ChenDRAG/VeRL_math_validation huggingface dataset:
Problem: 已知正四棱雉 $V-A B C D$ 的高等于 $A B$ 长度的一半， $M$ 是侧棱 $V B$ 的中点， $N$ 是侧棱 $V D$ 上点，满足 $D N=2 V N$ ，则异面直线 $A M, B N$ 所成角的余弦值为\(\frac{\sqrt{m}}{n}\)的格式，请给出m + n的值。
Answer: 22
Problem: 在平面直角坐标系 $x O y$ 中，圆 $C_{1}: x^{2}+y^{2}-a=0$ 关于直线 $l$ 对称的圆为 $C_{2}: x^{2}+y^{2}+2 x-2 a y+3=0$, 则直线 $l$ 的方程为 ax + by + c = 0。请给出 a + |b| + c 的值。
Answer: 11
Problem: 已知 $f(x)$ 是 $\mathbf{R}$ 上的奇函数, $f(1)=1$, 且对任意 $x<0$, 均有 $f\left(\frac{x}{x-1}\right)$ $=x f(x)$ 。 求 $f(1) f\left(\frac{1}{100}\right)+f\left(\frac{1}{2}\right) f\left(\frac{1}{99}\right)+f\left(\frac{1}{3}\right) f\left(\frac{1}{98}\right)+\cdots+f\left(\frac{1}{50}\right) f\left(\frac{1}{51}\right)$ 的值。原始答案为 \frac{m^{n}}{k!} 格式，请给出 m + n+k 的值。
Answer: 199
Problem: 设实数 $a$ 满足 $a<9 a^{3}-11 a<|a|$, 则 $a$ 的取值范围是一个区间，区间的两端点和形如\frac{a\sqrt{b}+c\sqrt{d}}{e}，请给出a+b+c+d+e
Answer: 13
Problem: 在 $\triangle A B C$ 中, 已知 $\overrightarrow{A B} \cdot \overrightarrow{A C}+2 \overrightarrow{B A} \cdot \overrightarrow{B C}=3 \overrightarrow{C A} \cdot \overrightarrow{C B}$。求 $\sin C$ 的最大值。原始答案的形式为$\frac{\sqrt{m}}{n}$，请给出m + n的值。
Answer: 10
Problem: 双曲线 $C$ 的方程为 $x^{2}-\frac{y^{2}}{3}=1$ ，左、右焦点分别为 $F_{1} 、 F_{2}$ 。过点 $F_{2}$ 作一直线与双曲线 $C$ 的右半支交于点 $P, Q$, 使得 $\angle F_{1} P Q=90^{\circ}$, 则 $\triangle F_{1} P Q$ 的内切圆半径的原始答案为 \sqrt{m} + n的形式，请给出m+n的值。
Answer: 6
Problem: 设 $P$ 为一圆锥的顶点, $A, B, C$ 是其底面圆周上的三点, 满足 $\angle A B C=90^{\circ}$, $M$ 为 $A P$ 的中点. 若 $A B=1, A C=2, A P=\sqrt{2}$, 则二面角 $M-B C-A$ 的大小以 \arctan \frac{m}{n} 的形式表示，请给出 m + n 的值。
Answer: 5
Problem: 在平面直角坐标系 $x O y$ 中, 圆 $C_{1}: x^{2}+y^{2}-a=0$ 关于直线 $l$ 对称的圆为 $C_{2}: x^{2}+y^{2}+2 x-2 a y+3=0$, 请给出直线 $l$ 的方程系数之和。
Answer: 3
Problem: 已知 $\left\{a_{n}\right\}$ 是各项均为正数的等比数列，且 $a_{50}, a_{51}$ 是方程 $100 \lg ^{2} x=\lg (100 x)$的两个不同的解。求 $a_{1} a_{2} \cdots a_{100}$ 的值。已知原始答案为$\sqrt{m}$的形式，请给出m的值。
Answer: 10
Problem: 设两个严格递增的正整数数列 $\left\{a_{n}\right\},\left\{b_{n}\right\}$ 满足: $a_{10}=b_{10}<2017$ ，对任意正整数 $n$ ，有 $a_{n+2}=a_{n+1}+a_{n}, b_{n+1}=2 b_{n}$, 请给出 $a_{1}+b_{1}$ 的所有可能值之和。
Answer: 33
