ChenDRAG/VeRL_math_validation huggingface dataset:
Problem: Keiko walks once around a track at exactly the same constant speed every day. The sides of the track are straight, and the ends are semicircles. The track has a width of \(6\) meters, and it takes her \(36\) seconds longer to walk around the outside edge of the track than around the inside edge. The original answer for Keiko's speed is in the format \(\frac{k\pi}{m}\), where k and m are integers. Please find the value of k + m.
Answer: 4
Problem: $A$ power boat and a raft both left dock $A$ on a river and headed downstream. The raft drifted at the speed of the river current. The power boat maintained a constant speed with respect to the river. The power boat reached dock $B$ downriver, then immediately turned and traveled back upriver. It eventually met the raft on the river $9 ~hours$ after leaving dock $A.$ If the answer is $x$ hours, what is the value of $10x$?
Answer: 45
Problem: A dart board is a regular octagon divided into regions as shown below. Suppose that a dart thrown at the board is equally likely to land anywhere on the board. Find the probability that the dart lands within the center square. The original answer is in the form \(\frac{k\sqrt{m} - n}{p}\), please find the value of k + m + n.
Answer: 4
Problem: Angie, Bridget, Carlos, and Diego are seated at random around a square table, one person to a side. Find the probability that Angie and Carlos are seated opposite each other. Express your answer in the form \(\frac{k}{m}\), where the fraction is in simplest form. Please provide the value of \(k + m\).
Answer: 4
Problem: Triangle $ABC$ has side-lengths $AB = 12, BC = 24,$ and $AC = 18.$ The line through the incenter of $\triangle ABC$ parallel to $\overline{BC}$ intersects $\overline{AB}$ at $M$ and $\overline{AC}$ at $N.$ What is the perimeter of $\triangle AMN?$
Answer: 30
Problem: Brian writes down four integers $w > x > y > z$ whose sum is $44$. The pairwise positive differences of these numbers are $1, 3, 4, 5, 6$ and $9$. What is the sum of the possible values of $w$?
Answer: 31
Problem: Two congruent squares, $ABCD$ and $PQRS$, have side length $15$. They overlap to form the $15$ by $25$ rectangle $AQRD$ shown. What is the percentage of the area of rectangle $AQRD$ that is shaded? If the answer is $x\%$, what is the value of $x$?
Answer: 20
Problem: Suppose $a$ and $b$ are single-digit positive integers chosen independently and at random. Find the probability that the point $(a,b)$ lies above the parabola $y=ax^2-bx$. The original answer is in \frac{k}{m} format, please give the value of k + m.
Answer: 100
Problem: A segment through the focus $F$ of a parabola with vertex $V$ is perpendicular to $\overline{FV}$ and intersects the parabola in points $A$ and $B$. Find $\cos\left(\angle AVB\right)$. The original answer is in the form -\frac{k}{m}, where k and m are integers. Please give the value of k + m.
Answer: 8
Problem: The circular base of a hemisphere of radius \(2\) rests on the base of a square pyramid of height \(6\). The hemisphere is tangent to the other four faces of the pyramid. The original answer is in the form of \(\sqrt{k}\). What is the value of \(k\) if it represents the edge-length squared of the base of the pyramid?
Answer: 18
Problem: A dart board is a regular octagon divided into regions as shown. Suppose that a dart thrown at the board is equally likely to land anywhere on the board. Find the probability that the dart lands within the center square. The original answer is in the form \(\frac{\sqrt{k} - 1}{m}\). Please give the value of k + m.
Answer: 4
Problem: Each vertex of convex pentagon $ABCDE$ is to be assigned a color. There are $6$ colors to choose from, and the ends of each diagonal must have different colors. Find the number of different colorings possible.
Answer: 3120
Problem: Rhombus \(ABCD\) has side length \(2\) and \(\angle B = 120^\circ\). Region \(R\) consists of all points inside the rhombus that are closer to vertex \(B\) than any of the other three vertices. The area of \(R\) is given in the form \(\frac{k\sqrt{n}}{m}\). Please find the value of \(k + m + n\).
