \(A 、 B\) 两点，弦 \(AB\) 的中垂线与 \(x\) 轴交于点 \(P\)，则线段 \(PF\) 的长等于多少？答案的形式为\(\frac{k}{m}\)，请给出k+m的值。
Answer: 19
user: 若 \(x \in\left[-\frac{5 \pi}{12},-\frac{\pi}{3}\right]\), 则 \(y=\tan \left(x+\frac{2 \pi}{3}\right)-\tan \left(x+\frac{\pi}{6}\right)+\cos \left(x+\frac{\pi}{6}\right)\) 的最大值是多少？答案的形式为 \(\frac{m}{n} \sqrt{p}\)，请给出m+n+p的值。
Answer: 20
user: 已知 \(x, y\) 都在区间 \((-2,2)\) 内，且 \(x y=-1\)，求函数 \(u=\frac{4}{4-x^{2}}+\frac{9}{9-y^{2}}\) 的最小值。答案的形式为\(\frac{k}{m}\)，请给出\(k+m\)的值。
Answer: 17
user: 在四面体 \(A B C D\) 中，设 \(A B=1\)，\(C D=\sqrt{3}\)，直线 \(A B\) 与 \(C D\) 的距离为 2，夹角为 \(\frac{\pi}{3}\)。求四面体 \(A B C D\) 的体积。答案的形式为\(\frac{a}{b}\)，请给出a+b的值。
Answer: 3
user: 删去正整数数列 $1,2,3, \cdots$ 中的所有完全平方数, 得到一个新数列. 求这个新数列的第2003项。
Answer: 2048
user: 过抛物线 \(y^{2}=8(x+2)\) 的焦点 \(F\) 作倾斜角为 \(60^{\circ}\) 的直线。若此直线与抛物线交于 \(A\)、\(B\) 两点，弦 \(AB\) 的中垂线与 \(x\) 轴交于点 \(P\)，求线段 \(PF\) 的长度。答案的形式为\(\frac{k}{m}\)，请给出k+m的值。
Answer: 19
user: 在四面体 $A B C D$ 中, 设 $A B=1, C D=\sqrt{3}$, 直线 $A B$ 与 $C D$ 的距离为 2, 夹角为 $\frac{\pi}{3}$, 求四面体 $A B C D$ 的体积。答案的形式为\frac{k}{m}，请给出k+m的值。
Answer: 3
user: 如果在区间 $[1,2]$ 上函数 $f(x)=x^{2}+p x+q$ 与 $g(x)=x+\frac{1}{x^{2}}$ 在同一点取相同的最小值，那么 $f(x)$ 在该区间上的最大值是多少？答案的格式为 \(a - \frac{b}{c} \sqrt[3]{d} + \sqrt[3]{e}\)，请计算a + b + c + d + e的值。
Answer: 17
user: 如果在区间 $[1,2]$ 上, 函数 $f(x)=x^{2}+p x+q$ 与 $g(x)=x+\frac{1}{x^{2}}$ 在同一点取相同的最小值，那么，求 $f(x)$ 在该区间上的最大值。原始答案的格式为 \(k - \frac{m}{n} \sqrt[3]{2} + \sqrt[3]{p}\)，请给出k + m + n + p的值。
Answer: 15
user: 设等差数列的首项及公差均为非负整数，项数不少于 3 ，且各项的和为 $97^{2}$ ，求这样的数列共有多少个。
Answer: 4
user: 已知数列 $\left\{x_{n}\right\}$ 满足 $x_{n+1}=x_{n}-x_{n-1}(n \geqslant 2), x_{1}=a, x_{2}=b$, 记 $S_{n}=$ $x_{1}+x_{2}+\cdots+x_{n}$. 求$x_{100}$和$S_{100}$的值。答案形式为$kx_{100}, mS_{100}$，请给出k+m的值。
Answer: 1
user: 设等差数列的首项及公差均为非负整数，项数不少于 3 ，且各项的和为 $97^{2}$ 。求这样的数列的个数。
Answer: 4
user: 设 $E, F, G$ 分别是正四面体 $A B C D$ 的棱 $A B, B C, C D$ 的中点，求二面角 $C-F G-E$ 的大小。如果答案为 $\pi-\operatorname{arccot} x$，其原始答案的形式为\frac{k\sqrt{n}}{m}，请计算k+m+n的值。