Problem: 正三棱雉 $P-A B C$ 中, $A B=1, A P=2$, 过 $A B$ 的平面 $\alpha$ 将其体积平分, 则棱 $P C$与平面 $\alpha$ 所成角的余弦值的原始答案是 \(\frac{\sqrt{m} + n}{k}\) 的形式，请给出 m + n + k 的值。
Answer: 18
Problem: 设数列 $\left\{a_{n}\right\}$ 定义为 $a_{1}=1$ ，$$a_{n+1}=\begin{cases}a_n+n,&a_n\leqslant n,\\[2ex]a_n-n,&a_n>n,\end{cases}n=1,2,\cdots.$$求满足 $a_{r}<r \leqslant 3^{2017}$ 的正整数 $r$ 的个数。原始答案是 \frac{a^{b}-c}{d} 的形式，请给出a+b+c+d的值。
Answer: 4041
Problem: 定义在 $\mathbb{R}$ 上的函数 $f(x)$ 对任意实数 $x$ 有 $f(x+3) \cdot f(x-4)=-1$, 又当 $0 \leqslant x<7$ 时, $f(x)=\log _{2}(9-x)$, 则 $f(-100)$ 的值为分数形式，请给出分子与分母之和。
Answer: 1
Problem: 设 $x_{1}, x_{2}, x_{3}$ 是非负实数, 满足 $x_{1}+x_{2}+x_{3}=1$, 求 $\left(x_{1}+3 x_{2}+5 x_{3}\right)\left(x_{1}+\frac{x_{2}}{3}+\frac{x_{3}}{5}\right)$的最小值和最大值。已知最小值为整数，最大值为\frac{m}{n}的格式，请给出m + n+最小值的值。
Answer: 15
Problem: 若实数 $x, y$ 满足 $x^{2}+2 \cos y=1$, 则 $x-\cos y$ 的取值范围是。请给出区间端点的和的平方。
Answer: 3
Problem: 在平面直角坐标系 $x O y$ 中, 椭圆 $C$ 的方程为 $\frac{x^{2}}{9}+\frac{y^{2}}{10}=1, F$ 为 $C$ 的上焦点, $A$为 $C$ 的右顶点, $P$ 是 $C$ 上位于第一象限内的动点, 则四边形 $O A P F$ 的面积的最大值为 \qquad。原始答案为\frac{\sqrt{m} + n}{k}的格式，请给出m + n + k的值。
Answer: 16
Problem: 设 $f(x)$ 是定义在 $\mathbf{R}$ 上的函数, 对任意实数 $x$ 有 $f(x+3) \cdot f(x-4)=-1$. 又当 $0 \leqslant x<7$ 时, $f(x)=\log _{2}(9-x)$, 则 $f(-100)$ 的值为 \qquad. 原始答案是以-\frac{m}{n}的形式给出的，请提供m + n的值。
Answer: 1
Problem: 在平面直角坐标系 $x O y$ 中, 椭圆 $C$ 的方程为 $\frac{x^{2}}{9}+\frac{y^{2}}{10}=1, F$ 为 $C$ 的上焦点, $A$为 $C$ 的右顶点, $P$ 是 $C$ 上位于第一象限内的动点, 则四边形 $O A P F$ 的面积的最大值为。原始答案为\frac{m\sqrt{n}}{k}的格式，请给出m + n + k的值。
Answer: 16
Problem: 设两个严格递增的正整数数列 $\left\{a_{n}\right\},\left\{b_{n}\right\}$ 满足: $a_{10}=b_{10}<2017$ ，对任意正整数 $n$ ，有 $a_{n+2}=a_{n+1}+a_{n}, b_{n+1}=2 b_{n}$, 则 $a_{1}+b_{1}$ 的所有可能值为多少？请提供所有可能值的和。
Answer: 33
Problem: 在平面直角坐标系 $x O y$ 中, 曲线 $C_{1}: y^{2}=4 x$, 曲线 $C_{2}:(x-4)^{2}+y^{2}=8$. 经过 $C_{1}$上一点 $P$ 作一条倾斜角为 $45^{\circ}$ 的直线 $l$, 与 $C_{2}$ 交于两个不同的点 $Q, R$, 求 $|P Q| \cdot|P R|$的取值范围。请给出所有区间上下限值之和。
Answer: 220
Problem: 设 $f(x)$ 是定义在 $\mathbf{R}$ 上的函数, 若 $f(x)+x^{2}$ 是奇函数, $f(x)+2^{x}$ 是偶函数,则 $f(1)$ 的值为 \qquad. 原始答案为 $-\frac{m}{n}$ 的格式，请给出 $|m + n|$ 的值。
Answer: 11
Problem: 设不等式 $\left|2^{x}-a\right|<\left|5-2^{x}\right|$ 对所有 $x \in[1,2]$ 成立, 求实数 $a$ 的取值范围。请给出区间端点的和。
Answer: 8
