ChenDRAG/VeRL_math_validation huggingface dataset:
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: Find the total number of solutions of the equation $xyz=2520$.
Answer: 540
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
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 \(\frac{k}{m}\) format, please give the value of k + m.
Answer: 29
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: 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
Problem: If \( t = \frac{1}{1 - \sqrt[4]{2}} \), find the expression that \( t \) equals to. The original answer is in the form of \(-(k + \sqrt[m]{n})(p + \sqrt{q})\). Please give the value of k + m + n + p + q.
Answer: 10
Problem: Let $\lfloor x \rfloor$ be the greatest integer less than or equal to $x$. Determine the number of real solutions to the equation:

$$4x^2 - 40 \lfloor x \rfloor + 51 = 0$$

Fill in the blank with the number of real solutions.
Answer: 4
Problem: Carl is exploring a cave and finds three types of rocks:
- 5-pound rocks worth $\$14 each,
- 4-pound rocks worth $\$11 each,
- 1-pound rocks worth $\$2 each.

There are at least 20 of each size available. Carl can carry at most 18 pounds. What is the maximum value, in dollars, of the rocks he can carry out of the cave?
Answer: 50
Problem: Andrea inscribed a circle inside a regular pentagon and circumscribed a circle around it, calculating the area of the region between the two circles. Bethany performed the same operations with a regular heptagon (7 sides). The areas of the two regions were $A$ and $B$, respectively. Each polygon had a side length of $2$. The relationship between $A$ and $B$ can be expressed in the form $A = kB$. Please find the value of $k$.
Answer: 1
Problem: In the year $2001$, the United States will host the International Mathematical Olympiad. Let $I$, $M$, and $O$ be distinct positive integers such that the product $I \cdot M \cdot O = 2001$. Find the largest possible value of the sum $I + M + O$.
Answer: 671
Problem: Two integers have a sum of $26$. When two more integers are added to the first two integers, the sum is $41$. Finally, when two more integers are added to the sum of the previous four integers, the sum is $57$. Find the minimum number of even integers among the $6$ integers.
Answer: 1
Problem: In parallelogram \(ABCD\) of the accompanying diagram, line \(DP\) is drawn bisecting \(BC\) at \(N\) and meeting \(AB\) (extended) at \(P\). From vertex \(C\), line \(CQ\) is drawn bisecting side \(AD\) at \(M\) and meeting \(AB\) (extended) at \(Q\). Lines \(DP\) and \(CQ\) meet at \(O\). If the area of parallelogram \(ABCD\) is \(k\), then find the area of the triangle \(QPO\) in terms of \(k\). The original answer is in \(\frac{m k}{n}\) format, please give the value of \(m + n\).
Answer: 17
Problem: Let $h_1$ and $h_2$ be the altitudes of a triangle drawn to the sides with lengths $5$ and $2\sqrt{6}$, respectively. If $5 + h_1 \leq 2\sqrt{6} + h_2$, find the length of the third side of the triangle.
Answer: 7
Problem: A circular grass plot with a diameter of 12 feet is intersected by a straight gravel path that is 3 feet wide. One edge of the path passes through the center of the plot. Calculate the number of square feet in the remaining grass area. The original answer is in the form $k\pi - m\sqrt{n}$, please provide the value of k + m + n.
Answer: 42
Problem: If $a$ and $b$ are digits for which:

\[
\begin{array}{ccc}
  & 2 & a \\
\times & b & 3 \\
\hline
  & 6 & 9 \\
9 & 2 \\
\hline
9 & 8 & 9
\end{array}
\]

Find the value of $a + b$.
Answer: 7
Problem: $(a_n)_{n = 0}^\infty$ is a sequence of integers. For every $n \ge 0$, $a_{n + 1} = a_n^3 + a_n^2$. Find the maximum number of distinct residues of $a_i$ in $\pmod{11}$.
Answer: 3
Problem: The stronger Goldbach conjecture states that any even integer greater than 7 can be written as the sum of two different prime numbers. For such representations of the even number 126, find the largest possible difference between the two primes.