Answer: 5
user: 设 $E 、 F 、 G$ 分别是正四面体 $A B C D$ 的棱 $A B 、 B C 、 C D$ 的中点。求二面角 $C-F G-E$ 的大小，并用表达式 $\pi-\operatorname{arccot} \frac{\sqrt{m}}{n}$ 表示，其中$m$和$n$是整数，且$\frac{\sqrt{m}}{n}$是最简分数。请给出$m + n$的值。
Answer: 4
user: 在坐标平面上有两个区域 $M$ 和 $N$，$M$ 是由 $y \geqslant 0, y \leqslant x$ 和 $y \leqslant 2-x$ 这三个不等式确定，$N$ 是随 $t$ 变化的区域，它由不等式 $t \leqslant x \leqslant t+1$ 确定，$t$ 的取值范围是 $0 \leqslant t \leqslant 1$。设 $M$ 和 $N$ 的公共面积是函数 $f(t)$，求 $f(t)$。答案形式为 $-at^{2}+bt+\frac{c}{d}$，请给出a+b+c+d的值。
Answer: 5
user: 在坐标平面上有两个区域 $M$ 和 $N$。$M$ 是由 $y \geqslant 0, y \leqslant x$ 和 $y \leqslant 2-x$ 这三个不等式确定的，$N$ 是随 $t$ 变化的区域，它由不等式 $t \leqslant x \leqslant t+1$ 所确定，$t$ 的取值范围是 $0 \leqslant t \leqslant 1$。设 $M$ 和 $N$ 的公共面积是函数 $f(t)$。原始答案的形式是 $-at^{2}+bt+\frac{c}{d}$，请给出 $a+b+c+d$ 的值，其中 $\frac{c}{d}$ 是最简分数。
Answer: 5
user: 平面上整点（纵、横坐标都是整数的点）到直线 $y=\frac{5}{3} x+\frac{4}{5}$ 的距离中的最小值的原始答案形式为\frac{\sqrt{n}}{k}，请计算n+k的值。
Answer: 119
user: 设 $\omega=\cos \frac{\pi}{5}+i \sin \frac{\pi}{5}$, 找到以 $\omega, \omega^{3}, \omega^{7}, \omega^{9}$ 为根的方程。请将答案写成 $ax^4 + bx^3 + cx^2 + dx + e = 0$ 的形式，并求出$a + b + c + d + e$的值。
Answer: 1
user: 在平面上，整点（纵、横坐标都是整数的点）到直线 $y=\frac{5}{3} x+\frac{4}{5}$ 的距离中的最小值是多少？原始答案的形式为\frac{\sqrt{n}}{m}，请给出n+m的值。
Answer: 119
user: 设 $\omega=\cos \frac{\pi}{5}+\mathrm{i} \sin \frac{\pi}{5}$. 求以 $\omega 、 \omega^{3} 、 \omega^{7} 、 \omega^{9}$ 为根的方程。答案应为$ax^{4}+bx^{3}+cx^{2}+dx+e=0$的形式，请给出a+b+c+d+e的值。
Answer: 1
user: 若 $\left(1+x+x^{2}\right)^{1000}$ 的展开式为 $a_{0}+a_{1} x+a_{2} x^{2}+\cdots+a_{2000} x^{2000}$，求 $a_{0}+a_{3}+a_{6}+a_{9}+\cdots+a_{1998}$ 的值。原始答案的形式为 $k^{n}$，请给出 $k + n$ 的值。
Answer: 1002
user: 若 \(0<x<\frac{\pi}{2}\)，且 \(\frac{\sin ^{4} x}{9}+\frac{\cos ^{4} x}{4}=\frac{1}{13}\)，求 \(\tan x\) 的值。答案形式为 \(\frac{k}{m}\)，请给出 \(k + m\) 的值。
Answer: 5
user: 已知四面体 $ABCD$ 内接于球 $O$，且 $AD$ 为球 $O$ 的直径。若$\triangle A B C$与$\triangle B C D$均是边长为 1 的等边三角形，求四面体 $ABCD$ 的体积。答案的形式为$\frac{\sqrt{k}}{m}$，请给出k+m的值。