Problem: 设 $a$ 为非零实数，在平面直角坐标系 $x O y$ 中，二次曲线 $x^{2}+a y^{2}+a^{2}=0$ 的焦距为 4 ，则 $a$ 的值为 $\qquad$。已知原始答案为 $\frac{m-\sqrt{n}}{k}$ 的格式，请给出 m + n + k 的值。
Answer: 20
Problem: 设复数 $z$ 满足 $|z|=1$, 使得关于 $x$ 的方程 $z x^{2}+2 \bar{z} x+2=0$ 有实根, 则这样的复数 $z$ 的和为 $-\frac{m}{n}$ 的形式，请给出 $m + n$ 的值。
Answer: 5
Problem: 在平面直角坐标系 $x O y$ 中, 设 $A B$ 是抛物线 $y^{2}=4 x$ 的过点 $F(1,0)$ 的弦, $\triangle A O B$ 的外接圆交抛物线于点 $P$ (不同于点 $O, A, B$ ). 若 $P F$ 平分 $\angle A P B$, 求 $|P F|$ 的所有可能值。原始答案为$\sqrt{m}-n$格式，请给出$m+n$的值。
Answer: 14
Problem: 设 $O$ 为 $\triangle A B C$ 的外心, 若 $\overrightarrow{A O}=\overrightarrow{A B}+2 \overrightarrow{A C}$, 则 $\sin \angle B A C$ 的值为多少。原始答案为 $\frac{\sqrt{m}}{n}$的形式，请给出 $m + n$ 的值。
Answer: 14
Problem: 设 $f(x)$ 是定义在 R 上的以 2 为周期的偶函数, 在区间 $[0,1]$ 上严格递减, 且满足 $f(\pi)=1, f(2 \pi)=2$, 不等式组 $\left\{\begin{array}{l}1 \leqslant x \leqslant 2 \\ 1 \leqslant f(x) \leqslant 2\end{array}\right.$ 的解集为$[\pi-a,b-c \pi]$的形式，求$a+b+c$的值。
Answer: 12
Problem: 已知定义在 $\mathrm{R}^{*}$ 上的函数 $f(x)=\left\{\begin{array}{l}\left|\log _{3} x-1\right|, 0<x \leqslant 9 \\ 4-\sqrt{x}, x>9\end{array}\right.$ 设 $a, b, c$ 是三个互不相同的实数，满足 $f(a)=f(b)=f(c)$ ，求 $a b c$ 的取值范围。请提供区间端点的和。
Answer: 225
Problem: 设集合 $A=\{1,2, \cdots, n\}, X, Y$ 均为 $A$ 的非空子集（允许 $X=Y$ ）。 $X$ 中的最大元与 $Y$ 中的最小元分别记为 $\max X, \min Y$ 。求满足 $\max X>\min Y$ 的有序集合对 $(X, Y)$ 的数目。请计算当 $n=8$ 时的结果。
Answer: 63232
Problem: 设 $f(x)$ 是定义在 R 上的以 2 为周期的偶函数，在区间 $[1,2]$ 上严格递减，且满足 $f(\pi)=1, f(2 \pi)=0$, 则不等式组 $\left\{\begin{array}{l}0 \leqslant x \leqslant 1 \\ 0 \leqslant f(x) \leqslant 1\end{array}\right.$ 的解集为为$[a\pi-b,c- \pi]$的形式，求$a+b+c$的值。
Answer: 12
Problem: 已知圆锥的顶点为 $P$, 底面半径长为 $2$, 高为 $1$. 在圆锥底面上取一点 $Q$, 使得直线 $P Q$与底面所成角不大于 $45^{\circ}$ ，满足条件的点 $Q$ 所构成的区域的面积为 $m\pi$ 的形式，请给出 $m$ 的值。
Answer: 3
Problem: 已知复数 $z_{1}, z_{2}, z_{3}$ 满足 $\left|z_{1}\right|=\left|z_{2}\right|=\left|z_{3}\right|=1,\left|z_{1}+z_{2}+z_{3}\right|=r$, 其中 $r$ 是给定实数,则 $\frac{z_{1}}{z_{2}}+\frac{z_{2}}{z_{3}}+\frac{z_{3}}{z_{1}}$ 的实数是$\frac{r^{2}-a}{b}$的形式，求$a+b$的值。
Answer: 5
Problem: 在 $\triangle A B C$ 中, $B C=a, C A=b, A B=c$. 若 $b$ 是 $a$ 与 $c$ 的等比中项, 且 $\sin A$ 是 $\sin (B-A)$ 与 $\sin C$ 的等差中项, 求 $\cos B$ 的值。原始答案的形式为$\frac{\sqrt{m} - n}{k}$，请给出 $m + n + k$ 的值。
