meetforever/GRPO_CL huggingface dataset:
user: Pascal's triangle is an array of positive integers, as illustrated in the figure. The first row consists of $1$, the second row contains two $1$'s, and each row starts and ends with $1$. For any given row, the $k^\text{th}$ number, when it is not $1$, is the sum of the $k^\text{th}$ and $(k-1)^\text{th}$ numbers from the row directly above it. Determine the quotient of the count of numbers in the first $n$ rows that are not $1$'s and the count of $1$'s. The original answer is in \frac{kn^2 - mn + p}{qn - r} format, please give the value of k + m + p + q + r.
Answer: 12
user: A truck is initially moving at velocity $v$. The driver presses the brake in order to slow the truck to a stop. The brake applies a constant force $F$ to the truck. The truck rolls a distance $x$ before coming to a stop, and the time it takes to stop is $t$.

Find the expression that represents the initial kinetic energy of the truck (i.e., the kinetic energy before the driver starts braking). The answer should be in the form of $kF \cdot mx$, where $k$ and $m$ are integers. Please give the value of $k + m$.
Answer: 2
user: Let \(f(x)\) be a real-valued function satisfying \(af(x) + bf(-x) = px^2 + qx + r\). Here, \(a\) and \(b\) are distinct real numbers, and \(p\), \(q\), \(r\) are non-zero real numbers. Determine the condition under which \(f(x) = 0\) will have real solutions. The original answer is in the form \(\left(\frac{k(a+b)}{m(a-b)}\right) \leq \frac{q^2}{4pr}\). Please provide the value of \(k + m\).
Answer: 2
user: 求值: \(\lim _{n \rightarrow+\infty}\left(\sum_{k=1}^{n} \arctan \frac{2}{k^{2}}\right) = x\pi\)，其中答案为\(\frac{k}{m}\)的形式，请计算k+m的值。
Answer: 7
user: 随机变量 $X(=1,2,3, \cdots), \quad Y(=0,1,2)$, 满足 $P(X=k)=\frac{1}{2^{k}}$, 且 $Y \equiv x(\bmod 3)$ ，求 $E(Y)$ 的值。答案的形式为$\frac{k}{m}$，请给出$k + m$的值。
Answer: 15
user: 设复数 $z_{1}, z_{2}$ 在复平面内对应的点分别为 $Z_{1}, Z_{2}, O$ 为坐标原点，若 $\left|z_{1}\right|=1, 5z_{1}^{2}+z_{2}^{2}-2z_{1}z_{2}=0$，求 $\triangle O Z_{1} Z_{2}$ 的面积。
Answer: 1
user: 设实数 $x_{1}, x_{2}, \cdots, x_{21}$ 满足 $0 \leqslant x_{i} \leqslant 1(1,2, \cdots, 21)$ ，求 $\sum_{i=1}^{21} \sum_{k=1}^{21}\left|x_{i}-x_{k}\right|$ 的最大值。
Answer: 220
user: 设复数 $z$ 满足 $|3 z-7 i|=3$ ，求 $\left|\frac{z^{2}-2 z+2}{z-1+\mathrm{i}}\right|$ 的最大值。答案的形式为\frac{k}{m}，请计算k+m的值。
Answer: 11
user: 设 \(\alpha, \beta\) 为锐角，且 \(\cos (\alpha+\beta)=\frac{\sin \alpha}{\sin \beta}\)，求 \(\tan \alpha\) 的最大值。答案的形式为\(\frac{\sqrt{k}}{m}\)，请给出k和m的和。
Answer: 6