Answer: 14
user: 对于实数 $t$, 已知等比数列 $\left\{a_n\right\}$ 的前三项依次为 $2 t, 5 t-1,6 t+2$, 且该数列的前 $n$ 项和为 $S_n$. 求满足不等式 $|S_n-\frac{1}{65}|<1$ 的最大整数 $n$ 的值。
Answer: 2
user: 定义在 $\mathbf{R}$ 上的函数 $f(x)$ 满足 $f(0)=0, f(x)+f(1-x)=1, f\left(\frac{x}{5}\right)=\frac{1}{2} f(x)$, 且当 $0 \leqslant x_1<x_2 \leqslant 1$ 时, $f\left(x_1\right) \leqslant f\left(x_2\right)$. 求 $f\left(\frac{1}{2007}\right)$ 的值。答案的形式为 \frac{k}{m}，请给出k+m的值。
Answer: 33
user: 已知 \(0<k<4\)，直线 \(l_1: k x-2 y-2 k+8=0\) 和直线 \(l_2: 2 x+k^2 y-4 k^2-4=0\) 与两坐标轴围成一个四边形。求使这个四边形面积最小的 \(k\) 值。答案的形式为\(\frac{m}{n}\)，请给出\(m+n\)的值。
Answer: 9
user: 长度分别为 $1, a, a, a, a, a$ 的线段能成为同一个四面体的 6 条棱的充要条件是 $a$ 的值满足什么条件？原始答案的形式为 a>\frac{\sqrt{k}}{m}，请计算k+m的值。
Answer: 6
user: 对于实数 $t$, 已知等比数列 $\left\{a_{n}\right\}$ 的前三项依次为 $2 t, 5 t-1,6 t+2$, 且该数列的前 $n$ 项和为 $S_{n}$. 求满足不等式 $\left|S_{n}-\frac{1}{65}\right|<1$ 的最大整数 $n$ 的值。设答案为$m$，请给出$m$的值。
Answer: 2
user: 已知函数 $f(x)=|2 x-4|-3 x+8$ $(x \in \mathbf{R})$. 求 $f(x)$ 的反函数 $f^{-1}(x)$ 的解析式。请将答案写成如下分段函数的形式：\begin{cases}-ax+b,\:x\leqslant2\\-\frac{c}{d}x+\frac{e}{f},\:x>2\end{cases}，其中a, b, c, d, e, f是待定系数。请给出a + b + c + d + e + f的值。
Answer: 28
user: 定义在 \(\mathbf{R}\) 上的函数 \(f(x)\) 满足 \(f(0)=0\), \(f(x)+f(1-x)=1\), \(f\left(\frac{x}{5}\right)=\frac{1}{2} f(x)\)，且当 \(0 \leqslant x_{1}<x_{2} \leqslant 1\) 时, \(f\left(x_{1}\right) \leqslant f\left(x_{2}\right)\)。求 \(f\left(\frac{1}{2007}\right)\) 的值。答案应为\(\frac{k}{m}\)的形式，请给出k+m的值。
Answer: 33
user: 长度分别为 $1, a, a, a, a, a$ 的线段能成为同一个四面体的 6 条棱的充要条件是 $a$ 满足的条件是 \frac{\sqrt{k}}{m}，请给出k+m的值。
Answer: 6
user: 已知 $0<k<4$, 直线 $l_{1}: k x-2 y-2 k+8=0$ 和直线 $l_{2}: 2 x+k^{2} y-4 k^{2}-4=0$ 与两坐标轴围成一个四边形。求使这个四边形面积最小的 $k$ 值。原始答案的形式为$\frac{m}{n}$，请给出$m+n$的值。
Answer: 9
user: 函数 $y=\tan x-\frac{2}{\cos x}$ 的最大值是多少？请以$-\sqrt{m}$的格式给出答案，并求出m的值。
Answer: 3
user: 已知数列 $\left\{a_{n}\right\}$ 的通项公式为
$$
a_{n}=\frac{1}{(n+1) \sqrt{n}+n \sqrt{n+1}}\left(n \in \mathbf{N}_{+}\right),
$$