Answer: 8
Problem: In the eight term sequence $A$, $B$, $C$, $D$, $E$, $F$, $G$, $H$, the value of $C$ is $5$ and the sum of any three consecutive terms is $30$. What is $A+H$?
Answer: 25
Problem: In the given circle, the diameter $\overline{EB}$ is parallel to $\overline{DC}$, and $\overline{AB}$ is parallel to $\overline{ED}$. The angles $AEB$ and $ABE$ are in the ratio $4 : 5$. What is the degree measure of angle $BCD$? If the answer is $x^\circ$, what is the value of $x$?
Answer: 130
Problem: Circles with radii \(1\), \(2\), and \(3\) are mutually externally tangent. Find the area of the triangle determined by the points of tangency. The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
Answer: 11
Problem: Let $f(x) = 10^{10x}, g(x) = \log_{10}\left(\frac{x}{10}\right), h_1(x) = g(f(x))$, and $h_n(x) = h_1(h_{n-1}(x))$ for integers $n \geq 2$. Find the sum of the digits of $h_{2011}(1)$.
Answer: 16089
Problem: Circles $A, B,$ and $C$ each have radius $1$. Circles $A$ and $B$ share one point of tangency. Circle $C$ has a point of tangency with the midpoint of $\overline{AB}.$ Find the area inside circle $C$ but outside circle $A$ and circle $B$.
Answer: 2
Problem: A pyramid has a square base with side of length \(1\) and has lateral faces that are equilateral triangles. A cube is placed within the pyramid so that one face is on the base of the pyramid and its opposite face has all its edges on the lateral faces of the pyramid. The volume of this cube is in the form \(k\sqrt{2} - m\). Find the value of \(k + m\).
Answer: 12
Problem: At a competition with $N$ players, the number of players given elite status is equal to $2^{1+\lfloor \log_{2} (N-1) \rfloor}-N$. Suppose that $19$ players are given elite status. Find the sum of the two smallest possible values of $N$.
Answer: 154
Problem: A lattice point in an $xy$-coordinate system is any point $(x, y)$ where both $x$ and $y$ are integers. The graph of $y = mx + 2$ passes through no lattice point with $0 < x \leq 100$ for all $m$ such that $\frac{1}{2} < m < a$. The original answer is in \frac{k}{m} format, please give the value of k + m. What is the maximum possible value of $a$?
Answer: 149
Problem: Rhombus $ABCD$ has side length $2$ and $\angle B = 120^\circ$. Region $R$ consists of all points inside the rhombus that are closer to vertex $B$ than any of the other three vertices. The area of region $R$ is given in the form \(\frac{k\sqrt{m}}{n}\). Please find the value of \(k + m + n\).
Answer: 8
Problem: Let $f(x)=ax^2+bx+c$, where $a$, $b$, and $c$ are integers. Suppose that $f(1)=0$, $50<f(7)<60$, $70<f(8)<80$, $5000k<f(100)<5000(k+1)$ for some integer $k$. Find the value of $k$.
Answer: 3
Problem: Quadrilateral $ABCD$ is a trapezoid, $AD = 15$, $AB = 50$, $BC = 20$, and the altitude is $12$. What is the area of the trapezoid?
Answer: 750
Problem: Let \(f_{1}(x)=\sqrt{1-x}\), and for integers \(n \geq 2\), let \(f_{n}(x)=f_{n-1}(\sqrt{n^2 - x})\). If \(N\) is the largest value of \(n\) for which the domain of \(f_{n}\) is nonempty, the domain of \(f_{N}\) is \([c]\). Find the value of \(N+c\).
Answer: -226
Problem: The arithmetic mean of two distinct positive integers $x$ and $y$ is a two-digit integer. The geometric mean of $x$ and $y$ is obtained by reversing the digits of the arithmetic mean. What is the absolute value of the difference $|x - y|$?
Answer: 66
Problem: Students guess that Norb's age is $24, 28, 30, 32, 36, 38, 41, 44, 47$, and $49$. Norb says, "At least half of you guessed too low, two of you are off by one, and my age is a prime number." How old is Norb?