user: 设多项式 \(f(x)\) 的各项系数都是非负实数，且 \(f(1)=f^{\prime}(1)=f^{\prime \prime}(1)=f^{\prime \prime \prime}(1)=1\)。求 \(f(x)\) 的常数项的最小值。答案的形式为\(\frac{k}{m}\)，请给出k+m的值。
Answer: 4
user: 已知集合 $A, B, C \subseteq\{1,2,3, \cdots, 2020\}$ ，且 $A \subseteq B \subseteq C$ ，求有序集合组 $(A, B, C)$的个数。答案的形式为x^y，请给出x+y的值。
Answer: 2024
user: 在 Rt $\triangle A B C$ 中, $\angle A B C=\frac{\pi}{2}, A B=\sqrt{3}, B C=1, \frac{\overrightarrow{P A}}{|\overrightarrow{P A}|}+\frac{\overrightarrow{P B}}{|\overrightarrow{P B}|}+\frac{\overrightarrow{P C}}{|\overrightarrow{P C}|}=\mathbf{0}$。求 $\angle B P C$ 的弧度数。答案的形式为$\frac{k\pi}{m}$，请给出k+m的值。
Answer: 5
user: 设复数 $w=\left(\frac{a+\mathrm{i}}{1+\mathrm{i}}\right)^{2}$, 其中 $a$ 为实数, 若 $w$ 的实部为 2, 求 $w$ 的虚部。答案的形式为$\frac{k}{m}$，请给出$k+m$的值。
Answer: -1
user: 设定点 $A, B, C, D$ 是以点 $O$ 为中心的正四面体的顶点，用 $\sigma$ 表示空间以直线 $O A$ 为轴且满足条件 $\sigma(B)=C$ 的旋转，用 $\tau$ 表示空间关于点 $O, C, D$ 所在平面的镜面反射，设 $l$ 为过 $A B$ 的中点与 $C D$ 的中点的直线，用 $\omega$ 表示空间以 $l$ 为轴的 $180^{\circ}$ 旋转。设 $\sigma \circ \tau$ 表示变换的复合，先作 $\tau$ ，再作 $\sigma$ ，则 $\omega$ 可以表示为 $k \circ m \circ k \circ k \circ m \circ k$。请计算 $k$ 和 $m$ 出现的次数之和。
Answer: 6
user: 如果平面 \(\alpha, \beta\), 直线 \(m, n\), 点 \(A, B\) 满足: \(\alpha \parallel \beta, m \subset \alpha, n \subset \beta, A \in \alpha, B \in \beta\)，且 \(AB\) 与 \(\alpha\) 所成的角为 \(\frac{\pi}{4}\), \(m \perp AB\), \(n\) 与 \(AB\) 所成的角为 \(\frac{\pi}{3}\)，那么 \(m\) 与 \(n\) 所成的角大小为 \(\frac{k}{m}\)，请计算k+m的值。
Answer: 5
user: 设 \(f(x)=\mathrm{e}^{a x}(a>0)\)。若过点 \(P(a, 0)\) 且平行于 \(y\) 轴的直线与曲线 \(C: y=f(x)\) 的交点为 \(Q\)，曲线 \(C\) 过点 \(Q\) 的切线交 \(x\) 轴于点 \(R\)，则 \(\triangle P Q R\) 的面积的最小值是多少？答案的形式为\(\frac{\sqrt{k \mathrm{e}}}{m}\)，请给出k + m的值。
Answer: 4
user: 设双曲线 $C_{1}: \frac{x^{2}}{a^{2}}-\frac{y^{2}}{4}=k(a>2, k>0)$, 椭圆 $C_{2}: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{4}=1$. 若 $C_{2}$ 的短轴长与 $C_{1}$ 的实轴长的比值等于 $C_{2}$ 的离心率，求 $C_{1}$ 在 $C_{2}$ 的一条准线上截得线段的长。
Answer: 4
user: 若 $A+B=\frac{2 \pi}{3}$, 求 $\cos ^{2} A+\cos ^{2} B$ 的最小值和最大值，分别记为 $x$ 和 $y$。答案的形式分别为x = \frac{m}{n}和y = \frac{p}{q}，请给出m+n+p+q的值。
Answer: 8
user: 若 \(A B\) 为过抛物线 \(y^{2}=4 x\) 的焦点 \(F\) 的弦, \(O\) 为坐标原点, 且 \(\angle O F A=135^{\circ}\), \(C\) 为抛物线的准线与 \(x\) 轴的交点, 求 \(\angle A C B\) 的正切值。原始答案的形式为 \(k \sqrt{m}\)，请给出 \(k + m\) 的值。
Answer: 4
user: 如图 1 所示, 在锐角 $\triangle A B C$ 中, $A B$ 边上的高 $C E$ 与 $A C$ 边上的高 $B D$ 交于点 $H$, 以 $D E$ 为直径作圆与 $A C$ 的另一个交点为 $G$, 若 $B C=25, B D=20, B E=7$, 求 $A G$ 的长。答案的形式为\frac{k}{m}，请给出k+m的值。
Answer: 59
user: 若数列 $\left\{a_{n}\right\}$ 的通项公式为 $a_{n}=\lg \left(1+\frac{2}{n^{2}+3 n}\right), n=1,2, \cdots, S_{n}$ 是数列 $\left\{a_{n}\right\}$ 的前 $n$ 项和，则求 $\lim _{n \rightarrow \infty} S_{n}$ 的值。答案的形式为 $\lg k$，请给出 $k$ 的值。
