ChenDRAG/VeRL_math_validation huggingface dataset:
Problem: A number is chosen at random from among the first 100 positive integers, and a positive integer divisor of that number is then chosen at random. What is the probability that the chosen divisor is divisible by 11? Express your answer as a fraction in simplest form, \(\frac{m}{n}\). Please provide the value of m + n.
Answer: 209
Problem: How many ordered pairs of integers $(m, n)$ satisfy the equation $m^2+mn+n^2 = m^2n^2$?
Answer: 3
Problem: How many complex numbers satisfy the equation $z^5=\overline{z}$, where $\overline{z}$ is the conjugate of the complex number $z$? Fill in the number of such complex numbers.
Answer: 7
Problem: Nicolas is planning to send a package to his friend Anton, who is a stamp collector. To pay for the postage, Nicolas would like to cover the package with a large number of stamps. Suppose he has a collection of $5$-cent, $10$-cent, and $25$-cent stamps, with exactly $20$ of each type. What is the greatest number of stamps Nicolas can use to make exactly $7.10$ in postage?  
(Note: The amount $7.10$ corresponds to $7$ dollars and $10$ cents. One dollar is worth $100$ cents.)  
Fill in the blank with the greatest number of stamps: _______
Answer: 55
Problem: An even number of circles are nested, starting with a radius of $1$ and increasing by $1$ each time, all sharing a common point. The region between every other circle is shaded, starting with the region inside the circle of radius $2$ but outside the circle of radius $1.$ An example showing $8$ circles is displayed below. What is the least number of circles needed to make the total shaded area at least $2023\pi$?
Answer: 64
Problem: What is the least positive integer $m$ such that $m\cdot2!\cdot3!\cdot4!\cdot5!\cdots16!$ is a perfect square? Find the value of $m$.
Answer: 70
Problem: Usain is walking for exercise by zigzagging across a \(100\)-meter by \(30\)-meter rectangular field, beginning at point \(A\) and ending on the segment \(\overline{BC}\). He wants to increase the distance walked by zigzagging as shown in the figure below \((APQRS)\). What angle \(\theta = \angle PAB=\angle QPC=\angle RQB=\cdots\) will produce a path length that is \(120\) meters? (This figure is not drawn to scale. Do not assume that the zigzag path has exactly four segments as shown; there could be more or fewer.) The original answer is in the form \(\arccos\frac{k}{m}\), please give the value of \(k+m\).
Answer: 11
Problem: Viswam walks half a mile to get to school each day. His route consists of $10$ city blocks of equal length and he takes $1$ minute to walk each block. Today, after walking $5$ blocks, Viswam discovers he has to make a detour, walking $3$ blocks of equal length instead of $1$ block to reach the next corner. From the time he starts his detour, at what speed, in mph, must he walk, in order to get to school at his usual time? If the answer is \(x\) mph, what is the value of \(10x\)?
Answer: 42
Problem: Define an $upno$ to be a positive integer of $2$ or more digits where the digits are strictly increasing moving left to right. Similarly, define a $downno$ to be a positive integer of $2$ or more digits where the digits are strictly decreasing moving left to right. For instance, the number $258$ is an $upno$ and $8620$ is a $downno$. Let $U$ equal the total number of $upnos$ and let $D$ equal the total number of $downnos$. Find the value of $|U-D|$.
Answer: 511
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 $\tfrac{\sqrt{m}}{n}$, where $m$ and $n$ are relatively prime positive integers. Find the value of $m+n$.
Answer: 21
Problem: The letters $\text{P}, \text{Q},$ and $\text{R}$ are entered into a $20\times20$ table according to the pattern shown below. Determine how many times each of the letters $\text{P}$, $\text{Q}$, and $\text{R}$ will appear in the completed table. Provide your answer in the format: number of Ps, number of Qs, number of Rs. Then, calculate the sum of these three numbers.