Answer: 8
Problem: 在平面直角坐标系 $x O y$ 中, 圆 $\Omega$ 与抛物线 $\Gamma: y^{2}=4 x$ 恰有一个公共点, 且圆 $\Omega$ 与 $x$轴相切于 $\Gamma$ 的焦点 $F$ 。求圆 $\Gamma$ 的半径。原始答案为 $\frac{m \sqrt{3}}{n}$ 的格式，请给出 $m + n$ 的值。
Answer: 13
Problem: 平面直角坐标系中, $\vec{e}$ 是单位向量, 向量 $\vec{a}$ 满足 $\vec{a} \cdot \vec{e}=2$, 且 $|\vec{a}|^{2} \leqslant 5|\vec{a}+t \vec{e}|$对任意实数 $t$ 成立，求 $|\vec{a}|$ 的取值范围. 请给出该区间端点之和的整数部分。
Answer: 6
Problem: 设三棱雉 $P-A B C$ 满足 $P A=P B=3, A B=B C=C A=2$, 则该三棱雉的体积的最大值是多少。原始答案的格式为 $\frac{\sqrt{m} + n}{k}$ ，请给出 $m + n + k$ 的值。
Answer: 11
Problem: 称一个复数数列 $\left\{z_{n}\right\}$ 为"有趣的"，若 $\left|z_{1}\right|=1$ ，且对任意正整数 $n$ ，均有 $4 z_{n+1}^{2}+2 z_{n} z_{n+1}+z_{n}^{2}=0$ 。存在最大的常数 $C$ ，使得对一切有趣的数列 $\left\{z_{n}\right\}$ 及正整数 $m$ ，均有 $\left|z_{1}+z_{2}+\cdots+z_{m}\right| \geqslant C$，若$C=\frac{\sqrt{a}}{b}$，则$a+b$的值是多少。
Answer: 6
Problem: 对任意闭区间 $I$ ，用 $M_{I}$ 表示函数 $y=\sin x$ 在 $I$ 上的最大值。若正数 $a$ 满足 $M_{[0, a]}=2 M_{[a, 2 a]}$ ，则 $a$ 的值为 $\frac{m \pi}{n}$ 或 $\frac{p \pi}{q}$ 的形式，请给出 $m + n + p + q$ 的值。
Answer: 36
Problem: 在六条棱长分别为 $2,3,3,4,5,5$ 的所有四面体中, 最大的体积是多少，原始答案为$\frac{m \sqrt{2}}{n}$，请给出$m + n$的值。
Answer: 11
Problem: 给定整数 $n \geqslant 3$, 设 $a_{1}, a_{2}, \cdots, a_{2 n}, b_{1}, b_{2}, \cdots, b_{2 n}$ 是 $4 n$ 个非负实数, 满足$$a_{1}+a_{2}+\cdots+a_{2 n}=b_{1}+b_{2}+\cdots+b_{2 n}>0$$且对任意 $i=1,2, \cdots, 2 n$, 有 $a_{i} a_{i+2} \geqslant b_{i}+b_{i+1}$ （这里 $a_{2 n+1}=a_{1}, a_{2 n+2}=a_{2}$, $b_{2 n+1}=b_{1}$ ), 求 $a_{1}+a_{2}+\cdots+a_{2 n}$ 的最小值。请提供当 $n=3$ 和 $n\geqslant 4$ 时最小值的和。
Answer: 28
Problem: 在粗圆 $\Gamma$ 中, $A$ 为长轴的一个端点, $B$ 为短轴的一个端点, $F_{1}, F_{2}$ 为两个焦点, 若 $\overrightarrow{A F_{1}} \cdot \overrightarrow{A F_{2}}+\overrightarrow{B F_{1}} \cdot \overrightarrow{B F_{2}}=0$, 则 $\frac{|A B|}{\left|F_{1} F_{2}\right|}$ 的值为多少，原始答案的格式为 $\frac{\sqrt{m}}{n}$ ，请给出 $m + n$ 的值。
Answer: 4
Problem: Let the *taxicab distance* between points $(x_1, y_1)$ and $(x_2, y_2)$ in the coordinate plane be given by $|x_1 - x_2| + |y_1 - y_2|$. For how many points $P$ with integer coordinates is the taxicab distance between $P$ and the origin less than or equal to $20$?