Answer: 100
Problem: Polynomial $f(x)$ satisfies $(x - 1)f(x + 1) - (x + 2)f(x) = 0$ for every $x \in \mathbb{R}$. If $f(2) = 6$, find the value of $f\left(\frac{3}{2}\right)$. The original answer is in \frac{k}{m} format, please give the value of k + m.
Answer: 23
Problem: A box contains $2$ red marbles, $2$ green marbles, and $2$ yellow marbles. Carol takes $2$ marbles from the box at random; then Claudia takes $2$ of the remaining marbles at random; and then Cheryl takes the last two marbles. Find the probability that Cheryl gets $2$ marbles of the same color. The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
Answer: 6
Problem: At each step, we change the positions of exactly two numbers from the sequence $1, 2, 3, 4, 5, 6, 7$. How many different arrangements can be formed after two steps?
Answer: 176
Problem: Find the value of $\frac{\log_5 9 \cdot \log_7 5 \cdot \log_3 7}{\log_2 \sqrt{6}} + \frac{1}{\log_9 \sqrt{6}}$.
Answer: 4
Problem: Let $D$ be a point on side $[BC]$ of triangle $ABC$ such that $[AD]$ is an angle bisector, $|BD|=4$, and $|DC|=3$. Let $E$ be a point on side $[AB]$ and different than $A$ such that $m(\widehat{BED})=m(\widehat{DEC})$. If the perpendicular bisector of segment $[AE]$ meets the line $BC$ at $M$, what is $|CM|$?
Answer: 9
Problem: For each positive integer $n$, let $S(n)$ denote the sum of the digits of $n$. Determine how many values of $n$ satisfy the equation \( n + S(n) + S(S(n)) = 2007 \).
Answer: 4
Problem: A circle with radius 2 is centered at $(2,0)$. Another circle with radius 1 is centered at $(5,0)$. A line is tangent to both circles at points located in the first quadrant. The original answer is in the form k\sqrt{m}. Please find the value of k + m, where k and m are integers.
Answer: 4
Problem: Cozy the Cat and Dash the Dog are going up a staircase with a certain number of steps. Instead of walking up the steps one at a time, both Cozy and Dash jump. Cozy goes two steps up with each jump (though if necessary, he will just jump the last step). Dash goes five steps up with each jump (though if necessary, he will just jump the last steps if there are fewer than 5 steps left). Suppose Dash takes 19 fewer jumps than Cozy to reach the top of the staircase. Let $s$ denote the sum of all possible numbers of steps this staircase can have. What is the sum of the digits of $s$?
Answer: 13
Problem: How many ten-digit positive integers with distinct digits are multiples of $11111$? Provide your answer as an integer.
Answer: 3456
Problem: A triangular corner with side lengths $DB=EB=1$ is cut from equilateral triangle $ABC$ of side length $3$. Find the perimeter of the remaining quadrilateral.
Answer: 8
Problem: What is the maximum area of an isosceles trapezoid that has legs of length \(1\) and one base twice as long as the other? The maximum area can be expressed in the form \(\frac{k}{m}\), where \(k\) and \(m\) are coprime integers. Please find the value of \(k + m\).
Answer: 5
Problem: $ABC$ and $A'B'C'$ are equilateral triangles with parallel sides and the same center, as shown in the figure. The distance between side $BC$ and side $B'C'$ is $\frac{1}{6}$ the altitude of $\triangle ABC$. Find the ratio of the area of $\triangle A'B'C'$ to the area of $\triangle ABC$. The answer should be in the form \(\frac{k}{m}\), where the fraction is in simplest form. Please give the value of \(k + m\).
Answer: 5
Problem: Lucky Larry's teacher asked him to substitute numbers for $a$, $b$, $c$, $d$, and $e$ in the expression $a - (b - (c - (d + e)))$ and evaluate the result. Larry ignored the parentheses but added and subtracted correctly and obtained the correct result by coincidence. The numbers Larry substituted for $a$, $b$, $c$, and $d$ were $1$, $2$, $3$, and $4$, respectively. What number did Larry substitute for $e$?