Answer: 3
user: 用蓝色和红色给一排10个方格涂色，要求至多有两个蓝色方格相邻的涂法种数是多少？
Answer: 504
user: 将函数 $y=\sqrt{4+6 x-x^{2}}-2(x \in[0,6])$ 的图象逆时针方向旋转 $\theta(0 \leqslant \theta \leqslant \alpha)$, 得到曲线 C。若对于每一个旋转角 $\theta$ ，曲线 C 都是一个函数的图象，则 $\alpha$ 的最大值为多少？答案的形式为\arctan \frac{k}{m}，请给出k+m的值。
Answer: 5
user: 有 $n$ 个质点, 每个质点质量为 $m_{k}$, 则质心位置 $x=\frac{\sum m_{k} x_{k}}{\sum m_{k}}$。对于一杆，长 3 m，放于 $x \in[-1,2]$ 间，且线密度满足 $\beta=2+x$，求质心的位置。答案的形式为\frac{k}{m}，请给出k+m的值。
Answer: 9
user: 已知 $[x]$ 为高斯函数，求满足 $\left[\frac{x}{2}\right]+\left[\frac{x}{3}\right]+\left[\frac{x}{5}\right]=x$ 的解的组数。
Answer: 30
user: 已知 $m, n$ 的最大公约数为 $10!$，最小公倍数为 $50!$，求数对 $(m, n)$ 的组数。答案是 $2^x$ 的形式，请求出 $x$ 的值。
Answer: 15
user: 已知四面体$D-ABC$中，$AC=BC=AD=BD=1$，求$D-ABC$体积的最大值。答案的形式为\frac{k\sqrt{m}}{n}，请给出k + m + n的值。
Answer: 32
user: 已知 $|\mathrm{z}|=1, \mathrm{zx}^{2}+2 \overline{\mathrm{z}} \mathrm{x}+2=0$，求 $z$ 的虚部的所有可能值。如果答案有多个，请用逗号分隔。答案的形式为\frac{\sqrt{m}}{n}, -\frac{\sqrt{m}}{n}, \frac{p}{q}，请计算m+n+p+q的值。
Answer: 20
user: 设复数 \(z\) 满足 \(|3z - 7i| = 3\)，求 \(\left|\frac{z^2 - 2z + 2}{z - 1 + i}\right|\) 的最大值和最小值。如果最大值为 \(\frac{m}{n}\)，最小值为 \(\frac{p}{q}\)，请给出 \(m + n + p + q\) 的值。
Answer: 16
user: 已知平面向量 $\boldsymbol{a}, \boldsymbol{b}, \boldsymbol{c}$ 满足 $|\boldsymbol{a}| \leq 2,|\boldsymbol{b}| \leq 1,|\boldsymbol{a}-2 \boldsymbol{b}-\boldsymbol{c}| \leq|\boldsymbol{a}+2 \boldsymbol{b}|$ ，则对所有可能的 $\boldsymbol{c},|\boldsymbol{c}|$ 的最大值和最小值分别是多少？请分别以 $x$ 和 $y$ 表示。答案的格式为 $x = k\sqrt{m}, y = 0$，请给出$k + m$的值。
Answer: 6
user: 已知 $f(x)=\sin x \cos x+\sin x+\frac{2}{5} \cos x, x \in\left[0, \frac{\pi}{2}\right]$, 设 $f(x)$ 的最大值为 $M=\frac{k}{n}$，最小值为 $m=\frac{p}{q}$。请计算 $k+n+p+q$ 的值。
Answer: 70
user: 若复数 $z$ 满足 $\left|\frac{1}{z}+z\right|=1$, 则求 $|z|$ 的两个可能的值。答案的形式分别为 \frac{\sqrt{k}-m}{n} 和 \frac{\sqrt{p}+q}{r}，请计算k+m+n+p+q+r的值。
Answer: 16
user: 已知向量 $\boldsymbol{a}, \boldsymbol{b}, \boldsymbol{c}$ 满足 $|\boldsymbol{a}| \leqslant 1,|\boldsymbol{b}| \leqslant 1,|\boldsymbol{a}+2 \boldsymbol{b}+\boldsymbol{c}|=|\boldsymbol{a}-2 \boldsymbol{b}|$。求 $|c|$ 的最大值和最小值。最大值的答案形如 $k \sqrt{m}$，请给出 $k + m$ 的值。
Answer: 7
user: 若非负实数 $x, y$ 满足 $2 x+y=1$, 求 $x+\sqrt{x^{2}+y^{2}}$ 的最小值和最大值。最小值的答案形式为\(\frac{k}{m}\)，最大值的答案形式为\(n\)。请计算k+m+n的值。
Answer: 10
user: 由 \(y \geqslant 2|x|-1\) 和 \(y \leqslant-3|x|+5\) 所围成的平面区域的面积为多少？请注意，答案的形式为\(\frac{k}{m}\)，其中k和m是互质的正整数。请给出k+m的值。
Answer: 41
user: 已知实数 $a, b$ 满足：当 $|x| \leqslant 1$ 时，恒有 $\left|x^{2}+a x+b\right| \leqslant 2$ 。求 $a$ 的最大值、最小值，以及 $b$ 的最大值。答案的形式为m, n, p，其中m为$a$的最大值，n为$a$的最小值，p为$b$的最大值。请给出m + n + p的值。