Answer: 400
Problem: Let $ABCD$ be a rectangle with $AB = 30$ and $BC = 28$. Point $P$ and $Q$ lie on $\overline{BC}$ and $\overline{CD}$ respectively so that all sides of $\triangle{ABP}, \triangle{PCQ},$ and $\triangle{QDA}$ have integer lengths. Find the perimeter of $\triangle{APQ}$.
Answer: 84
Problem: Flora the frog starts at \(0\) on the number line and makes a sequence of jumps to the right. In any one jump, independent of previous jumps, Flora leaps a positive integer distance \(m\) with probability \(\frac{1}{2^m}\).

Find the probability that Flora will eventually land at \(10\). Express your answer as a fraction in simplest form \(\frac{k}{m}\), and then provide the value of \(k + m\).
Answer: 3
Problem: Triangle $ABC$ has side lengths in arithmetic progression, and the smallest side has length $6.$ If the triangle has an angle of $120^\circ,$ what is the area of $ABC$? Provide your answer in the form of $a\sqrt{b}$, where $a$ and $b$ are integers. Please find the value of a + b.
Answer: 18
Problem: Circle \(C_1\) and \(C_2\) each have radius \(1\), and the distance between their centers is \(\frac{1}{2}\). Circle \(C_3\) is the largest circle internally tangent to both \(C_1\) and \(C_2\). Circle \(C_4\) is internally tangent to both \(C_1\) and \(C_2\) and externally tangent to \(C_3\). The radius of \(C_4\) is in the form \(\frac{k}{m}\). Please find the value of \(k + m\).
Answer: 31
Problem: Greta Grasshopper sits on a long line of lily pads in a pond. From any lily pad, Greta can jump $5$ pads to the right or $3$ pads to the left. What is the fewest number of jumps Greta must make to reach the lily pad located $2023$ pads to the right of her starting position?
Answer: 411
Problem: The line segment formed by $A(1, 2)$ and $B(3, 3)$ is rotated to the line segment formed by $A'(3, 1)$ and $B'(4, 3)$ about the point $P(r, s)$. Find the value of $|r-s|$.
Answer: 1
Problem: Sonya the frog chooses a point uniformly at random lying within the square \([0, 6] \times [0, 6]\) in the coordinate plane and hops to that point. She then randomly chooses a distance uniformly at random from \([0, 1]\) and a direction uniformly at random from \{north, south, east, west\}. All of her choices are independent. Find the probability that she lands outside the square. The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
Answer: 13
Problem: What is the product of all solutions to the equation  
$$\log_{7x}2023\cdot \log_{289x}2023=\log_{2023x}2023?$$
Answer: 1
Problem: An equilateral triangle is placed inside a larger equilateral triangle so that the region between them can be divided into three congruent trapezoids, as shown below. The side length of the inner triangle is \(\frac{2}{3}\) the side length of the larger triangle. What is the ratio of the area of one trapezoid to the area of the inner triangle? The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
Answer: 17
Problem: Four congruent semicircles are drawn on the surface of a sphere with radius $2$, as shown, creating a close curve that divides the surface into two congruent regions. The length of the curve is $\pi\sqrt{n}$. What is the value of $n$?
Answer: 32
Problem: Rows $1, 2, 3, 4,$ and $5$ of a triangular array of integers are shown below.  
$$\begin{array}{cccccccc}&&&&1&&&&\\&&&1&&1&&&\\&&1&&3&&1&&\\&1&&5&&5&&1&\\1&&7&&11&&7&&1\end{array}$$  
Each row after the first row is formed by placing a $1$ at each end of the row, and each interior entry is $1$ greater than the sum of the two numbers diagonally above it in the previous row. Find the units digit of the sum of the $2023$ numbers in the $2023$rd row.
Answer: 5
Problem: Two integers are inserted into the list $3, 3, 8, 11, 28$ to double its range. The mode and median remain unchanged. What is the maximum possible sum of the two additional numbers?
