sungyub/dapo-math-17k-verl huggingface dataset:
user: The number obtained from the last two nonzero digits of $90!$ is equal to $n$. What is the value of $n$?

Answer: 12
user: Monic quadratic polynomial $P(x)$ and $Q(x)$ have the property that $P(Q(x))$ has zeros at $x=-23, -21, -17,$ and $-15$, and $Q(P(x))$ has zeros at $x=-59,-57,-51$ and $-49$. Find the sum of the minimum values of $P(x)$ and $Q(x)$.

Answer: -100
user: Semicircles $POQ$ and $ROS$ pass through the center $O$. What is the ratio of the combined areas of the two semicircles to the area of circle $O$? The answer is in the form of \frac{k}{m}, where k and m are integers. Please provide the value of k + m.

Answer: 3
user: $A$ high school basketball game between the Raiders and Wildcats was tied at the end of the first quarter. The number of points scored by the Raiders in each of the four quarters formed an increasing geometric sequence, and the number of points scored by the Wildcats in each of the four quarters formed an increasing arithmetic sequence. At the end of the fourth quarter, the Raiders had won by one point. Neither team scored more than $100$ points. What was the total number of points scored by the two teams in the first half?

Answer: 34
user: Let $f(x) = \log_{10} \left(\sin(\pi x) \cdot \sin(2 \pi x) \cdot \sin (3 \pi x) \cdots \sin(8 \pi x)\right)$. The intersection of the domain of $f(x)$ with the interval $[0,1]$ is a union of $n$ disjoint open intervals. What is the value of $n$?

Answer: 12
user: The set of real numbers $x$ for which 

$$\dfrac{1}{x-2009}+\dfrac{1}{x-2010}+\dfrac{1}{x-2011}\ge1$$ 

is the union of intervals of the form $a<x\le b$. What is the sum of the lengths of these intervals?

Answer: 3
user: Jim starts with a positive integer $n$ and creates a sequence of numbers. Each successive number is obtained by subtracting the largest possible integer square less than or equal to the current number until zero is reached. For example, if Jim starts with $n = 55$, then his sequence contains $5$ numbers:
$$\begin{array}{ccccc}
{}&{}&{}&{}&55\\
55&-&7^2&=&6\\
6&-&2^2&=&2\\
2&-&1^2&=&1\\
1&-&1^2&=&0\\
\end{array}$$
Let $N$ be the smallest number for which Jim’s sequence has $8$ numbers. What is the units digit of $N$?

Answer: 3
user: Let $a > 0$, and let $P(x)$ be a polynomial with integer coefficients such that 
$P(1) = P(3) = P(5) = P(7) = a$, and 
$P(2) = P(4) = P(6) = P(8) = -a$. 
Find the smallest possible value of $a$.

Answer: 315
user: Two quadrilaterals are considered the same if one can be obtained from the other by a rotation and a translation. How many different convex cyclic quadrilaterals are there with integer sides and perimeter equal to $32$? Provide your answer as an integer.

Answer: 568
user: Suppose [$a$ $b$] denotes the average of $a$ and $b$, and {$a$ $b$ $c$} denotes the average of $a$, $b$, and $c$. Calculate the value of $\{\{{1 1 0}\} { [0 1] } 0\}$. The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.

Answer: 25
user: Extend the square pattern of $8$ black and $17$ white square tiles by attaching a border of black tiles around the square. What is the ratio of black tiles to white tiles in the extended pattern? The original answer is in the form of a ratio, $a:b$. Please provide the value of $a + b$.

Answer: 49
user: Last summer $30\%$ of the birds living on Town Lake were geese, $25\%$ were swans, $10\%$ were herons, and $35\%$ were ducks. What is the percentage of the birds that were not swans and were geese? If the answer is x%, what is the value of x?

Answer: 40
user: Let $N$ be the second smallest positive integer that is divisible by every positive integer less than $7$. What is the sum of the digits of $N$?

Answer: 3
user: The players on a basketball team made some three-point shots, some two-point shots, and some one-point free throws. They scored as many points with two-point shots as with three-point shots. Their number of successful free throws was one more than their number of successful two-point shots. The team's total score was $61$ points. How many free throws did they make?

Answer: 13
user: Two tangents to a circle are drawn from a point $A$. The points of contact $B$ and $C$ divide the circle into arcs with lengths in the ratio $2 : 3$. What is the degree measure of $\angle{BAC}$? If the answer is $x^\circ$, what is the value of $x$?

Answer: 36
user: A majority of the $30$ students in Ms. Demeanor's class bought pencils at the school bookstore. Each of these students bought the same number of pencils, and this number was greater than $1$. The cost of a pencil in cents was greater than the number of pencils each student bought, and the total cost of all the pencils was $17.71$. What was the cost of a pencil in cents?

Answer: 11
user: Let \(x\) and \(y\) be two-digit positive integers with mean \(60\). Find the maximum value of the ratio \(\frac{x}{y}\). The answer is in the form \(\frac{m}{n}\), please provide the value of \(m + n\).

Answer: 40
user: 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 $30$. Find the value of $A+H$.

Answer: 25
user: At a twins and triplets convention, there were $9$ sets of twins and $6$ sets of triplets, all from different families. Each twin shook hands with all the twins except his/her siblings and with half the triplets. Each triplet shook hands with all the triplets except his/her siblings and with half the twins. How many handshakes took place?

Answer: 441
user: Two real numbers are selected independently and at random from the interval \([-20,10]\). Find the probability that the product of those numbers is greater than zero. Provide your answer as a simplified fraction \(\frac{k}{m}\), and then give the value of \(k + m\).

Answer: 14
user: A pair of standard 6-sided dice is rolled once. The sum of the numbers rolled determines the diameter of a circle. What is the probability that the numerical value of the area of the circle is less than the numerical value of the circle's circumference? Provide your answer as a fraction in simplest form, \(\frac{k}{m}\), and find the value of k + m.

Answer: 13
user: Rectangle $ABCD$ has $AB=6$ and $BC=3$. Point $M$ is chosen on side $AB$ so that $\angle AMD=\angle CMD$. If the degree measure of $\angle AMD$ is $x^\circ$, what is the value of $x$?

Answer: 75
user: 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
user: 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
user: $A$ 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: What is the hundreds digit of \(2011^{2011}\)?

Answer: 6
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: $A$ 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: How many sequences of zeros and ones of length $20$ have all the zeros consecutive, or all the ones consecutive, or both?

Answer: 380
user: 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
user: 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