Answer: 1
user: 已知集合 \( A=\{x\ |\ x>a\} \), \( B=\{-1,0,1\} \)，若 \( A\cap B=\{0,1\} \)，则实数 \( a \) 的取值范围是 \([a, b)\)。请给出a+b的值。
Answer: -1
user: 在 \(\triangle ABC\) 中，若 \(|\overrightarrow{AB}|=2\)，\(|\overrightarrow{AC}|=3\)，\(|\overrightarrow{BC}|=4\)，\(O\) 为 \(\triangle ABC\) 的内心，且 \(\overrightarrow{AO}=\lambda\overrightarrow{AB}+\mu\overrightarrow{BC}\)，则 \(\lambda+\mu\) 的答案形式为 \(\frac{k}{m}\)，请计算 \(k + m\) 的值。
Answer: 16
user: 满足 $a \geq b \geq c$ 且 $abc = 2(a-1)(b-1)(c-1)$ 的正整数组 $(a, b, c)$ 共有多少组？
Answer: 5
user: 设实数 $x,y$ 满足不等式组 \(\begin{cases} x-y\leq0\\ 4x-y\geq0\\ x+y\leq 3\end{cases}\)，求 \(z=x+2y-\frac{1}{x}\) 的最大值。
Answer: 4
user: 已知等腰梯形 ABCD 中，AD\parallel BC，BC=2AD=4，\angle ABC=60^\circ，\overrightarrow{CE}=\frac{1}{3}\overrightarrow{CD}，求 \overrightarrow{CA}\cdot\overrightarrow{BE} 的值。
Answer: -10
user: 设函数 \( f(x) = x^2 - x + \sqrt{2x^4 - 6x^2 + 8x + 16} \)，求 \( f(x) \) 的最小值。
Answer: 4
user: 记 \( S = \frac{3^2+4}{3^2-4} + \frac{4^2+4}{4^2-4} + \frac{5^2+4}{5^2-4} + \cdots + \frac{13^2+4}{13^2-4} \)，则与 \( S \) 最接近的整数为多少？
Answer: 15
user: 在四边形 ABCD 中，\(AB \parallel CD\)，\(\overrightarrow{AC} = \lambda \overrightarrow{AB} + \mu \overrightarrow{AD}\) (\(\lambda, \mu \in \mathbb{R}\))。若 \(\lambda + \mu = \frac{3}{2}\)，求 \(\frac{|\overrightarrow{CD}|}{|\overrightarrow{AB}|}\) 的值。原始答案是以\(\frac{k}{m}\)的形式给出的，请给出k+m的值。
Answer: 3
user: 在正四面体 ABCD 中，棱 AD 的中点和面 BCD 的中心的连线为 MN，棱 CD 的中点和面 ABC 的中心的连线为 PQ。求 MN 与 PQ 所成角的余弦值。答案应为\(\frac{k}{m}\)的形式，请给出k+m的值。
Answer: 19
user: 已知函数 \( f(x) = \sqrt{2x^4-18x^2+12x+68}+x^2-x+1 \)，求 \( f(x) \) 的最小值。
Answer: 9
user: 已知 \(A\), \(B\), \(C\) 是平面上三个不同点，且 \(BC = a\), \(CA = b\), \(AB = c\)，求 \(\frac{c}{a+b}+\frac{b}{c}\) 的最小值。答案应为\(\sqrt{k} - \frac{m}{n}\)的形式，请给出k+m+n的值。
Answer: 5
user: 双曲线 \(x^2-y^2=2022\) 上格点（横纵坐标均为整数的点）的个数为多少？
Answer: 0
user: 设 $a, b \in \mathbb{N}_+$，且满足 $\frac{1}{a}-\frac{1}{b}=\frac{1}{2021}$，则所有正整数对 $(a, b)$ 的个数为多少？
Answer: 4
user: How many complex numbers satisfy the equation $z^{5}=\overline{z}$, where $\overline{z}$ is the conjugate of the complex number $z$?
Answer: 7
user: For how many real values of $x$ is $\sqrt{120 - \sqrt{x}}$ an integer?
Answer: 11
user: If \( f(x) = \frac{x(x-1)}{2} \), then what is the expression for \( f(x+2) \)? The original answer is in the form \(\frac{(x+k)f(x+m)}{n*x}\). Please find the value of k + m + n.
Answer: 4
user: What is the $111^{\text{st}}$ smallest positive integer which does not have $3$ and $4$ in its base-$5$ representation?