Answer: 60
Problem: Let $P(x)$ be the unique polynomial of minimal degree with the following properties:  
$P(x)$ has a leading coefficient $1$,  
$1$ is a root of $P(x)-1$,  
$2$ is a root of $P(x-2)$,  
$3$ is a root of $P(3x)$, and  
$4$ is a root of $4P(x)$.  
The roots of $P(x)$ are integers, with one exception. The root that is not an integer can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime integers. Find $m+n$.
Answer: 47
Problem: 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{p}{q} format, please give the value of p + q.
Answer: 29
Problem: A lampshade is made in the form of the lateral surface of the frustum of a right circular cone. The height of the frustum is \(3\sqrt{3}\) inches, its top diameter is 6 inches, and its bottom diameter is 12 inches. A bug is at the bottom of the lampshade and there is a glob of honey on the top edge of the lampshade at the spot farthest from the bug. The bug wants to crawl to the honey, but it must stay on the surface of the lampshade. The original answer is in the form \(k\sqrt{m} + n\pi\). Please give the value of k + m + n.
Answer: 10
Problem: Alina writes the numbers $1, 2, \dots , 9$ on separate cards, one number per card. She wishes to divide the cards into $3$ groups of $3$ cards so that the sum of the numbers in each group will be the same. In how many ways can this be done?
Answer: 2
Problem: Circle \(C_1\) and \(C_2\) each have radius \(1\), and the distance between their centers is \(\frac{1}{2}\). Circle \(C_3\) is the largest circle internally tangent to both \(C_1\) and \(C_2\). Circle \(C_4\) is internally tangent to both \(C_1\) and \(C_2\) and externally tangent to \(C_3\). The radius of \(C_4\) is given in the form \(\frac{k}{m}\). Please find the value of \(k + m\).
Answer: 31
Problem: How many distinct values of $x$ satisfy 
$\lfloor{x}\rfloor^2-3x+2=0$, where $\lfloor{x}\rfloor$ denotes the largest integer less than or equal to $x$?
Answer: 4
Problem: Let $f$ be the unique function defined on the positive integers such that \[\sum_{d\mid n}d\cdot f\left(\frac{n}{d}\right)=1\] for all positive integers $n$. What is the value of $f(2023)$?
Answer: 96
Problem: In a sequence of positive integers, each term after the second is the product of the previous two terms. The sixth term is $4000$. What is the first term?
Answer: 5
Problem: If the positive integer $n$ has positive integer divisors $a$ and $b$ with $n = ab$, then $a$ and $b$ are said to be complementary divisors of $n$. Suppose that $N$ is a positive integer that has one complementary pair of divisors that differ by $20$ and another pair of complementary divisors that differ by $23$. Find the sum of the digits of $N$.
Answer: 15
Problem: An arithmetic sequence of positive integers has $n \ge 3$ terms, initial term $a$, and common difference $d > 1$. Carl wrote down all the terms in this sequence correctly except for one term, which was off by $1$. The sum of the terms he wrote was $222$. What is the value of $a + d + n$?
Answer: 20
Problem: 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
Problem: Suppose that $a$, $b$, $c$ and $d$ are positive integers satisfying all of the following relations.
$abcd=2^6\cdot 3^9\cdot 5^7$
${lcm}(a,b)=2^3\cdot 3^2\cdot 5^3$
${lcm}(a,c)=2^3\cdot 3^3\cdot 5^3$
${lcm}(a,d)=2^3\cdot 3^3\cdot 5^3$
${lcm}(b,c)=2^1\cdot 3^3\cdot 5^2$
${lcm}(b,d)=2^2\cdot 3^3\cdot 5^2$
${lcm}(c,d)=2^2\cdot 3^3\cdot 5^2$
Find the greatest common divisor ${gcd}(a,b,c,d)$ and provide the answer as an integer.