Answer: 37
Problem: Each vertex of convex pentagon $ABCDE$ is to be assigned a color. There are $6$ colors to choose from, and the ends of each diagonal must have different colors. How many different colorings are possible?
Answer: 3120
Problem: A pyramid has a square base with sides of length \(1\) and has lateral faces that are equilateral triangles. A cube is placed within the pyramid so that one face is on the base of the pyramid and its opposite face has all its edges on the lateral faces of the pyramid. The volume of this cube can be expressed in the form \(k\sqrt{2} - m\). Please calculate the value of \(k + m\).
Answer: 12
Problem: Let \(T_1\) be a triangle with side lengths \(2011\), \(2012\), and \(2013\). For \(n \geq 1\), if \(T_n = \triangle ABC\) and \(D, E\), and \(F\) are the points of tangency of the incircle of \(\triangle ABC\) to the sides \(AB\), \(BC\), and \(AC\), respectively, then \(T_{n+1}\) is a triangle with side lengths \(AD, BE\), and \(CF\), if it exists. Find the perimeter of the last triangle in the sequence \(\left(T_n\right)\). The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
Answer: 1637
Problem: Seven students count from $1$ to $1000$ as follows:  
  
Alice says all the numbers, except she skips the middle number in each consecutive group of three numbers. That is, Alice says $1, 3, 4, 6, 7, 9, . . ., 997, 999, 1000$.  
  
Barbara says all of the numbers that Alice doesn't say, except she also skips the middle number in each consecutive group of three numbers.  
  
Candice says all of the numbers that neither Alice nor Barbara says, except she also skips the middle number in each consecutive group of three numbers.  
  
Debbie, Eliza, and Fatima say all of the numbers that none of the students with the first names beginning before theirs in the alphabet say, except each also skips the middle number in each of her consecutive groups of three numbers.  
  
Finally, George says the only number that no one else says.  
  
What number does George say?
Answer: 365
Problem: What is the hundreds digit of \(2011^{2011}\)?
Answer: 6
Problem: Let \(f(z)= \frac{z+a}{z+b}\) and \(g(z)=f(f(z))\), where \(a\) and \(b\) are complex numbers. Suppose that \(\left| a \right| = 1\) and \(g(g(z))=z\) for all \(z\) for which \(g(g(z))\) is defined. Find the difference between the largest and smallest possible values of \(\left| b \right|\). The original answer is in the form \(\sqrt{m} - n\). Please give the value of m + n.
Answer: 4
Problem: How many $4$-digit positive integers have four different digits, where the leading digit is not zero, the integer is a multiple of $5$, and $5$ is the largest digit?
Answer: 84
Problem: A lattice point in an $xy$-coordinate system is any point $(x, y)$ where both $x$ and $y$ are integers. The graph of $y = mx + 2$ passes through no lattice point with $0 < x \le 100$ for all $m$ such that $\frac{1}{2} < m < a$. What is the maximum possible value of $a$? The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
Answer: 149
Problem: Consider all quadrilaterals $ABCD$ such that $AB=14$, $BC=9$, $CD=7$, and $DA=12$. Find the radius of the largest possible circle that fits inside or on the boundary of such a quadrilateral. The original answer is in the form $k\sqrt{m}$, please give the value of k + m.
Answer: 8
Problem: Let $P(z) = z^8 + \left(4\sqrt{3} + 6\right)z^4 - \left(4\sqrt{3} + 7\right)$. Find the minimum perimeter among all the $8$-sided polygons in the complex plane whose vertices are precisely the zeros of $P(z)$. The original answer is in the form of $k\sqrt{m}$, please give the value of k + m.
Answer: 10
Problem: Let \(T_1\) be a triangle with side lengths \(2011, 2012,\) and \(2013\). For \(n \ge 1\), if \(T_n = \triangle ABC\) and \(D, E,\) and \(F\) are the points of tangency of the incircle of \(\triangle ABC\) to the sides \(AB, BC\), and \(AC,\) respectively, then \(T_{n+1}\) is a triangle with side lengths \(AD, BE,\) and \(CF,\) if it exists. Find the perimeter of the last triangle in the sequence \(( T_n )\). The original answer is in \(\frac{k}{m}\) format, please give the value of \(k + m\).
