sungyub/dapo-math-17k-verl huggingface dataset:
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: What is the smallest positive integer that is neither prime nor square and that has no prime factor less than $50$?

Answer: 3127
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: 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
user: A square with area $4$ is inscribed in a square with area $5$, with each vertex of the smaller square on a side of the larger square. A vertex of the smaller square divides a side of the larger square into two segments, one of length $a$, and the other of length $b$. The original answer is in the form \frac{k}{m}, where \frac{k}{m} is a fully reduced fraction. Please find the value of k + m.

Answer: 3
user: When counting from $3$ to $201$, $53$ is the $51^{st}$ number counted. When counting backwards from $201$ to $3$, $53$ is the $n^{th}$ number counted. What is the value of $n$?

Answer: 149
user: $A$ softball team played ten games, scoring $1$, $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, and $10$ runs. They lost by one run in exactly five games. In each of their other games, they scored twice as many runs as their opponent. How many total runs did their opponents score?

Answer: 45
user: Joey and his five brothers are ages $3$, $5$, $7$, $9$, $11$, and $13$. One afternoon two of his brothers whose ages sum to $16$ went to the movies, two brothers younger than $10$ went to play baseball, and Joey and the $5$-year-old stayed home. How old is Joey?

Answer: 11
user: The number in each box below is the product of the numbers in the two boxes that touch it in the row above. For example, $30 = 6\times5$. What is the missing number in the top row?

unitsize(0.8cm); draw((-1,0)--(1,0)--(1,-2)--(-1,-2)--cycle); draw((-2,0)--(0,0)--(0,2)--(-2,2)--cycle); draw((0,0)--(2,0)--(2,2)--(0,2)--cycle); draw((-3,2)--(-1,2)--(-1,4)--(-3,4)--cycle); draw((-1,2)--(1,2)--(1,4)--(-1,4)--cycle); draw((1,2)--(1,4)--(3,4)--(3,2)--cycle); label("600",(0,-1)); label("30",(-1,1)); label("6",(-2,3)); label("5",(0,3)); [/asy]

Find the missing number in the top row.

Answer: 4
user: A student must choose a program of four courses from a menu of courses consisting of English, Algebra, Geometry, History, Art, and Latin. This program must contain English and at least one mathematics course. In how many ways can this program be chosen?

Answer: 9
user: Six points are equally spaced around a circle of radius \(1\). Three of these points are the vertices of a triangle that is neither equilateral nor isosceles. What is the area of this triangle? The original answer is in \(\frac{\sqrt{k}}{m}\) format, please give the value of k + m.

Answer: 5
user: Jo and Blair take turns counting from $1$ to one more than the last number said by the other person. Jo starts by saying $``1"$, so Blair follows by saying $``1, 2"$. Jo then says $``1, 2, 3"$, and so on. What is the $53^{{rd}}$ number said?

Answer: 8
user: Line \(l_1\) has equation \(3x - 2y = 1\) and goes through \(A = (-1, -2)\). Line \(l_2\) has equation \(y = 1\) and meets line \(l_1\) at point \(B\). Line \(l_3\) has positive slope, goes through point \(A\), and meets \(l_2\) at point \(C\). The area of \(\triangle ABC\) is \(3\). The original answer for the slope of \(l_3\) is in the form \(\frac{k}{m}\). Please find the value of \(k + m\).

Answer: 7
user: In $\triangle ABC$, $AB=AC=28$ and $BC=20$. Points $D,E,$ and $F$ are on sides $\overline{AB}$, $\overline{BC}$, and $\overline{AC}$, respectively, such that $\overline{DE}$ and $\overline{EF}$ are parallel to $\overline{AC}$ and $\overline{AB}$, respectively. What is the perimeter of parallelogram $ADEF$?

Answer: 56
user: What is the sum of the exponents of the prime factors of the square root of the largest perfect square that divides $12!$?

Answer: 8
user: $A$ flower bouquet contains pink roses, red roses, pink carnations, and red carnations. One third of the pink flowers are roses, three fourths of the red flowers are carnations, and six tenths of the flowers are pink. What is the percentage of the flowers that are carnations? Provide your answer as a percentage.

Answer: 70
user: Let $S$ be the set of positive integers $n$ for which $\frac{1}{n}$ has the repeating decimal representation $0.\overline{ab} = 0.ababab\cdots,$ with $a$ and $b$ different digits. Find the sum of the elements of $S$.

Answer: 143
user: Alex has $75$ red tokens and $75$ blue tokens. There is a booth where Alex can give two red tokens and receive in return a silver token and a blue token and another booth where Alex can give three blue tokens and receive in return a silver token and a red token. Alex continues to exchange tokens until no more exchanges are possible. How many silver tokens will Alex have at the end?