Answer: 3
Problem: A regular pentagon with area \(\sqrt{5}+1\) is printed on paper and cut out. The five vertices of the pentagon are folded into the center of the pentagon, creating a smaller pentagon. What is the area of the new pentagon? The original answer is in the form of \(n - \sqrt{m}\). Please find the value of m + n.
Answer: 6
Problem: A regular pentagon with area \(\sqrt{5}+1\) is printed on paper and cut out. The five vertices of the pentagon are folded into the center of the pentagon, creating a smaller pentagon. What is the area of the new pentagon? The original answer is in the form \(\sqrt{k} - m\). Please give the value of k + m.
Answer: 6
Problem: Fifteen integers $a_1, a_2, a_3, \dots, a_{15}$ are arranged in order on a number line. The integers are equally spaced and have the property that
\[1 \le a_1 \le 10, \thickspace 13 \le a_2 \le 20, \thickspace \text{ and } \thickspace 241 \le a_{15}\le 250.\]
Find the sum of the digits of $a_{14}$.
Answer: 8
Problem: Aaliyah rolls two standard $6$-sided dice. She notices that the product of the two numbers rolled is a multiple of $6$. Find the integer that cannot be the sum of the two numbers.
Answer: 6
Problem: On Monday, Taye has $2. Every day, he either gains $3 or doubles the amount of money he had on the previous day. How many different dollar amounts could Taye have on Thursday, 3 days later?
Answer: 6
Problem: Buzz Bunny is hopping up and down a set of stairs, one step at a time. In how many ways can Buzz start on the ground, make a sequence of $6$ hops, and end up back on the ground? (For example, one sequence of hops is up-up-down-down-up-down.)
Answer: 5
Problem: A chess king is said to attack all the squares one step away from it, horizontally, vertically, or diagonally. For instance, a king on the center square of a $3 \times 3$ grid attacks all $8$ other squares, as shown below. Suppose a white king and a black king are placed on different squares of a $3 \times 3$ grid so that they do not attack each other (in other words, not right next to each other). In how many ways can this be done?
Answer: 32
Problem: Jordan owns $15$ pairs of sneakers. Three fifths of the pairs are red and the rest are white. Two thirds of the pairs are high-top and the rest are low-top. What is the least possible value of the fraction of the collection that consists of red high-top sneakers? The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
Answer: 19
Problem: Any three vertices of the cube $PQRSTUVW$, shown in the figure below, can be connected to form a triangle. (For example, vertices $P$, $Q$, and $R$ can be connected to form isosceles $\triangle PQR$.) How many of these triangles are equilateral and contain $P$ as a vertex? Fill in the blank with the number of such triangles.
Answer: 3
Problem: A group of frogs (called an army) is living in a tree. A frog turns green when in the shade and turns yellow when in the sun. Initially, the ratio of green to yellow frogs was $3 : 1$. Then $3$ green frogs moved to the sunny side and $5$ yellow frogs moved to the shady side. Now the ratio is $4 : 1$. What is the difference between the number of green frogs and the number of yellow frogs now?
Answer: 24
Problem: Rodrigo has a very large sheet of graph paper. First he draws a line segment connecting point $(0,4)$ to point $(2,0)$ and colors the $4$ cells whose interiors intersect the segment, as shown below. Next Rodrigo draws a line segment connecting point $(2000,3000)$ to point $(5000,8000)$. How many cells will he color this time?
Answer: 7000
Problem: Jean has made a piece of stained glass art in the shape of two mountains, as shown in the figure below. One mountain peak is $8$ feet high while the other peak is $12$ feet high. Each peak forms a $90^\circ$ angle, and the straight sides form a $45^\circ$ angle with the ground. The artwork has an area of $183$ square feet. The sides of the mountain meet at an intersection point near the center of the artwork, $h$ feet above the ground. What is the value of $h$?