Answer: 755
user: A bakery owner turns on his doughnut machine at 8:30 AM. At 11:10 AM, the machine has completed one-third of the day's job. At what time will the doughnut machine complete the job? If the answer is in the format "x:xx PM", please provide the numerical value of x:xx without the colon.
Answer: 430
user: Suppose $a$ and $b$ are digits which are not both nine and not both zero. If the repeating decimal $0.\overline{AB}$ is expressed as a fraction in lowest terms, find how many possible denominators there are.
Answer: 5
user: On an examination of $n$ questions, a student answers correctly $15$ of the first $20$. Of the remaining questions, he answers one third correctly. All the questions have the same credit. If the student's mark is $50\%$, how many different values of $n$ can there be?
Answer: 1
user: Given $0 \le x_0 <1$, let \[ x_n= \begin{cases} 2x_{n-1} & \text{if}\ 2x_{n-1} <1 \\ 2x_{n-1}-1 & \text{if}\ 2x_{n-1} \ge 1 \end{cases} \] for all integers $n>0$. Find the number of values of $x_0$ for which $x_0=x_5$.
Answer: 31
user: In the adjoining figure, $ABCD$ is a square and $CMN$ is an equilateral triangle. If the area of $ABCD$ is one square inch, find the area of $CMN$ in square inches. The original answer is in the form \(k\sqrt{n} - m\), please give the value of k + n + m.
Answer: 8
user: A list of five positive integers has a mean of $12$ and a range of $18$. The mode and median are both $8$. Find how many different values are possible for the second largest element of the list.
Answer: 6
user: A bug (of negligible size) starts at the origin on the coordinate plane. First, it moves one unit right to \((1,0)\). Then it makes a \(90^\circ\) counterclockwise turn and travels \(\frac{1}{2}\) a unit to \(\left(1, \frac{1}{2} \right)\). If it continues in this fashion, each time making a \(90^\circ\) degree turn counterclockwise and traveling half as far as the previous move, to which point \((x, y)\) will it come closest? Provide the answer in the form of two fractions \(\left(\frac{a}{b}, \frac{c}{d}\right)\), and calculate the value of a + b + c + d.
Answer: 16
user: A function \(f\left(a \frac{b}{c}\right)\) for a simplified mixed fraction \(a \frac{b}{c}\) returns \(\frac{a + b}{c}\). For instance, \(f\left(2 \frac{5}{7}\right) = 1\) and \(f\left(\frac{4}{5}\right) = \frac{4}{5}\). Find the sum of the three smallest positive rational \(x\) where \(f(x) = \frac{2}{9}\). The original answer is in \(\frac{m}{n}\) format, please give the value of \(m + n\).