Answer: 3
Problem: The triangle \(ABC\) has vertices such that \(AB = 3\), \(BC = 4\), and \(AC = 5\). The inscribed circle is tangent to \(AB\) at \(C'\), \(BC\) at \(A'\), and \(AC\) at \(B'\). Find the ratio between the area of the triangles \(A'B'C'\) and \(ABC\). The answer should be in the form \(\frac{k}{m}\). Please calculate the value of \(k + m\).
Answer: 6
Problem: In $\triangle ABC$, $AB = 8$, $BC = 7$, $CA = 6$, and side $BC$ is extended to a point $P$ such that $\triangle PAB$ is similar to $\triangle PCA$. Find the length of $PC$.
Answer: 9
Problem: Let $S$ be the set of all those 2007-place decimal integers $\overline{2s_1a_2a_3 \ldots a_{2006}}$ which contain an odd number of digit $9$ in each sequence $a_1, a_2, a_3, \ldots, a_{2006}$. The cardinal number of $S$ is given in the form \frac{1}{x}(y^{2006} - z^{2006}). Please calculate the value of x + y + z.
Answer: 20
Problem: In the equation \(\frac{x(x - 1) - (m + 1)}{(x - 1)(m - 1)} = \frac{x}{m}\), find the value of \(m\) such that the roots are equal. The original answer is in the form \(-\frac{a}{b}\). Please provide the value of \(a + b\).
Answer: 3
Problem: A particle of mass $m$ moving at speed $v_0$ collides with a particle of mass $M$ which is originally at rest. The fractional momentum transfer $f$ is the absolute value of the final momentum of $M$ divided by the initial momentum of $m$.

The fractional energy transfer is the absolute value of the final kinetic energy of $M$ divided by the initial kinetic energy of $m$.

If the collision is perfectly elastic, under what condition will the fractional energy transfer between the two objects be a maximum? Express your answer as a relationship between $m$ and $M$ in the form k*m = n*M, and provide the sum of k and n.
Answer: 2
Problem: A sphere with center \(O\) has a radius of \(6\). A triangle with sides of lengths \(15\), \(15\), and \(24\) is situated in space such that each of its sides is tangent to the sphere. The distance between \(O\) and the plane determined by the triangle is in the form \(m\sqrt{n}\). Find the value of \(m + n\).
Answer: 7
Problem: A vertical mass-spring oscillator is displaced $2.0\, \text{cm}$ from equilibrium. The $100\, \text{g}$ mass passes through the equilibrium point with a speed of $0.75\, \text{m/s}$. What is the spring constant of the spring? If the spring constant is $k\, \text{N/m}$, what is the value of $k$?
Answer: 140
Problem: If one can find a student with at least $k$ friends in any class which has 21 students such that at least two of any three of these students are friends, what is the largest possible value of $k$?
Answer: 10
Problem: If $ABCD$ is a $2 \times 2$ square, $E$ is the midpoint of $\overline{AB}$, $F$ is the midpoint of $\overline{BC}$, $\overline{AF}$ and $\overline{DE}$ intersect at $I$, and $\overline{BD}$ and $\overline{AF}$ intersect at $H$, find the area of quadrilateral $BEIH$. The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
Answer: 22
Problem: A frog is at the center of a circular shaped island with radius $r$. The frog jumps $\frac{1}{2}$ meters at first. After the first jump, it turns right or left at exactly $90^\circ$, and it always jumps one half of its previous jump. After a finite number of jumps, what is the least $r$ that ensures the frog can never fall into the water? The original answer is in \frac{\sqrt{m}}{n} format, please give the value of m + n.
Answer: 8
Problem: Find the value of $2^{-(2k+1)} - 2^{-(2k-1)} + 2^{-2k}$. The original answer is in the form $-x*2^{-(y*z+k)}$, where x, y, and z are integers. Please provide the sum of x, y, and z.