Answer: 1637
Problem: In a bag of marbles, \(\frac{3}{5}\) of the marbles are blue and the rest are red. If the number of red marbles is doubled and the number of blue marbles stays the same, what fraction of the marbles will be red? Express your answer as a fraction in simplest form. The original answer is in the form \(\frac{k}{m}\). Please find the value of k + m.
Answer: 11
Problem: Suppose that the euro is worth $1.3$ dollars. If Diana has $500$ dollars and Etienne has $400$ euros, by what percent is the value of Etienne's money greater than the value of Diana's money? Provide your answer as a percentage.
Answer: 4
Problem: Peter's family ordered a 12-slice pizza for dinner. Peter ate one slice and shared another slice equally with his brother Paul. What fraction of the pizza did Peter eat? Express your answer as a fraction in simplest form, in the format \(\frac{k}{m}\), and give the value of k + m.
Answer: 9
Problem: Two integers have a sum of $26$. When two more integers are added to the first two, the sum is $41$. Finally, when two more integers are added to the sum of the previous $4$ integers, the sum is $57$. Find the minimum number of even integers among the $6$ integers.
Answer: 1
Problem: In a bag of marbles, \(\frac{3}{5}\) of the marbles are blue and the rest are red. If the number of red marbles is doubled and the number of blue marbles stays the same, what fraction of the marbles will be red? The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
Answer: 11
Problem: Mary divides a circle into $12$ sectors. The central angles of these sectors, measured in degrees, are all integers and they form an arithmetic sequence. What is the degree measure of the smallest possible sector angle?
Answer: 8
Problem: Small lights are hung on a string $6$ inches apart in the order red, red, green, green, green, red, red, green, green, green, and so on continuing this pattern of $2$ red lights followed by $3$ green lights. How many feet separate the 3rd red light and the 21st red light? If the answer is $x$ feet, what is the value of $10x$?
Answer: 225
Problem: An iterative average of the numbers \(1, 2, 3, 4\), and \(5\) is computed the following way. Arrange the five numbers in some order. Find the mean of the first two numbers, then find the mean of that with the third number, then the mean of that with the fourth number, and finally the mean of that with the fifth number. Find the difference between the largest and smallest possible values that can be obtained using this procedure. The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
Answer: 25
Problem: $A$ pair of six-sided dice are labeled so that one die has only even numbers (two each of $2, 4$, and $6$), and the other die has only odd numbers (two of each $1, 3$, and $5$). The pair of dice is rolled. Find the probability that the sum of the numbers on the tops of the two dice is $7$. Express your answer in the form \(\frac{k}{m}\) and provide the value of \(k + m\).
Answer: 4
Problem: A triangle has an area of $30$, one side of length $10$, and the median to that side has a length of $9$. Let $\theta$ be the acute angle formed by that side and the median. The original answer for $\sin{\theta}$ is in the form \frac{k}{m}. Please provide the value of k + m.
Answer: 5
Problem: What is the area of the polygon whose vertices are the points of intersection of the curves $x^2 + y^2 = 25$ and $(x-4)^2 + 9y^2 = 81$? Provide your answer as an integer.
Answer: 27
Problem: Externally tangent circles with centers at points $A$ and $B$ have radii of lengths $5$ and $3$, respectively. A line externally tangent to both circles intersects ray $AB$ at point $C$. What is the length of $BC$?
Answer: 12
Problem: Alex, Mel, and Chelsea play a game that has $6$ rounds. In each round there is a single winner, and the outcomes of the rounds are independent. For each round, the probability that Alex wins is $\frac{1}{2}$, and Mel is twice as likely to win as Chelsea. What is the probability that Alex wins three rounds, Mel wins two rounds, and Chelsea wins one round? The original answer is in \frac{m}{n} format, please give the value of m + n.