Answer: 841
Problem: Let \(ABC\) be an equilateral triangle. \(D\) and \(E\) are midpoints of \([AB]\) and \([AC]\). The ray \([DE\) cuts the circumcircle of \(\triangle ABC\) at \(F\). The original answer is in the form \(\frac{\sqrt{k} - m}{n}\). Please find the value of \(k + m + n\).
Answer: 8
Problem: The equiangular convex hexagon $ABCDEF$ has $AB = 1$, $BC = 4$, $CD = 2$, and $DE = 4$. Find the area of the hexagon. The original answer is in the form \frac{m}{n}\sqrt{k}, where \frac{m}{n} is a simplified fraction. Please provide the value of m + n + k.
Answer: 50
Problem: Given \( 1 \le n \le 455 \) and \( n^3 \equiv 1 \pmod{455} \), find the number of solutions.
Answer: 9
Problem: The numbers from 1 to 1996 are written sequentially as follows: 12345678910111213... up to 1996. Find how many zeros appear in this sequence.
Answer: 489
Problem: For how many positive integers $n$ is $n^3 - 8n^2 + 20n - 13$ a prime number? Provide your answer as an integer.
Answer: 3
Problem: A big $L$ is formed as shown. Find its area.

\[ \text{[asy]} \text{unitsize}(4\text{mm}); \\
\text{defaultpen}(\text{linewidth}(0.8\text{pt})+\text{fontsize}(12\text{pt})); \\
\text{draw}((0,0)--(5,0)--(5,2)--(2,2)--(2,8)--(0,8)--\text{cycle}); \\
\text{label}("5",(2.5,0),S); \\
\text{label}("2",(5,1),E); \\
\text{label}("2",(1,8),N); \\
\text{label}("8",(0,4),W); \]
Answer: 22
Problem: For all positive integers $n$ less than $2002$, let
\[
a_n = \begin{cases} 
11 & \text{if } n \text{ is divisible by } 13 \text{ and } 14 \\
13 & \text{if } n \text{ is divisible by } 11 \text{ and } 14 \\
14 & \text{if } n \text{ is divisible by } 11 \text{ and } 13 \\
0 & \text{otherwise} \end{cases}
\]
Calculate $\sum_{n = 1}^{2001} a_n$.
Answer: 448
Problem: Given distinct straight lines $OA$ and $OB$. From a point on $OA$, a perpendicular is drawn to $OB$; from the foot of this perpendicular, a line is drawn perpendicular to $OA$. From the foot of this second perpendicular, a line is drawn perpendicular to $OB$; and so on indefinitely. The lengths of the first and second perpendiculars are $a$ and $b$, respectively. Find the limit of the sum of the lengths of the perpendiculars as the number of perpendiculars grows beyond all bounds. The original answer is in the form \frac{k \cdot a^2}{k \cdot a - m \cdot b}, please give the value of k + m.
Answer: 2
Problem: The sum of the base-\( 10 \) logarithms of the divisors of \( 10^n \) is \( 792 \). What is the value of \( n \)?
Answer: 11
Problem: Chloé chooses a real number uniformly at random from the interval \([0, 2017]\). Independently, Laurent chooses a real number uniformly at random from the interval \([0, 4034]\). Find the probability that Laurent's number is greater than Chloé's number. The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
Answer: 7
Problem: Triangle $ABC$ with $AB=50$ and $AC=10$ has area $120$. Let $D$ be the midpoint of $\overline{AB}$, and let $E$ be the midpoint of $\overline{AC}$. The angle bisector of $\angle BAC$ intersects $\overline{DE}$ and $\overline{BC}$ at $F$ and $G$, respectively. What is the area of quadrilateral $FDBG$?