Answer: 4
Problem: 设\(\boldsymbol{a}, \boldsymbol{b}, \boldsymbol{c}\)在同一平面内的三个单位向量，且\(\boldsymbol{a}\perp\boldsymbol{b}\)，求\((\boldsymbol{c}-\boldsymbol{a})\cdot(\boldsymbol{c}-\boldsymbol{b})\)的最大值。答案的形式为\(k + \sqrt{m}\)，请给出\(k + m\)的值。
Answer: 3
Problem: 设\(\boldsymbol{a}、\boldsymbol{b}、\boldsymbol{c}\)为同一平面内的三个单位向量，且 \(\boldsymbol{a}\perp\boldsymbol{b}\)。求\((\boldsymbol{c}-\boldsymbol{a})\cdot(\boldsymbol{c}-\boldsymbol{b})\)的最大值。答案的形式为\(m + \sqrt{n}\)，请给出\(m+n\)的值。
Answer: 3
Problem: 如图 1, 设 $P$ 为 $\triangle A B C$ 内一点, 且 $\boldsymbol{A P}$ $=\frac{2}{5} \boldsymbol{A B}+\frac{1}{5} \boldsymbol{A} \boldsymbol{C}$ 。求 $\frac{S_{\triangle A B P}}{S_{\triangle A B C}}$ 的值。答案应为\frac{k}{m}的形式，请给出k+m的值。
Answer: 6
Problem: 如图, 在棱长为 1 的正四面体 $A B C D$ 中, $G$ 为 $\triangle B C D$ 的重心, $M$ 是线段 $A G$ 的中点, 求三棱锥 $M-B C D$ 的外接球的表面积除以 $\pi$ 的值。答案的形式为\(\frac{k}{m}\)，请给出\(k+m\)的值。
Answer: 5
Problem: 如图, 在棱长为 1 的正方体 \(A B C D-A_{1} B_{1} C_{1} D_{1}\) 中, \(P, Q, R\) 分别是棱 \(A B, A D, A A_{1}\) 的中点。以 \(\triangle P Q R\) 为底面作一个直三棱柱，使其另一个底面的三个顶点也都在正方体的表面上。求这个直三棱柱的体积。原始答案的形式为\(\frac{k}{m}\)，请给出k+m的值。
Answer: 19
Problem: 设 \(x, y\in\mathbf{R}\), 且\(\log _{4}(x+2 y)+\log _{4}(x-2y)=1\), 求\(x-|y|\)的最小值。原始答案是以\(\sqrt{k}\)的形式给出，请给出k的值。
Answer: 3
Problem: 设 \(x 、 y \in \mathbf{R}\), 且\(\log _{4}(x+2 y)+\log _{4}(x-2 y)=1\)。求 \(x-|y|\) 的最小值。答案的形式为\(\sqrt{k}\)，请给出k的值。
Answer: 3
Problem: 设 \(x\)、\(y\in\mathbf{R}\)，且 \(\log_{4}(x+2y)+\log_{4}(x-2y)=1\)。求 \(x-|y|\) 的最小值。原始答案的形式为\(\sqrt{m}\)，请给出m的值。
Answer: 3
Problem: 设函数$f(x)=x^{3}+a x^{2}+b x+c$（$a, b, c$均为非零整数），若 $f(a)=a^{3}, f(b)=b^{3}$，则 $c$ 的值是多少？
Answer: 16
Problem: 在棱长为1的正四面体$ABCD$中, $G$为$\triangle BCD$的重心，$M$为线段$AG$的中点。若三棱锥$M-BCD$的外接球的表面积为$x\pi$，求$x$的值。答案应为\frac{k}{m}的形式，请给出k+m的值。
Answer: 5
Problem: 设集合$A=\left\{n\left|\frac n3\in\mathbf{N}_+\right\},\right.$
$B= \{$ $y|$ $y= x+ 4$ + $\sqrt {5- x^{2}}$ $\} .$
求集合$A\cap B$ 中元素的个数。
Answer: 2
Problem: 设集合 $A=\left\{n \left\lvert\, \frac{n}{3} \in \mathbf{N}_{+}\right.\right\}$，
$B=\left\{y \mid y=x+4+\sqrt{5-x^{2}}\right\}$.