Answer: 41
Problem: Point $B$ is due east of point $A$. Point $C$ is due north of point $B$. The distance between points $A$ and $C$ is $10\sqrt{2}$, and $\angle BAC = 45^\circ$. Point $D$ is $20$ meters due north of point $C$. Calculate the distance $AD$ and determine the interval in which this distance falls, expressed as (a, b) where a and b are consecutive integers. Please find the value of a + b.
Answer: 63
Problem: A square region \(ABCD\) is externally tangent to the circle with equation \(x^2+y^2=1\) at the point \((0,1)\) on the side \(CD\). Vertices \(A\) and \(B\) are on the circle with equation \(x^2+y^2=4\). The side length of this square can be expressed in the form \(\frac{k\sqrt{m}-n}{p}\), where \(\sqrt{m}\) is a square root of a prime number. Find the value of \(k + m + n + p\).
Answer: 30
Problem: How many sequences of zeros and ones of length $20$ have all the zeros consecutive, or all the ones consecutive, or both?
Answer: 380
Problem: Two parabolas have equations \(y= x^2 + ax + b\) and \(y= x^2 + cx + d\), where \(a, b, c,\) and \(d\) are integers, each chosen independently by rolling a fair six-sided die. Find the probability that the parabolas will have at least one point in common. The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
Answer: 67
Problem: Chubby makes nonstandard checkerboards that have $31$ squares on each side. The checkerboards have a black square in every corner and alternate red and black squares along every row and column. How many black squares are there on such a checkerboard?
Answer: 481
Problem: Two equilateral triangles are contained in a square whose side length is \(2\sqrt{3}\). The bases of these triangles are the opposite sides of the square, and their intersection is a rhombus. The area of the rhombus is in the form \(k\sqrt{3} - m\). Please find the value of k + m.
Answer: 20
Problem: Bernardo and Silvia play the following game. An integer between $0$ and $999$ inclusive is selected and given to Bernardo. Whenever Bernardo receives a number, he doubles it and passes the result to Silvia. Whenever Silvia receives a number, she adds $50$ to it and passes the result to Bernardo. The winner is the last person who produces a number less than $1000$. Let $N$ be the smallest initial number that results in a win for Bernardo. What is the sum of the digits of $N$?
Answer: 7
Problem: A \(3 \times 3\) square is partitioned into \(9\) unit squares. Each unit square is painted either white or black with each color being equally likely, chosen independently and at random. The square is then rotated \(90\,^{\circ}\) clockwise about its center, and every white square in a position formerly occupied by a black square is painted black. The colors of all other squares are left unchanged. Find the probability that the grid is now entirely black. The original answer is in \(\frac{m}{n}\) format, please give the value of m + n.
Answer: 561
Problem: Jesse cuts a circular disk of radius \(12\), along \(2\) radii to form \(2\) sectors, one with a central angle of \(120^\circ\). He makes two circular cones using each sector to form the lateral surface of each cone. The ratio of the volume of the smaller cone to the larger cone is in the form \(\frac{\sqrt{k}}{n}\). Please give the value of \(k + n\).
Answer: 20
Problem: 三名跑步者同时从一个500米的圆形跑道上的同一点出发，分别以4.4、4.8和5.0米每秒的恒定速度顺时针方向跑动。当他们再次在圆形跑道上相遇时，跑步者停止跑步。请计算跑步者跑步的时间（以秒为单位）。
Answer: 2500
Problem: Three circles with radius $2$ are mutually tangent. What is the total area of the circles and the region bounded by them, as shown in the figure? Express your answer in the form $a\pi + b\sqrt{3}$, where $a$ and $b$ are constants. Please find the value of $a + b$.
Answer: 14
Problem: Circle $C_1$ has its center $O$ lying on circle $C_2$. The two circles meet at $X$ and $Y$. Point $Z$ in the exterior of $C_1$ lies on circle $C_2$ and $XZ=13$, $OZ=11$, and $YZ=7$. The radius of circle $C_1$ is in the form \(\sqrt{k}\). Find the value of \(k\).
Answer: 30
Problem: Amy, Beth, and Jo listen to four different songs and discuss which ones they like. No song is liked by all three. Furthermore, for each of the three pairs of the girls, there is at least one song liked by those two girls but disliked by the third. In how many different ways is this possible?