Answer: 75
Problem: Each of 2010 boxes in a line contains a single red marble, and for $1 \le k \le 2010$, the box in the $k^{th}$ position also contains $k$ white marbles. Isabella begins at the first box and successively draws a single marble at random from each box, in order. She stops when she first draws a red marble. Let $P(n)$ be the probability that Isabella stops after drawing exactly $n$ marbles. Find the smallest value of $n$ for which $P(n) < \frac{1}{2010}$.
Answer: 45
Problem: Jane and Jena sit at non-adjacent chairs of a four-chair circular table. In a turn, one person can move to an adjacent chair without a person. Jane moves in the first turn, and alternates with Jena afterwards. After nine moves, in how many ways can Jena be adjacent to Jane?
Answer: 32
Problem: A rectangle with perimeter $176$ is divided into five congruent rectangles as shown in the diagram. What is the perimeter of one of the five congruent rectangles?
Answer: 80
Problem: If $\sin 2x \sin 3x = \cos 2x \cos 3x$, then one value for $x$ is $x^\circ$. What is the value of $x$?
Answer: 18
Problem: What is the sum of the last two digits of $7^{42} + 7^{43}$ in base $10$?
Answer: 11
Problem: Let $N = 123456789101112\dots4344$ be the $79$-digit number obtained by writing the integers from $1$ to $44$ in order, one after the other. Find the remainder when $N$ is divided by $45$.
Answer: 9
Problem: A baseball is dropped on top of a basketball. The basketball hits the ground, rebounds with a speed of $4.0 \, \text{m/s}$, and collides with the baseball as it is moving downward at $4.0 \, \text{m/s}$. After the collision, the baseball moves upward as shown in the figure, and the basketball is instantaneously at rest right after the collision. The mass of the baseball is $0.2 \, \text{kg}$ and the mass of the basketball is $0.5 \, \text{kg}$. Ignore air resistance and any changes in velocities due to gravity during the very short collision times. If the speed of the baseball right after colliding with the upward-moving basketball is $x \, \text{m/s}$, what is the value of $x$?
Answer: 6
Problem: How many distinct four-tuples $(a, b, c, d)$ of rational numbers are there such that:

$$a \log_{10} 2 + b \log_{10} 3 + c \log_{10} 5 + d \log_{10} 7 = 2005$$

Find the number of such distinct four-tuples.
Answer: 1
Problem: A circle is inscribed in a triangle with side lengths $8$, $13$, and $17$. Let the segments of the side of length $8$, made by a point of tangency, be $r$ and $s$, with $r < s$. What is the ratio $r:s$? Express your answer in the form of $x:y$ where $x$ and $y$ are integers. Please give the value of x + y.
Answer: 4
Problem: What is the sum of all real numbers $x$ for which the median of the numbers $4, 6, 8, 17,$ and $x$ is equal to the mean of those five numbers? Provide your answer as an integer.
Answer: -5
Problem: In triangle $ABC$, point $F$ divides side $AC$ in the ratio $1:2$. Let $E$ be the point of intersection of side $BC$ and $AG$, where $G$ is the midpoint of $BF$. Find the ratio in which point $E$ divides side $BC$. The answer is in the form a:b, please give the value of a + b.
Answer: 4
Problem: Consider the set of complex numbers $z$ satisfying $|1+z+z^{2}|=4$. The maximum value of the imaginary part of $z$ can be written in the form $\frac{\sqrt{m}}{n}$, where $m$ and $n$ are relatively prime positive integers. Find the value of $m+n$.
Answer: 21
Problem: In triangle $\triangle ABC$ with $|BC| > |BA|$, $D$ is a point inside the triangle such that $\angle ABD = \angle DBC$, $\angle BDC = 150^\circ$, and $\angle DAC = 60^\circ$. If the measure of $\angle BAD$ is $x^\circ$, what is the value of $x$?
Answer: 60
Problem: Quadrilateral $ABCD$ is inscribed in a circle with the following properties:
- $\angle BAC = 70^{\circ}$
- $\angle ADB = 40^{\circ}$
- $AD = 4$
- $BC = 6$

What is the length of $AC$?
Answer: 6
Problem: Find the number of real numbers which satisfy the equation $x|x-1|-4|x|+3=0$.
Answer: 3
Problem: Triangle $ABC$ is inscribed in a circle. The measure of the non-overlapping minor arcs $AB$, $BC$, and $CA$ are, respectively, $x+75^\circ$, $2x+25^\circ$, $3x-22^\circ$. Find one interior angle of the triangle, in degrees, as an integer.