其前 $n$ 项和为 $S_{n}$. 求在数列 $S_{1}, S_{2}, \cdots, S_{2008}$中，有理数项共有多少项。
Answer: 43
user: 设$k、m、n$均为整数，过圆$x^2+y^2=$ $(3k+1)^2$ 外一点 $P(m^3-m,n^3-n)$向该圆引两条切线，切点分别为A、B.求直线$AB$上整点(横、纵坐标都是整数的点)的个数。
Answer: 0
user: 已知函数
$$
f(x)=\frac{x^{4}+\left(k^{2}+2 k-4\right) x^{2}+4}{x^{4}+2 x^{2}+4}
$$
的最小值是 $0$. 求非零实数 $k$ 的值。
Answer: -2
user: 在 $5$ 件产品中有 $4$ 件正品，$1$ 件次品。从中任取两件，记其中含正品的个数为随机变量 $\xi$。求 $\xi$ 的数学期望 $\mathrm{E}\xi$。答案的形式为 \(\frac{k}{m}\)，请给出 \(k+m\) 的值。
Answer: 13
user: 记 \(F(x, y) = (x-y)^2 + \left( \frac{x}{3} + \frac{3}{y} \right)^2 \quad (y \neq 0)\)。求 \(F(x, y)\) 的最小值。原始答案的形式为\(\frac{k}{m}\)，请计算\(k + m\)的值。
Answer: 23
user: 长方体 $A B C D-A_{1} B_{1} C_{1} D_{1}$ 的八个顶点都在球 $O$ 的球面上，其中，$A A_{1}=1, A B=2 \sqrt{2}, A D=3 \sqrt{3}$。求 $B、C$ 两点的球面距离除以 $\pi$ 的值。
Answer: 2
user: 函数
$$
f(x)=\sqrt{x-5}+\sqrt{24-3 x}
$$
的最大值为多少？答案的格式为$k \sqrt{m}$，请给出k+m的值。
Answer: 5
user: 设正三角形\(\triangle_1\)的面积为 \(S_1\)，作\(\triangle_1\)的内切圆，再作内切圆的内接正三角形\(\triangle_2\)，面积为\(S_2\)，如此下去作一系列的正三角形\(\triangle_3, \triangle_4,\cdots\)，其面积相应为\(S_3,S_4,\cdots.\)设\(S_1=1,T_n=S_{1}+S_{2}+\cdots+S_{n}.\)求\(\lim _{n\to + \infty} T_{n}\)的值。原始答案的形式为\(\frac{k}{m}\)，请给出k+m的值。
Answer: 7
user: If the ratio of the legs of a right triangle is $1:2$, then what is the ratio of the corresponding segments of the hypotenuse made by a perpendicular from the vertex? Express your answer in the form $x:y$, and calculate the value of x + y.
Answer: 5
user: If $b$ men take $c$ days to lay $f$ bricks, find the number of days it will take $c$ men working at the same rate to lay $b$ bricks. The original answer is in the format \(\frac{x*b^2}{y*f}\). Please provide the value of x + y.
Answer: 2
user: On the same side of a straight line, three circles are drawn as follows: a circle with a radius of $4$ inches is tangent to the line. The other two circles are equal, and each is tangent to the line and to the other two circles. What is the radius of the equal circles in inches?