Answer: 132
Problem: Jesse cuts a circular paper disk of radius \(12\) along two radii to form two sectors, the smaller having a central angle of \(120\) degrees. He makes two circular cones, using each sector to form the lateral surface of a cone. What is the ratio of the volume of the smaller cone to that of the larger? Express your answer in the form \(\frac{\sqrt{k}}{m}\), where k and m are integers, and provide the value of k + m.
Answer: 20
Problem: Triangle $ABC$ has $AB=27$, $AC=26$, and $BC=25$. Let $I$ be the intersection of the internal angle bisectors of $\triangle ABC$. What is the length of $BI$?
Answer: 15
Problem: What is the smallest positive integer that is neither prime nor square and that has no prime factor less than $50$?
Answer: 3127
Problem: In rectangle \(ABCD\), \(AB=6\), \(AD=30\), and \(G\) is the midpoint of \(\overline{AD}\). Segment \(AB\) is extended \(2\) units beyond \(B\) to point \(E\), and \(F\) is the intersection of \(\overline{ED}\) and \(\overline{BC}\). The area of quadrilateral \(BFDG\) is originally given in the form \(\frac{k}{m}\). Please calculate the value of \(k + m\).
Answer: 137
Problem: Adam, Benin, Chiang, Deshawn, Esther, and Fiona have internet accounts. Some, but not all, of them are internet friends with each other, and none of them has an internet friend outside this group. Each of them has the same number of internet friends. In how many different ways can this happen?
Answer: 170
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 \(1\le i\le 4,\) 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 \(P_1'P_4'\). The original answer is in the form \(\frac{k\sqrt{n}}{m}\). Please find the value of k + m + n.
Answer: 9
Problem: Consider the polynomial
\[ P(x)=\prod\limits_{k=0}^{10}(x^{2^k}+2^k)=(x+1)(x^2+2)(x^4+4)\cdots (x^{1024}+1024) \]
The coefficient of \( x^{2012} \) is equal to \( 2^a \). What is the value of \( a \)?
Answer: 6
Problem: Let points $A$ = $(0 ,0 ,0)$, $B$ = $(1, 0, 0)$, $C$ = $(0, 2, 0)$, and $D$ = $(0, 0, 3)$. Points $E$, $F$, $G$, and $H$ are midpoints of line segments $\overline{BD},\text{ }  \overline{AB}, \text{ } \overline {AC},$ and $\overline{DC}$ respectively. Find the area of quadrilateral $EFGH$. The original answer is in the form \frac{k\sqrt{m}}{n}, please give the value of k + m + n.
Answer: 12
Problem: Let $a$, $b$, and $c$ be positive integers with $a \ge b \ge c$ such that
$a^2-b^2-c^2+ab=2011$ and
$a^2+3b^2+3c^2-3ab-2ac-2bc=-1997$.

What is the value of $a$?
Answer: 253
Problem: The sum of the first $m$ positive odd integers is $212$ more than the sum of the first $n$ positive even integers. Find the sum of all possible values of $n$.
Answer: 255
Problem: Let $R$ be a set of nine distinct integers. Six of the elements are $2$, $3$, $4$, $6$, $9$, and $14$. What is the number of possible values of the median of $R$?
Answer: 7
Problem: A solid tetrahedron is sliced off a solid wooden unit cube by a plane passing through two nonadjacent vertices on one face and one vertex on the opposite face not adjacent to either of the first two vertices. The tetrahedron is discarded and the remaining portion of the cube is placed on a table with the cut surface face down. What is the height of this object in the form of \(\frac{a \sqrt{b}}{c}\), where \(a\), \(b\), and \(c\) are integers? Please find the value of a + b + c.
Answer: 8
Problem: A circle of radius \(2\) is cut into four congruent arcs. The four arcs are joined to form the star figure shown. Find the ratio of the area of the star figure to the area of the original circle. The original answer is in the form \(\frac{k-\pi}{m\pi}\). Please provide the value of \(k + m\).
Answer: 5