Answer: 5
Problem: 设 $A=[-2,4), B=\left\{x \mid x^{2}-a x-4 \leq 0\right\}$ ，若 $B \subseteq A$ ，则实数 $a$ 的取值范围为 [a, b)。请给出a+b的值。
Answer: 3
Problem: 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
Problem: 已知 \(M=\left\{(x, y) \mid y \geqslant x^{2}\right\}, N=\left\{(x, y) \mid x^{2}+(y-a)^{2} \leqslant 1\right\}\)。求使 \(M \cap N=N\) 成立的充要条件中 \(a\) 的最小值。原始答案的形式为\(\frac{k}{m}\)，请给出\(k + m\)的值。
Answer: 9
Problem: 已知两个实数集合 $A=\left\{a_{1}, a_{2}, a_{3}, \cdots, a_{100}\right\}$ ，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？请以 $C_{m}^{n}$ 的形式给出，并求出m+n的值。
Answer: 148
Problem: 若 \(\left(x^{2}-2 x+m\right)\left(x^{2}-2 x+n\right)=0\) 的 \(4\) 个根组成首项为 \(\frac{1}{4}\) 的等差数列，则 \(|m-n|\) 的值为多少？答案的形式为 \(\frac{k}{m}\)，请给出\(k+m\)的值。
Answer: 3
Problem: 数列 $\left\{a_{n}\right\}$ 满足 $a_{1}=1$, 前 $n$ 项和为 $S_{n}, S_{n+1}=4 a_{n}+2$, 求 $a_{2013}$ 的值。原始答案为$x \times y^{z}$的形式，其中$y$不能是2的倍数，请给出$x + y + z$的值。
Answer: 5033
Problem: 若 $x^{2}=2 y+5, y^{2}=2 x+5(x \neq y)$, 求 $x^{3}-2 x^{2} y^{2}+y^{3}$ 的值。
Answer: -16
Problem: 在数列 $\left\{a_{n}\right\}$ 中, $a_{1}=1$, 若前 $n$ 项和 $S_{n}$ 满足 $S_{n+1}=4 a_{n}+2$, 求 $a_{2013}$ 的值。答案的形式为 x \times y^{z}，请给出 x + y + z 的值。
Answer: 5033
Problem: $x 、 y$ 均为负数, 且 $x+y=-1$, 求 $x y+\frac{1}{x y}$ 的最小值。答案的形式为$\frac{k}{m}$，请给出k和m的和，其中k和m是最简分数形式下的整数。
Answer: 21
Problem: 已知 $x+y=-1$, 且 $x, y$ 均为负实数，求 $x y+\frac{1}{x y}$ 的最小值。答案的形式为 $\frac{k}{m}$，请给出 $k + m$ 的值。
Answer: 21
Problem: 已知 $n$ 为不超过 $2015$ 的正整数且 $1^{n}+2^{n}+3^{n}+4^{n}$ 的个位数为 $0$，求满足条件的正整数 $n$ 的个数。
Answer: 1512
Problem: 已知 $n$ 为不超过 2015 的正整数，且 $1^{n}+2^{n}+3^{n}+4^{n}$ 的个位数为 0 ，求满足条件的正整数 $n$ 的个数。
Answer: 1512
Problem: 圆内接四边形 $A B C D$ 中, $A B=136, B C=80, C D=150, D A=102$, 求它的外接圆直径。
Answer: 170
Problem: 三角形 $A B C$ 的三个顶点分别对应复数 $z_{1}, z_{2}, z_{3}$, 已知 $\frac{z_{2}-z_{1}}{z_{3}-z_{1}}=1+2 i$, 求三角形 $A B C$ 的面积与其最长边长的平方的比。答案形如\frac{k}{m}，请给出k+m的值。
Answer: 6