Answer: 16
user: Brianna is using part of the money she earned on her weekend job to buy several equally-priced CDs. She used one fifth of her money to buy one third of the CDs. What fraction of her money will she have left after she buys all the CDs? Express your answer in the form \(\frac{k}{m}\), where \(\frac{k}{m}\) is in simplest form. Find the value of \(k + m\).
Answer: 7
user: 求$(x-1)(x-2)(x-3)(x-4)$的最小值，其中$x$为实数。请给出最小值的整数部分。
Answer: -1
user: From a point $P$ outside of a circle with center $O$, tangent segments $\overline{PA}$ and $\overline{PB}$ are drawn. If $\frac{1}{|\overline{OA}|^2} + \frac{1}{|\overline{PA}|^2} = \frac{1}{16}$, find the length of $|\overline{AB}|$.
Answer: 8
user: There are lily pads in a row numbered 0 to 11, in that order. There are predators on lily pads 3 and 6, and a morsel of food on lily pad 10. Fiona the frog starts on pad 0, and from any given lily pad, has a \(\frac{1}{2}\) chance to hop to the next pad, and an equal chance to jump 2 pads. Find the probability that Fiona reaches pad 10 without landing on either pad 3 or pad 6. The original answer is in \(\frac{k}{m}\) format, please give the value of \(k + m\).
Answer: 271
user: The solution of the equation \(7^{x+7} = 8^x\) can be expressed in the form \(x = \log_b 7^7\). The original answer is in the format \(\frac{k}{m}\). Please find the value of \(k + m\).
Answer: 15
user: Let \(M\) be the intersection of diagonals of the convex quadrilateral \(ABCD\), where \(m(\widehat{AMB})=60^\circ\). Let the points \(O_1\), \(O_2\), \(O_3\), \(O_4\) be the circumcenters of the triangles \(ABM\), \(BCM\), \(CDM\), \(DAM\), respectively. The original answer is in the form \(\frac{k}{m}\), where k and m are integers. Please give the value of k + m.
Answer: 5
user: A person standing on the edge of a fire escape simultaneously launches two apples, one straight up with a speed of $7\,\text{m/s}$ and the other straight down at the same speed. How far apart are the two apples $2$ seconds after they were thrown, assuming that neither has hit the ground? If the answer is $x\,\text{m}$, what is the value of $x$?
Answer: 28
user: Equilateral triangle \(ABC\) has been creased and folded so that vertex \(A\) now rests at \(A'\) on \(\overline{BC}\) as shown. If \(BA' = 1\) and \(A'C = 2\), then find the length of the crease \(\overline{PQ}\). The original answer is in the form \(\frac{k}{m}\sqrt{n}\), please give the value of k + m + n.
Answer: 48
user: Given a right triangle $ABC$ with legs $BC=3$ and $AC=4$, find the length of the shorter angle trisector from $C$ to the hypotenuse. Provide your answer in the form \(\frac{k\sqrt{m}-n}{p}\), and then find the value of k + m + n + p.
Answer: 72
user: A rise of $600$ feet is required to get a railroad line over a mountain. The grade can be kept down by lengthening the track and curving it around the mountain peak. Find the additional length of track required to reduce the grade from $3\%$ to $2\%$. If the answer is $x$ ft., what is the value of $x$?
Answer: 10000
user: What is the maximum value of the function \(\frac{1}{|x+1| + |x+2| + |x-3|}?\) The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
Answer: 6
user: Find the coefficient of $x^7$ in the polynomial expansion of 
\[ (1+2x-x^2)^4 \].
Answer: -8
user: How many integer quadruples \(a, b, c, d\) are there such that \(7\) divides \(ab - cd\) where \(0 \leq a, b, c, d < 7\)? Find the total number of such quadruples.
Answer: 385
user: A farmer bought $749$ sheep. He sold $700$ of them for the price paid for the $749$ sheep. The remaining $49$ sheep were sold at the same price per head as the other $700$. Based on the cost, what is the percent gain on the entire transaction? Provide your answer as a number.