Answer: 95
user: Let $f(x) = x^2 + 6x + 1$, and let $R$ denote the set of points $(x,y)$ in the coordinate plane such that
\[ f(x) + f(y) \leq 0 \text{ and } f(x) - f(y) \leq 0 \]
Find the integer that is closest to the area of $R$.
Answer: 25
user: Two circles of radius 1 are to be constructed as follows. The center of circle \(A\) is chosen uniformly and at random from the line segment joining \((0,0)\) and \((2,0)\). The center of circle \(B\) is chosen uniformly and at random, and independently of the first choice, from the line segment joining \((0,1)\) to \((2,1)\). Find the probability that circles \(A\) and \(B\) intersect. The original answer is in the form \(\frac{k \sqrt{m} - n}{p}\), please give the value of k + m + n + p.
Answer: 14
user: A convex polyhedron $Q$ has vertices $V_1, V_2, \ldots, V_n$, and 100 edges. The polyhedron is cut by planes $P_1, P_2, \ldots, P_n$ in such a way that plane $P_k$ cuts only those edges that meet at vertex $V_k$. In addition, no two planes intersect inside or on $Q$. The cuts produce $n$ pyramids and a new polyhedron $R$. Find the number of edges of the polyhedron $R$.
Answer: 300
user: Evaluate the following sum: 
\[ \binom{n}{1} \sin(a) + \binom{n}{2} \sin(2a) + \ldots + \binom{n}{n} \sin(na) \]
Find the value of the expression in terms of \( n \) and \( a \). The answer is in the form \( k \cdot m^n \cos^n \left(\frac{a}{2}\right)\sin \left(\frac{n \cdot a}{2}\right) \). Please provide the sum of k and m.
Answer: 3
user: Let $K$ be the number of sequences $A_1$, $A_2$, \dots, $A_n$ such that $n$ is a positive integer less than or equal to $10$, each $A_i$ is a subset of \{1, 2, 3, \dots, 10\}, and $A_{i-1}$ is a subset of $A_i$ for each $i$ between $2$ and $n$, inclusive. For example, \{\}, \{5, 7\}, \{2, 5, 7\}, \{2, 5, 7\}, \{2, 5, 6, 7, 9\} is one such sequence, with $n = 5$. What is the remainder when $K$ is divided by $10$?
Answer: 5
user: On a certain math exam, $10\%$ of the students got 70 points, $25\%$ got 80 points, $20\%$ got 85 points, $15\%$ got 90 points, and the rest got 95 points. What is the difference between the mean and the median score on this exam?
Answer: 1
user: A softball team played ten games, scoring $1, 2, 3, 4, 5, 6, 7, 8, 9$, and $10$ runs. They lost by one run in exactly five games. In each of the other games, they scored twice as many runs as their opponent. How many total runs did their opponents score?
Answer: 45
user: How many pairs of positive integers $(a,b)$ are there such that the roots of the polynomial $x^2-ax-b$ are not greater than $5$? Provide the number of such pairs.
Answer: 50
user: Find the sum of the digits of the largest even three-digit number (in base ten representation) which is not changed when its units and hundreds digits are interchanged.
Answer: 25
user: Regular hexagon \(ABCDEF\) has an area of \(1\). Starting with edge \(AB\) and moving clockwise, a new point is drawn exactly one half of the way along each side of the hexagon. For example, on side \(AB\), the new point, \(G\), is drawn so that \(AG = \frac{1}{2} AB\). This forms hexagon \(GHIJKL\), as shown. What is the area of this new hexagon? The answer is in the form \(\frac{k}{m}\), where \(k\) and \(m\) are integers. Please find the value of \(k + m\).
Answer: 7
user: In triangle $ABC$, lines $CE$ and $AD$ are drawn such that \( \frac{CD}{DB} = \frac{3}{1} \) and \( \frac{AE}{EB} = \frac{3}{2} \). Let \( r = \frac{CP}{PE} \), where $P$ is the intersection point of $CE$ and $AD$. Find the value of \( r \).