Answer: 61
Problem: $A, B, C$ are playing a backgammon tournament. Initially, $A$ plays against $B$. The winner of this game then plays against $C$. The tournament continues such that the last winner plays against the player who did not play in the previous game. The tournament concludes when a player wins two consecutive games. If each player has an equal chance of winning a game, what is the probability that $C$ wins the tournament? The original answer is in \(\frac{k}{m}\) format, where the fraction is in simplest form. Please provide the value of k + m.
Answer: 9
Problem: Jo and Blair take turns counting from $1$ to one more than the last number said by the other person. Jo starts by saying "1", so Blair follows by saying "1, 2". Jo then says "1, 2, 3", and so on. What is the 53rd number said?
Answer: 8
Problem: A unit equilateral triangle is given. Divide each side into three equal parts. Remove the equilateral triangles whose bases are the middle one-third segments. Now we have a new polygon. Remove the equilateral triangles whose bases are the middle one-third segments of the sides of the polygon. After repeating these steps infinitely, find the area of the new shape. The original answer is in \(\dfrac{\sqrt{k}}{m}\) format, please give the value of k + m.
Answer: 13
Problem: Three equally spaced parallel lines intersect a circle, creating three chords of lengths $38$, $38$, and $34$. What is the distance between two adjacent parallel lines?
Answer: 6
Problem: Five times $A$'s money added to $B$'s money is more than $51.00$. Three times $A$'s money minus $B$'s money is $21.00$. If $a$ represents $A$'s money in dollars and $b$ represents $B$'s money in dollars, find the smallest integer value of $a$ and $b$ that satisfy these conditions. The answer should be in the form $a > m, b > n$. Please give the value of m + n.
Answer: 15
Problem: Phillip and Paula both pick a rational number, and they notice that Phillip's number is greater than Paula's number by $12$. They each square their numbers to get a new number, and see that the sum of these new numbers is half of $169$. Finally, they each square their new numbers and note that Phillip's latest number is now greater than Paula's by $5070$. What was the sum of their original numbers?
Answer: 5
Problem: If 1 pint of paint is needed to paint a statue 6 ft. high, find the number of pints it will take to paint (to the same thickness) 540 statues similar to the original but only 1 ft. high.
Answer: 15
Problem: What is the value of 
\[ \log_3 7 \cdot \log_5 9 \cdot \log_7 11 \cdot \log_9 13 \cdots \log_{21} 25 \cdot \log_{23} 27 \]

Find the value of the expression.
Answer: 6
Problem: A regular octahedron is formed by joining the centers of adjoining faces of a cube. What is the ratio of the volume of the octahedron to the volume of the cube? The original answer is in the form \(\frac{k}{m}\), where \(\frac{k}{m}\) is a simplified fraction. Please provide the value of k + m.
Answer: 7
Problem: What is the maximum possible value of $5 - |6x - 80|$ over all integers $x$?
Answer: 3
Problem: If both \(x\) and \(y\) are integers, find the number of pairs of solutions for the equation \((x-8)(x-10) = 2^y\).
Answer: 2
Problem: Find the largest integer that divides each of the numbers in the sequence \(1^5 - 1, 2^5 - 2, 3^5 - 3, \cdots, n^5 - n, \cdots\).
Answer: 30
Problem: A box contains $3$ shiny pennies and $4$ dull pennies. One by one, pennies are drawn at random from the box and not replaced. If the probability is $\frac{a}{b}$ that it will take more than four draws until the third shiny penny appears and $\frac{a}{b}$ is in lowest terms, find the value of $a+b$.
Answer: 66
Problem: A unit cube has vertices \(P_1, P_2, P_3, P_4, P_1', P_2', P_3'\), and \(P_4'\). Vertices \(P_2, P_3\), and \(P_4\) are adjacent to \(P_1\), and for \(1\leq i\leq 4\), vertices \(P_i\) and \(P_i'\) are opposite to each other. A regular octahedron has one vertex in each of the segments \(P_1P_2, P_1P_3, P_1P_4, P_1'P_2', P_1'P_3'\), and \(P_1'P_4'\). Find the octahedron's side length. The original answer is in the form \(\frac{a\sqrt{b}}{c}\), please give the value of \(a + b + c\).