Answer: 7
user: A palindrome, such as $83438$, is a number that remains the same when its digits are reversed. The numbers $x$ and $x + 32$ are three-digit and four-digit palindromes, respectively. Find the sum of the digits of $x$.
Answer: 24
user: A barn with a flat roof is rectangular in shape, $10$ yd. wide, $13$ yd. long, and $5$ yd. high. It is to be painted inside and outside, and on the ceiling, but not on the roof or floor. What is the total number of square yards to be painted?
Answer: 590
user: Chord $[AD]$ is perpendicular to the diameter $[BC]$ of a circle. Let $E$ and $F$ be the midpoints of the arcs $AC$ and $CD$, respectively. If $AD \cap BE = \{G\}$, $AF \cap BC = \{H\}$ and $m(AC) = \alpha$, find the measure of angle $BHC$ in terms of $\alpha$. The answer is in the form $x^\circ$, where $x = k - \frac{m\alpha}{2}$. Please find the value of $k + m$.
Answer: 91
user: Which value of $x$ satisfies the equation $\sqrt[3]{6+\sqrt{x}} + \sqrt[3]{6-\sqrt{x}} = \sqrt[3]{3}$?
Answer: 45
user: With a rational denominator, express \(\frac{\sqrt{2}}{\sqrt{2} + \sqrt{3} - \sqrt{5}}\) in the form of a simplified fraction. The original answer is in the form \(\frac{k + m\sqrt{6} + n\sqrt{15}}{p}\). Please find the value of k + m + n + p.
Answer: 11
user: Find the number of digits in the number \( N=2^{12}\times 5^8 \).
Answer: 10
user: An \(8 \times 2\sqrt{2}\) rectangle has the same center as a circle of radius \(2\). Find the area of the region common to both the rectangle and the circle. The original answer is in the form \(k\pi + m\). Please provide the value of \(k + m\).
Answer: 6
user: Consider two concentric circles with radii \(17\) and \(19\). The larger circle has a chord, half of which lies inside the smaller circle. What is the length of the chord in the larger circle? The original answer is in the form \(k\sqrt{m}\). Please find the value of \(k + m\).
Answer: 14
user: Let $T_k$ be the transformation of the coordinate plane that first rotates the plane $k$ degrees counterclockwise around the origin and then reflects the plane across the $y$-axis. Find the least positive integer $n$ such that performing the sequence of transformations $T_1, T_2, T_3, \dots, T_n$ returns the point $(1,0)$ back to itself.
Answer: 359
user: Three digits are selected at random and without replacement from the set of digits $0$ through $9$. Find the probability that the three digits can be arranged to form a multiple of $5$. The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
Answer: 23
user: If \(b > 1\), \(x > 0\) and \((2x)^{\log_b 2} - (3x)^{\log_b 3} = 0\), find the value of \(x\). The original answer is in \(\frac{k}{m}\) format, please give the value of \(k + m\).
Answer: 7
user: Determine the number of digits in $4^{16} \times 5^{25}$ when written in the usual base 10 form.
Answer: 28
user: $A$ gives $B$ as many cents as $B$ has and $C$ as many cents as $C$ has. Similarly, $B$ then gives $A$ and $C$ as many cents as each then has. $C$, similarly, then gives $A$ and $B$ as many cents as each then has. If each finally has $16$ cents, with how many cents does $A$ start?
Answer: 26
user: How many real pairs $(x, y)$ are there such that
\[
x^2 + 2y = 2xy \\
x^3 + x^2y = y^2
\]
Find the number of real pairs $(x, y)$ that satisfy the equations.
Answer: 3
user: For every real number $x$, let $\lfloor x \rfloor$ denote the greatest integer not exceeding $x$, and let:
\[f(x) = \lfloor x \rfloor (2014^{x-\lfloor x \rfloor} - 1).\]
The set of all numbers $x$ such that $1 \leq x < 2014$ and $f(x) \leq 1$ is a union of disjoint intervals. What is the sum of the lengths of those intervals?