求集合 $A \cap B$ 中元素的个数。
Answer: 2
Problem: 在凸四边形 $A B C D$ 中, 若 $B C=4, \angle A D C=60^{\circ}, \angle B A D=90^{\circ}$, 四边形 $A B C D$ 的面积等于 $\frac{A B \cdot C D+B C \cdot A D}{2}$, 则 $C D$ 的长乘以10后的值（精确到小数点后1位）是多少？
Answer: 69
Problem: 正方体的八个顶点中任取 3 个构成三角形，求该三角形是等腰三角形的概率。答案的形式为\(\frac{k}{m}\)，请计算\(k + m\)的值。
Answer: 11
Problem: 已知 $\frac{\sin x}{\sqrt{1-\cos ^{2} x}}-\frac{\cos x}{\sqrt{1-\sin ^{2} x}}=2(0<x \leqslant 2 \pi)$, 求 $x$ 的取值范围。答案的形式为 \left(\frac{a\pi}{b}, c\pi\right)，请计算a+b+c的值。
Answer: 4
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？答案的形式为 $k(1+\sqrt{m})$，请给出 $k+m$ 的值。
Answer: 8
Problem: 令 $a_{n}$ 表示距离 $\sqrt{n}$ 最近的正整数，其中 $n \in \mathbf{N}^{*}$ 。若 $\frac{1}{a_{1}}+\frac{1}{a_{2}}+\cdots+\frac{1}{a_{n}}=2016$，则求正整数 $n$ 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: 设 $A B, C D$ 是圆 $O$ 的两条互相垂直的直径, 弦 $D F$ 交 $A B$ 于点 $E, D E=24$, $E F=18$，求 $O E$ 的值。答案的形式为$k \sqrt{m}$，请给出k+m的值。
Answer: 8
Problem: 已知 $k \neq 1$ ，则等比数列 $a+\log _{2} k, a+\log _{4} k, a+\log _{8} k$ 的公比是多少？答案的形式为\frac{k}{m}，请给出k+m的值。
Answer: 4
Problem: 两个圆内切于 $K$, 大圆的弦 $A B$ 与小圆切于 $L$, 已知 $A K: B K=2: 5, A L=10$ ，求 $B L$ 的长。
Answer: 25
Problem: $f(x)$ 是定义在实数集 $\mathbf{R}$ 上的函数，满足 $\forall x \in \mathbf{R}, 2 f(x)+f\left(x^{2}-1\right)=1$ ，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: 从一个正九边形的 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, $B D=6, \angle A B D=\angle C B D=30^{\circ}$, 设四边形 $A B C D$ 的面积为 $x \sqrt{3}$，求 $x$ 的值。
Answer: 9
Problem: 若实数 $a, b$ 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, 若 $P_{1}, P_{2}, P_{3}, P_{4}$ 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，则四边形 $P_{1} P_{2} P_{3} P_{4}$ 的面积等于多少？答案的形式为k-\sqrt{m}，请给出k+m的值。
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范围。请以区间形式 $(\frac{c}{d}, e)$ 表示，并给出 $c+d+e$ 的值。
Answer: 43
Problem: 已知正方形 $A B C D$ 与点 $P$ 在同一平面内，该正方形的边长为 1。若 $|P A|^{2}+|P B|^{2}=|P C|^{2}$，则 $|P D|$ 的最大值是多少？答案的形式为 $k+\sqrt{m}$，请给出k+m的值。
Answer: 4
Problem: 使得 $p^{3}+7 p^{2}$ 为完全平方数的不大于 100 的素数 $p$ 的个数为多少？
Answer: 2
Problem: 函数 $f(x)=x(x+1)(x+2)(x+3)$ 的最小值是多少？
Answer: -1
Problem: 不等式组 \(\left\{\begin{array}{l}y \geqslant 2|x|-1 \\ y \leqslant-3|x|+5\end{array}\right.\) 所表示的平面区域的面积是多少？答案应为\(\frac{k}{m}\)的形式，请给出k+m的值。
Answer: 41
Problem: Find the value of $\left(1+\cos \frac{\pi}{5}\right)\left(1+\cos \frac{3 \pi}{5}\right)$. The original answer is in the form of \frac{k}{m}, please provide the value of k + m.