Problem: 空间中点集定义如下: $A_{n}=\left\{\left.(x, y, z) \in \mathbf{R}^{3}|3| x\right|^{n}+|8 y|^{n}+|z|^{n} \leq 1\right\}, A=\bigcup_{n=1}^{\infty} A_{n}$, 求由 $A$ 中的点组成的图形的体积。
Answer: 1
Problem: 已知 $x+y+z=2016, \frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{1}{2016}$, 求 $(x-2016)(y-2016)(z-2016)$ 的值。
Answer: 0
Problem: 正方体的 $8$ 个顶点中任取 $3$ 个构成三角形，求三角形是等腰三角形的概率。答案的形式为\frac{k}{m}，请给出k+m的值。
Answer: 11
Problem: 用 $a_{n}$ 表示距离 $\sqrt{n}$ 最近的正整数, 已知 $\frac{1}{a_{1}}+\frac{1}{a_{2}}+\cdots+\frac{1}{a_{n}}=2016$, 求 $n$ 的值。
Answer: 1017072
Problem: 四个半径为 1 的球两两相切，则它们的外切正四面体的棱长为多少？答案的形式为k(1+\sqrt{m})，请给出k+m的值。
Answer: 8
Problem: 实系数方程 $x^{4}+a x^{3}+b x^{2}+c x+d=0$ 的根都不是实数, 其中两个根的和为 $2+i$, 另两根的积为 $5+6 i$,则 $b$ 等于多少？
Answer: 15
Problem: 圆内接四边形 $A B C D$ 中, $A B=136, B C=80, C D=150, D A=102$, 求它的外接圆直径的值。
Answer: 170
Problem: 在正方体的八个顶点中任取 3 个构成三角形，求所构成的三角形是等腰三角形的概率。答案的形式为\(\frac{k}{m}\)，请给出k+m的值。
Answer: 11
Problem: 令 $a_{n}$ 表示距离 $\sqrt{n}$ 最近的正整数，其中 $n \in \mathbf{N}^{*}$ ，若 $\frac{1}{a_{1}}+\frac{1}{a_{2}}+\cdots+\frac{1}{a_{n}}= 2016$，求正整数 $n$ 的值。
Answer: 1017072
Problem: $\triangle A B C$ 的三个顶点分别对应复数 $z_{1}, z_{2}, z_{3}$, 已知 $\frac{z_{2}-z_{1}}{z_{3}-z_{1}}=1+2 \mathrm{i}$。求 $\triangle A B C$ 的面积与其最长边的长的平方的比。答案的形式为\(\frac{k}{m}\)，请给出k+m的值。
Answer: 6
Problem: 实系数方程 $x^{4}+a x^{3}+b x^{2}+c x+d=0$ 的根都不是实数，其中两个根的和为 $2+\mathrm{i}$, 另两个根的积为 $5+6 \mathrm{i}$。求 $b$ 的值。
Answer: 15
Problem: 四个半径为 1 的球两两相切, 则它们外切正四面体的棱长为多少？请以 $k(1+\sqrt{m})$ 的形式给出答案，并求出k和m的和。
Answer: 8
Problem: 设 $A$ 是以 $B C$ 为直径的圆上的一点, $D, E$ 是线段 $B C$ 上的点, $F$ 是 $B C$ 延长线上的点,已知 $B F=4, B D=2, B E=5, \angle B A D=\angle A C D, \angle B A F=\angle C A E$, 求 $B C$ 的长。
Answer: 11
Problem: $f(x)$ 是一个定义在实轴 $R$ 上的函数，满足 $2 f(x)+f\left(x^{2}-1\right)=1, \forall x \in R$ ，求 $f(-\sqrt{2})$ 的值。答案的形式为\frac{k}{m}，请给出k+m的值。
Answer: 4
Problem: 若方程 $x^{2}-3 x-1=0$ 的根也是方程 $x^{4}+a x^{2}+b x+c=0$ 的根, 则 $a+b-2 c$ 的值为多少？
Answer: -13
Problem: 已知 \(k \neq 1\)，则等比数列 \(a+\log _{2} k, a+\log _{4} k, a+\log _{8} k\) 的公比为多少？答案应为 \(\frac{m}{n}\) 的形式，请给出 \(m+n\) 的值。