Answer: 1
user: How many triples of positive integers $(a,b,c)$ are there such that $a!+b^3 = 18+c^3$?
Answer: 1
user: A \(4 \times 4 \times h\) rectangular box contains a sphere of radius \(2\) and eight smaller spheres of radius \(1\). The smaller spheres are each tangent to three sides of the box, and the larger sphere is tangent to each of the smaller spheres. The original answer is in the form \(k + m\sqrt{n}\). Please find the value of \(k + m + n\).
Answer: 11
user: For how many primes $p$, does there exist an integer $m$ such that $m^3+3m-2 \equiv 0 \pmod{p}$ and $m^2+4m+5 \equiv 0 \pmod{p}$? Write your answer as an integer.
Answer: 2
user: Let $a$ and $b$ satisfy the conditions \( \begin{cases} a^3 - 6a^2 + 15a = 9 \\ b^3 - 3b^2 + 6b = -1 \end{cases} \). Find the value of \((a - b)^{2014}\).
Answer: 1
user: Two different positive numbers \(a\) and \(b\) each differ from their reciprocals by 1. The original answer is in the form \(\sqrt{k}\). Please find the value of \(k\).
Answer: 5
user: If \(a^2 + b^2 + c^2 + d^2 - ab - bc - cd - d + \frac{2}{5} = 0\) where \(a, b, c, d\) are real numbers, find the value of \(a\). The answer should be in the form \(\frac{k}{m}\), please provide the value of \(k + m\).
Answer: 6
user: Nine congruent spheres are packed inside a unit cube such that one sphere is centered at the cube's center, and each of the remaining spheres is tangent to the central sphere and three faces of the cube. What is the radius of each sphere? Provide your answer in the form of \(\frac{k\sqrt{n}-m}{p}\), where \(k\), \(m\), \(n\), and \(p\) are integers, and find the value of \(k + m + n + p\).
Answer: 10
user: The expression $1 - \frac{1}{1 + \sqrt{3}} + \frac{1}{1 - \sqrt{3}}$ equals what value? The answer should be in the form k - \sqrt{m}. Please find the value of k + m.
Answer: 4
user: Let $D$ be a point on side $[BC]$ of triangle $ABC$ where $|BC|=11$ and $|BD|=8$. The circle passing through the points $C$ and $D$ touches $AB$ at $E$. Let $P$ be a point on the line which is passing through $B$ and is perpendicular to $DE$. If $|PE|=7$, then what is $|DP|$?
Answer: 5
user: A thin, uniform rod has mass \(m\) and length \(L\). Let the acceleration due to gravity be \(g\). The rotational inertia of the rod about its center is \(md^2\).

The rod is suspended from a distance \(kd\) from the center and undergoes small oscillations with an angular frequency \(\beta \sqrt{\frac{g}{d}}\).

Find the maximum value of \(\beta\). The original answer is in \(\frac{x}{\sqrt{y}}\) format, please give the value of \(x + y\).
Answer: 3
user: Find the total number of solutions of the equation $xyz=2520$.
Answer: 540
user: If \( a \pm bi \) (\( b \neq 0 \)) are imaginary roots of the equation \( x^3 + qx + r = 0 \), where \( a \), \( b \), \( q \), and \( r \) are real numbers, find \( q \) in terms of \( a \) and \( b \). The original answer is in the form \( kb^2 - ma^2 \). Please calculate the value of k + m.
Answer: 4
user: If $ \log 2 = 0.3010$ and $ \log 3 = 0.4771$, find the value of $ x$ when $ 3^{x+3} = 135$. If the answer is $y$, what is the value of $100y$?
Answer: 147
user: A company sells peanut butter in cylindrical jars. Marketing research suggests that using wider jars would increase sales. If the diameter of the jars is increased by $25\%$ without altering the volume, by what percent must the height be decreased? If the answer is $y\%$, what is the value of $y$?