Answer: 9
Problem: In a narrow alley of width $w$, a ladder of length $a$ is placed with its foot at point $P$ between the walls. Resting against one wall at $Q$, the distance $k$ above the ground makes a $45^\circ$ angle with the ground. Resting against the other wall at $R$, a distance $h$ above the ground, the ladder makes a $75^\circ$ angle with the ground. The width $w$ can be expressed as w = x*h. Find the value of x.
Answer: 1
Problem: Each integer $1$ through $9$ is written on a separate slip of paper and all nine slips are put into a hat. Jack picks one of these slips at random and puts it back. Then Jill picks a slip at random. What digit is most likely to be the units digit of the **sum** of Jack's integer and Jill's integer?
Answer: 0
Problem: A drawer in a darkened room contains $100$ red socks, $80$ green socks, $60$ blue socks, and $40$ black socks. A youngster selects socks one at a time from the drawer but is unable to see the color of the socks drawn. What is the smallest number of socks that must be selected to guarantee that the selection contains at least $10$ pairs? Provide your answer as an integer.
Answer: 23
Problem: The roots of the equation $x^2 + 5x - 7 = 0$ are $x_1$ and $x_2$. Find the value of $x_1^3 + 5x_1^2 - 4x_1 + x_1^2 x_2 - 4x_2$.
Answer: 20
Problem: A father takes his twins and a younger child out to dinner on the twins' birthday. The restaurant charges $4.95$ for the father and $0.45$ for each year of a child's age, where age is defined as the age at the most recent birthday. If the total bill is $9.45$, find the age of the youngest child.
Answer: 2
Problem: Let $f$ be a function defined on positive integers such that $f(1)=4$, $f(2n)=f(n)$ and $f(2n+1)=f(n)+2$ for every positive integer $n$. Find the number of positive integers $k$ less than $2014$ for which $f(k)=8$.
Answer: 165
Problem: Let $ABCD$ be a trapezoid such that $AB \parallel CD$, $|AB|<|CD|$, and $\text{Area}(\triangle ABC)=30$. Let the line through $B$ parallel to $AD$ meet $[AC]$ at $E$. If $|AE|:|EC|=3:2$, then what is the area of trapezoid $ABCD$?
Answer: 80
Problem: The altitudes of a triangle are $12$, $15$, and $20$. If the largest angle in this triangle is $x^\circ$, determine the value of $x$.
Answer: 90
Problem: How many three-digit positive integers $x$ are there with the property that $x$ and $2x$ have only even digits? (One such number is $x = 220$, since $2x = 440$ and each of $x$ and $2x$ has only even digits.)
Answer: 18
Problem: A ladder is formed by removing some consecutive unit squares of a $10 \times 10$ chessboard such that for each $k$-th row ($k \in \{1,2,\dots, 10\}$), the leftmost $k-1$ unit squares are removed. Find the number of rectangles formed by the composition of unit squares that the ladder has.
Answer: 715
Problem: Before Ashley started a three-hour drive, her car’s odometer reading was $29792$, a palindrome. At her destination, the odometer reading was another palindrome. If Ashley never exceeded the speed limit of $75$ miles per hour, what was her greatest possible average speed? The original answer is in the form of \frac{m}{n}, please provide the value of m + n.
Answer: 214
Problem: Let $s_k$ denote the sum of the $k$th powers of the roots of the polynomial $x^3 - 5x^2 + 8x - 13$. In particular, $s_0 = 3$, $s_1 = 5$, and $s_2 = 9$. Let $a$, $b$, and $c$ be real numbers such that $s_{k+1} = a \cdot s_k + b \cdot s_{k-1} + c \cdot s_{k-2}$ for $k = 2$, $3$, $\ldots$ Find the value of $a + b + c$.
Answer: 10
Problem: Let $a_1, a_2, \dots, a_n$ be positive integers such that none of them is a multiple of $5$. What is the largest integer $n < 2005$, such that $a_1^4 + a_2^4 + \cdots + a_n^4$ is divisible by $5$? Find the value of $n$.
Answer: 2000
Problem: Doug and Dave shared a pizza with $8$ equally-sized slices. Doug wanted a plain pizza, but Dave wanted anchovies on half the pizza. The cost of a plain pizza was $8$, and there was an additional cost of $2$ for putting anchovies on one half. Dave ate all the slices of anchovy pizza and one plain slice. Doug ate the remainder. Each paid for what he had eaten. How many more dollars did Dave pay than Doug?
Answer: 4