Answer: 9
Problem: 若 $x$ 为实数, 使得 $2, x, x^{2}$ 互不相同，且其中有一个数恰为另一个数的 2倍，则这样的实数 $x$ 的个数为多少？
Answer: 4
Problem: 若 $S=\frac{1}{\log _{\frac{1}{2}} \pi}+\frac{1}{\log _{\frac{1}{3}} \pi}+\frac{1}{\log _{\frac{1}{5}} \pi}+\frac{1}{\log _{\frac{1}{7}} \pi}$, 求不超过 $S$ 且与 $S$ 最接近的整数。
Answer: -5
Problem: 正整数 $9+95+995+\cdots+\underbrace{99\cdots95}_{2016\text{个}9}$ 的十进制表示中数字 1 的个数为多少？
Answer: 2014
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 $\triangle D O A$ 的内切圆半径的形式为\(\frac{k}{m}\)，请给出k+m的值。
Answer: 11
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: 若某三角形的三边长为三个连续正整数，且该三角形有一个角是另一个角的 2 倍，请找出这个三角形的三边长。答案的格式为(k, m, n)，请给出k + m + n的值。
Answer: 15
Problem: 将等差数列 $1,5,9,13, \cdots, 2017$ 排成一个大数 $15913 \cdots 2017$ ，求该数值被 9 除的余数。
Answer: 1
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 $2a+3b+5c$ 的最大值和最小值之差。
Answer: 9
Problem: 若函数 $f(t)=t^{2}+2t$ ，则点集 $\left\{(x, y) \left\lvert\,\left\{\begin{array}{l}f(x)+f(y) \leqslant 2 \\ f(x) \geqslant f(y)\end{array}\right\}\right.\right.$ 所构成图形的面积是 $k\pi$，求 $k$ 的值。
Answer: 2
Problem: 若数列 \(\left\{a_{n}\right\}\) 的前 \(n\) 项和 \(S_{n}=n^{2} a_{n}\)，且 \(a_{1}=2019\)，求 \(a_{2018}\) 的值。原始答案的形式为\(\frac{k}{m}\)，请给出k+m的值。
Answer: 1010
Problem: 若正数 \(a, b\) 满足 \(a+b=1\)，则求 \(\frac{1}{a}+\frac{27}{b^{3}}\) 的最小值。答案的形式为\(\frac{k+m \sqrt{3}}{n}\)，请计算k+m+n的值。
Answer: 62
Problem: 设实数 $x ,y$ 满足 $\frac{x^{2}}{5}+\frac{y^{2}}{4}=1$, 求 $\sqrt{x^{2}+y^{2}-2 y+1}+\sqrt{x^{2}+y^{2}-2 x+1}$ 的最小值。答案的形式为 $k\sqrt{m} - \sqrt{n}$，请给出k+m+n的值。
Answer: 9
Problem: 把正整数中的非完全平方数从小到大排成一个数列 $\left\{a_{n}\right\}(n \geqslant 1)$, 例如, $a_{1}=2$, $a_{2}=3$, $a_{3}=5$, $a_{4}=6$, $\cdots$, 求 $a_{2018}$ 的值。
Answer: 2063
Problem: 15 人围坐在圆桌旁，选出 4 人使得其中任意两人都不相邻的选法数为多少？
Answer: 450
Problem: 设集合 $S$ 中有 10 个元素, 从 $S$ 中每次随机选取 1 个元素, 取出后还放回 $S$ 中, 则取5次后出现重复元素的概率是（保留两位有效数字）。如果答案是 $x$，那么请给出 $100x$ 的值。
Answer: 70