Answer: 4
Problem: 方程 $\left(\frac{x^{3}+x}{3}\right)^{3}+\frac{x^{3}+x}{3}=3 x$ 的所有实根的平方和等于多少？
Answer: 4
Problem: 设 $A B, C D$ 是圆 $O$ 的两条垂直直径，弦 $D F$ 交 $A B$ 于点 $E, D E=24, E F=18$，求 $O E$ 的值。原始答案的形式为 $k\sqrt{m}$，请给出 $k + m$ 的值。
Answer: 8
Problem: 从一个正 9 边形的 9 个顶点中选 3 个使得它们是一个等腰三角形的三个顶点的方法数是多少？
Answer: 30
Problem: 在圆内接四边形 $A B C D$ 中, $B D=6, \angle A B D=\angle C B D=30^{\circ}$，求四边形 $ABCD$ 的面积。答案的形式为$n \sqrt{m}$，请给出$m + n$的值。
Answer: 12
Problem: 实数 $a, b$ 满足 $\left(a^{2}+4\right)\left(b^{2}+1\right)=5(2 a b-1)$, 求 $b\left(a+\frac{1}{a}\right)$ 的值乘以10后的结果。
Answer: 35
Problem: 已知正方形 \(A B C D\) 的边长为 \(1\)，\(P_{1}, P_{2}, P_{3}, P_{4}\) 是正方形内部的 \(4\) 个点，使得 \(\triangle A B P_{1}, \triangle B C P_{2}, \triangle C D P_{3}\) 和 \(\triangle D A P_{4}\) 都是正三角形。求四边形 \(P_{1} P_{2} P_{3} P_{4}\) 的面积。答案的形式为 \(a-\sqrt{b}\)，请给出a和b的和。
Answer: 5
Problem: 已知某个三角形的两条高的长度分别为 10 和 20 ，求它的第三条高的长度的取值区间。如果该区间为 \((\frac{c}{d}, e)\)，请给出 \(c + d + e\) 的值。
Answer: 43
Problem: 设整数 $a, m, n$ 满足 $\sqrt{a^{2}-4 \sqrt{5}}=\sqrt{m}-\sqrt{n}$, 求这样的整数组 $(a, m, n)$ 的个数。
Answer: 2
Problem: 设 $S=\frac{1}{\log _{\frac{1}{2}} \pi}+\frac{1}{\log _{\frac{1}{3}} \pi}+\frac{1}{\log _{\frac{1}{5}} \pi}+\frac{1}{\log _{\frac{1}{7}} \pi}$, 求不超过 $S$ 且与 $S$ 最接近的整数。
Answer: -5
Problem: 函数 $f(x)=x(x+1)(x+2)(x+3)$ 的最小值为多少？
Answer: -1
Problem: 整数 $p, q$ 满足 $p+q=218, x^{2}+p x+q=0$ 有整数根，求满足这样条件的整数对 $(p, q)$ 的个数。
Answer: 4
Problem: 已知正整数 $n$ 满足 $n \neq 2017$ ，且 $n^{n}$ 与 $2017^{2017}$ 有相同的个位数字，则 $|2017-n|$ 的最小值为多少？
Answer: 6
Problem: 一群学生参加学科夏令营，每名同学至少参加一个学科考试。已知有 100 名学生参加了数学考试，50 名学生参加了物理考试，48 名学生参加了化学考试。学生总数是参加至少两门考试学生的两倍，也是参加三门考试学生数的三倍。求学生总数。
Answer: 108
Problem: 单位圆的内接五边形的所有边及所有对角线的长度的平方和的最大值为多少？
Answer: 25
Problem: Find the value of $\left(1+\cos \frac{\pi}{7}\right)\left(1+\cos \frac{3 \pi}{7}\right)\left(1+\cos \frac{5 \pi}{7}\right)$. The original answer is in the form \frac{k}{m}, where k and m are integers. Please provide the value of k + m.