Answer: 5
user: If $a, b,$ and $c$ are digits for which \[ \begin{array}{cccc} & 7 & a & 2 \\ - & 4 & 8 & b \\ \hline & c & 7 & 3 \end{array} \] then find the value of $a + b + c$.
Answer: 17
user: How many ordered pairs of integers $(x,y)$ are there such that $2011y^2 = 2010x + 3$?
Answer: 0
user: In base $R_1$, the expanded fraction $F_1$ becomes $0.373737\ldots$, and the expanded fraction $F_2$ becomes $0.737373\ldots$. In base $R_2$, fraction $F_1$, when expanded, becomes $0.252525\ldots$, while fraction $F_2$ becomes $0.525252\ldots$. Find the sum of $R_1$ and $R_2$, each written in base ten.
Answer: 19
user: Fermat determines that since his final exam counts as two tests, he only needs to score a $28$ on it for his test average to be $70$. If he gets a perfect $100$ on the final exam, his average will be $88$. What is the lowest score Fermat can receive on his final and still have an average of $80$?
Answer: 68
user: Let's write \(p\), \(q\), and \(r\) as three distinct prime numbers, where 1 is not a prime. Find the smallest positive perfect cube that leaves \(n = pq^2r^4\) as a divisor. The original answer is in the form \((p^a q^b r^c)^3\), please give the value of a + b + c.
Answer: 4
user: How many even three-digit integers have the property that their digits, read left to right, are in strictly increasing order? Provide your answer as an integer.
Answer: 34
user: In the adjoining figure, $AB$ and $BC$ are adjacent sides of square $ABCD$. Let $M$ be the midpoint of $AB$ and $N$ be the midpoint of $BC$. The lines $AN$ and $CM$ intersect at $O$. Determine the ratio of the area of quadrilateral $AOCD$ to the area of square $ABCD$. The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
Answer: 5
user: What is the sum of real roots of the equation $x^4-8x^3+13x^2 -24x + 9 = 0$?
Answer: 7
user: Regular hexagon $ABCDEF$ has vertices $A$ and $C$ at $(0,0)$ and $(7,1)$, respectively. What is the area of the hexagon? If the area is expressed as $x\sqrt{3}$, what is the value of $x$?
Answer: 25
user: In triangle $ABC$, $AB = AC$. If there is a point $P$ strictly between $A$ and $B$ such that $AP = PC = CB$, then if the answer is $x^\circ$, what is the value of $x$?
Answer: 36
user: If the (convex) area bounded by the x-axis and the lines $y=mx+4$, $x=1$, and $x=4$ is $7$, find the value of $m$. The original answer is in \frac{k}{m} format, please give the value of k + m.
Answer: 1
user: What is the perimeter of the smallest rectangle with integer side lengths that fits three non-overlapping squares with areas $4$, $9$, and $16$?
Answer: 24
user: Al and Barb start their new jobs on the same day. Al's schedule is $3$ work-days followed by $1$ rest-day. Barb's schedule is $7$ work-days followed by $3$ rest-days. On how many of their first $1000$ days do both have rest-days on the same day?
Answer: 100
user: Find the last 5 digits of the number $M = 5^{2010}$.
Answer: 65625
user: A mass is attached to an ideal spring. At time \(t = 0\) the spring is at its natural length and the mass is given an initial velocity; the period of the ensuing (one-dimensional) simple harmonic motion is \(T\). At what time \(t\) is the power delivered to the mass by the spring first a maximum, in terms of \(T\)? The answer is in the form \(\frac{kT}{m}\). Please find the value of \(k + m\).
Answer: 9
user: Joe and JoAnn each bought 12 ounces of coffee in a 16-ounce cup. Joe drank 2 ounces of his coffee and then added 2 ounces of cream. JoAnn added 2 ounces of cream, stirred the coffee well, and then drank 2 ounces. What is the resulting ratio of the amount of cream in Joe's coffee to that in JoAnn's coffee? The original answer is in the form \(\frac{k}{m}\). Please give the value of k + m.
Answer: 13
user: First, $a$ is chosen at random from the set $\{1, 2, 3, \ldots, 99, 100\}$, and then $b$ is chosen at random from the same set. Find the probability that the integer $3^a + 7^b$ has a units digit of $8$. The original answer is in the form of a common fraction \(\frac{k}{m}\). Please provide the value of k + m.