Answer: 36
user: Let \(P\) be an arbitrary point inside \(\triangle ABC\) with sides \(3, 7, 8\). Find the probability that the distance of \(P\) to at least one vertex of the triangle is less than \(1\). The original answer is in the form \(\frac{\pi}{k}\sqrt{m}\). Please find the value of \(k + m\).
Answer: 39
user: Three balls marked 1, 2, and 3, are placed in an urn. One ball is drawn, its number is recorded, then the ball is returned to the urn. This process is repeated twice more, and each ball is equally likely to be drawn on each occasion. If the sum of the numbers recorded is 6, find the probability that the ball numbered 2 was drawn all three times. Express your answer in the form of a simplified fraction \(\frac{k}{m}\). Please give the value of k + m.
Answer: 8
user: If $A$ and $B$ are vertices of a polyhedron, define the distance $d(A, B)$ to be the minimum number of edges of the polyhedron one must traverse in order to connect $A$ and $B$. For example, if $\overline{AB}$ is an edge of the polyhedron, then $d(A, B) = 1$, but if $\overline{AC}$ and $\overline{CB}$ are edges and $\overline{AB}$ is not an edge, then $d(A, B) = 2$. Let $Q$, $R$, and $S$ be randomly chosen distinct vertices of a regular icosahedron (regular polyhedron made up of 20 equilateral triangles). Find the probability that $d(Q, R) > d(R, S)$. The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
Answer: 29
user: Let $M$ be the least common multiple of all the integers $10$ through $30$, inclusive. Let $N$ be the least common multiple of $M$, $32$, $33$, $34$, $35$, $36$, $37$, $38$, $39$, and $40$. What is the value of $\frac{N}{M}$?
Answer: 74
user: Line segment $\overline{AB}$ is a diameter of a circle with $AB=24$. Point $C$, not equal to $A$ or $B$, lies on the circle. As point $C$ moves around the circle, the centroid (center of mass) of $\triangle{ABC}$ traces out a closed curve missing two points. To the nearest positive integer, find the area of the region bounded by this curve.
Answer: 50
user: What is the sum \( \frac{1}{1 \cdot 2 \cdot 3} + \frac{1}{2 \cdot 3 \cdot 4} + \cdots + \frac{1}{1996 \cdot 1997 \cdot 1998} \)? Provide your answer in the form of \(\frac{k}{m} - \frac{n}{p \cdot q \cdot r}\). Please find the value of \(k + m + n + p + q + r\).
Answer: 4003
user: Let $C_1$ and $C_2$ be circles defined by:
\[
(x - 10)^2 + y^2 = 36
\]
and
\[
(x + 15)^2 + y^2 = 81
\]
respectively. Find the length of the shortest line segment $\overline{PQ}$ that is tangent to $C_1$ at $P$ and to $C_2$ at $Q$.
Answer: 20
user: For how many integers $n$, does the equation system \[\begin{array}{rcl} 2x+3y &=& 7 \\ 5x + ny &=& n^2 \end{array}\] have a solution over integers? Write your answer as an integer.
Answer: 8
user: A quadrilateral has vertices $P(a,b)$, $Q(b,a)$, $R(-a, -b)$, and $S(-b, -a)$, where $a$ and $b$ are integers with $a > b > 0$. The area of $PQRS$ is $16$. What is the value of $a+b$?
Answer: 4
user: A rhombus is formed by two radii and two chords of a circle whose radius is $16$ feet. What is the area of the rhombus in square feet? The original answer is in the form k\sqrt{m}, please give the value of k + m.
Answer: 131
user: How many triangles with positive area are there whose vertices are points in the $xy$-plane whose coordinates are integers $(x,y)$ satisfying $1 \le x \le 4$ and $1 \le y \le 4$? Provide your answer as an integer.
Answer: 516
user: Let $a$, $b$, $c$, $d$, and $e$ be distinct integers such that
\[ (6 - a)(6 - b)(6 - c)(6 - d)(6 - e) = 45. \]
Find the value of $a + b + c + d + e$.
Answer: 25