Answer: 15
Problem: 已知存在正整数 $a, b, c$ 满足 $a+b+c=407,10^{n} \mid a b c$ ，则 $n$ 的最大值是多少？
Answer: 6
Problem: 正整数 $9+95+995+\cdots+\underbrace{99 \cdots 9}_{2016} 5$ 的十进制表示中数值 $1$ 的个数为多少？
Answer: 2014
Problem: 一个三位数等于它的个位数字的阶乘之和，则此三位数的个位数字之和为多少？
Answer: 10
Problem: 设 $a, b, c$ 和 $\left(a-\frac{1}{b}\right)\left(b-\frac{1}{c}\right)\left(c-\frac{1}{a}\right)$ 均为正整数，求 $2a + 3b + 5c$ 的最大值和最小值之差。
Answer: 9
Problem: $O$ 是凸四边形 $A B C D$ 对角线 $A C$ 和 $B D$ 的交点。已知三角形 $A O B, B O C, C O D, D O A$ 的周长相同，且三角形 $A O B, B O C, C O D$ 的内切圆半径分别为 $3, 4, 6$。求三角形 $D O A$ 的内切圆半径。答案的形式为\(\frac{k}{m}\)，请给出k+m的值。
Answer: 11
Problem: 假设三角形三边长为连续的三个正整数，且该三角形的一个角是另一个角的两倍，求这个三角形的三边长。答案的形式为k, m, n，请给出k + m + n的值。
Answer: 15
Problem: 若 $a, b, c$ 和 $\left(a-\frac{1}{b}\right)\left(b-\frac{1}{c}\right)\left(c-\frac{1}{a}\right)$ 均为正整数，求 $2 a+3 b+5 c$ 的最大值和最小值之差。
Answer: 9
Problem: 一群学生参加学科夏令营，每名同学至少参加数学、物理、化学中的一门学科考试. 已知有 $100$ 名学生参加了数学考试，$50$名学生参加了物理考试， $48$ 名学生参加了化学考试.若学生总数是参加至少两门考试的学生的 $2$ 倍，也是参加三门考试学生的 $3$ 倍，则学生总数为多少？
Answer: 108
Problem: 已知正整数 $n$ 满足 $n \neq 2017$ ，且 $n^{n}$ 与 $2017^{2017}$ 有相同的个位数字，求 $|2017-n|$ 的最小值。
Answer: 6
Problem: 若存在正整数 $a, b, c$ 满足 $a+b+c=407,10^{n} \mid a b c$ ，求 $n$ 的最大值。
Answer: 6
Problem: 已知 $O$ 是凸四边形 $A B C D$ 对角线 $A C$ 和 $B D$ 的交点, $\triangle A O B, \triangle B O C$, $\triangle C O D, \triangle D O A$ 的周长相等. 若 $\triangle A O B, \triangle B O C, \triangle C O D$ 的内切圆半径分别为 3, 4, 6, 则 $\triangle D O A$ 的内切圆半径的答案为\(\frac{k}{m}\)的形式，请给出k+m的值。
Answer: 11
Problem: 满足 $p+q=218$ 且关于 $x$ 的方程 $x^{2}+p x+q=0$ 有整数根的有序整数对 $(p, q)$ 的对数为多少？
Answer: 4
Problem: 若实数 $x, y$ 满足 $\frac{x^{2}}{4}+y^{2}=1$ ，则 $|3 x+4 y-12|$ 的取值范围是 $[c, d]$。请分别求出 $c$ 和 $d$ 的值，并给出 $c + d$ 的值。
Answer: 24
Problem: 满足 \(\left(y^{2}+4 x^{2}-1\right)\left(x^{2}+4 y^{2}-4\right) \leqslant 0\) 的所有点 \((x, y)\) 组成的平面图形的面积是多少？请以\(\pi\)为单位给出答案。答案的形式为\(\frac{k}{m}\)，请给出k和m的和。
Answer: 5