Answer: 19
user: Dave's Amazing Hotel has $3$ floors. If you press the up button on the elevator from the $3$rd floor, you are immediately transported to the $1$st floor. Similarly, if you press the down button from the $1$st floor, you are immediately transported to the $3$rd floor. Dave gets in the elevator at the $1$st floor and randomly presses up or down at each floor. After doing this $482$ times, the probability that Dave is on the first floor can be expressed as $\frac{m}{n}$ where $m$ and $n$ are relatively prime positive integers. What is the remainder when $m+n$ is divided by $1000$?
Answer: 803
user: An object in the plane moves from one lattice point to another. At each step, the object may move one unit to the right, one unit to the left, one unit up, or one unit down. If the object starts at the origin and takes a ten-step path, how many different points could be the final point? Write your answer as an integer.
Answer: 121
user: A square $ABCD$ with sides of length 1 is divided into two congruent trapezoids and a pentagon, which have equal areas, by joining the center of the square with points $E, F, G$ where $E$ is the midpoint of $BC$, $F$ and $G$ are on $AB$ and $CD$, respectively. They are positioned such that $AF < FB$, $DG < GC$, and $F$ is directly opposite $G$. If $FB = x$, the length of the longer parallel side of each trapezoid, find the value of $x$. The original answer is in the form of \frac{k}{m}, please give the value of k + m.
Answer: 11
user: The radius of Earth at the equator is approximately 4000 miles. Suppose a jet flies once around Earth at a speed of 500 miles per hour relative to Earth. If the flight path is a negligible height above the equator, find the best estimate of the number of hours of flight.
Answer: 50
user: Find the number of distinct integral solutions of $x^{4} + 2x^{3} + 3x^{2} - x + 1 \equiv 0 \pmod{30}$ where $0 \leq x < 30$.
Answer: 1
user: What is the minimum value of \((x^2+2x+8-4\sqrt{3})\cdot(x^2-6x+16-4\sqrt{3})\) where \(x\) is a real number? The answer should be in the form \(a - b\sqrt{3}\). Please find the value of \(a + b\).
Answer: 176
user: A sequence of numbers is defined recursively by $a_1 = 1$, $a_2 = \frac{3}{7}$, and
\[
a_n = \frac{a_{n-2} \cdot a_{n-1}}{2a_{n-2} - a_{n-1}}
\]
for all $n \geq 3$. Then $a_{2019}$ can be written as $\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. Find the value of $p+q$.
Answer: 8078
user: Circles with centers \(O\) and \(P\) have radii 2 and 4, respectively, and are externally tangent. Points \(A\) and \(B\) are on the circle centered at \(O\), and points \(C\) and \(D\) are on the circle centered at \(P\), such that \(\overline{AD}\) and \(\overline{BC}\) are common external tangents to the circles. Find the area of hexagon \(AOBCPD\). The original answer is in the form \(k\sqrt{m}\). Please give the value of \(k + m\).
Answer: 26
user: $(a_n)_{n = 1}^\infty$ is defined on real numbers with $a_n \neq 0$, $a_n a_{n + 3} = a_{n + 2} a_{n + 5}$ and $a_1 a_2 + a_3 a_4 + a_5 a_6 = 6$. Find the value of $a_1 a_2 + a_3 a_4 + \cdots + a_{41} a_{42}$.
Answer: 42
user: Let $d$ be one of the common tangent lines of externally tangent circles $k_1$ and $k_2$. $d$ touches $k_1$ at $A$. Let $[AB]$ be a diameter of $k_1$. The tangent from $B$ to $k_2$ touches $k_2$ at $C$. If $|AB|=8$ and the diameter of $k_2$ is $7$, then what is the length of $|BC|$?
Answer: 8
user: The Minnesota Twins face the New York Mets in the 2006 World Series. Assuming the two teams are evenly matched (each has a \(0.5\) probability of winning any game), find the probability that the World Series (a best of 7 series of games which lasts until one team wins four games) will require the full seven games to determine a winner. The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
Answer: 21
user: Let \(f(x) = \frac{x+1}{x-1}\). Then for \(x^2 \neq 1\), find the expression for \(f(-x)\). The original answer is in \(\frac{k}{m}\) format, please give the value of \(k + m\).
Answer: 2