Answer: 103
user: Triangle \(ABC\) is equilateral with \(AB=1\). Points \(E\) and \(G\) are on \(\overline{AC}\) and points \(D\) and \(F\) are on \(\overline{AB}\) such that both \(\overline{DE}\) and \(\overline{FG}\) are parallel to \(\overline{BC}\). Furthermore, triangle \(ADE\) and trapezoids \(DFGE\) and \(FBCG\) all have the same perimeter. Find the value of \(DE+FG\). The original answer is in \(\frac{k}{m}\) format, please give the value of \(k + m\).

Answer: 34
user: Ted's grandfather used his treadmill on $3$ days this week. He went $2$ miles each day. On Monday he jogged at a speed of $5$ miles per hour. He walked at the rate of $3$ miles per hour on Wednesday and at $4$ miles per hour on Friday. If Grandfather had always walked at $4$ miles per hour, he would have spent less time on the treadmill. How many minutes less did he spend?

Answer: 4
user: The angles in a particular triangle are in arithmetic progression, and the side lengths are $4,5,x$. The sum of the possible values of $x$ equals $a+\sqrt{b}+\sqrt{c}$ where $a, b$, and $c$ are positive integers. What is the value of $a+b+c$?

Answer: 36
user: In $\triangle ABC$, $AB=AC=28$ and $BC=20$. Points $D,E,$ and $F$ are on sides $\overline{AB}$, $\overline{BC}$, and $\overline{AC}$, respectively, such that $\overline{DE}$ and $\overline{EF}$ are parallel to $\overline{AC}$ and $\overline{AB}$, respectively. Find the perimeter of parallelogram $ADEF$.

Answer: 56
user: Let \(S\) be the set of sides and diagonals of a regular pentagon. A pair of elements of \(S\) are selected at random without replacement. Find the probability that the two chosen segments have the same length. The original answer is in \(\frac{k}{m}\) format, where \(\frac{k}{m}\) is the simplified form of the probability. Please give the value of \(k + m\).

Answer: 13
user: Cities $A$, $B$, $C$, $D$, and $E$ are connected by roads $\widetilde{AB}$, $\widetilde{AD}$, $\widetilde{AE}$, $\widetilde{BC}$, $\widetilde{BD}$, $\widetilde{CD}$, and $\widetilde{DE}$. Find the number of different routes from $A$ to $B$ that use each road exactly once. (Such a route will necessarily visit some cities more than once.)

Answer: 16
user: At the $2013$ Winnebago County Fair a vendor is offering a "fair special" on sandals. If you buy one pair of sandals at the regular price of $＄50$, you get a second pair at a $40\%$ discount, and a third pair at half the regular price. Javier took advantage of the "fair special" to buy three pairs of sandals. What is the percentage of the $＄150$ regular price that he saved? Provide your answer as a percentage.

Answer: 30
user: How many three-digit numbers are not divisible by $5$, have digits that sum to less than $20$, and have the first digit equal to the third digit?

Answer: 60
user: Jo and Blair take turns counting from $1$ to one more than the last number said by the other person. Jo starts by saying “$1$”, so Blair follows by saying “$1 , 2$”. Jo then says “$1,2,3$”, and so on. What is the $53^{rd}$ number said?

Answer: 8
user: Let points $A = (0,0) , \ B = (1,2), \ C = (3,3),$ and $D = (4,0)$. Quadrilateral $ABCD$ is cut into equal area pieces by a line passing through $A$. This line intersects $\overline{CD}$ at point $\left (\frac{p}{q}, \frac{r}{s} \right )$, where these fractions are in lowest terms. Find the value of $p + q + r + s$.

Answer: 58
user: The internal angles of quadrilateral $ABCD$ form an arithmetic progression. Triangles $ABD$ and $DCB$ are similar with $\angle DBA = \angle DCB$ and $\angle ADB = \angle CBD$. Moreover, the angles in each of these two triangles also form an arithmetic progression. In degrees, what is the largest possible sum of the two largest angles of $ABCD$? If the answer is \(x^\circ\), what is the value of \(x\)?

Answer: 240
user: The sequence 
$\log_{12}{162}$, $\log_{12}{x}$, $\log_{12}{y}$, $\log_{12}{z}$, $\log_{12}{1250}$ 
is an arithmetic progression. What is the value of $x$?

Answer: 270
user: Two non-decreasing sequences of nonnegative integers have different first terms. Each sequence has the property that each term beginning with the third is the sum of the previous two terms, and the seventh term of each sequence is $N$. Find the smallest possible value of $N$.

Answer: 104
user: A wire is cut into two pieces, one of length \(a\) and the other of length \(b\). The piece of length \(a\) is bent to form an equilateral triangle, and the piece of length \(b\) is bent to form a regular hexagon. The triangle and the hexagon have equal area. The original answer is in the form \(\frac{\sqrt{p}}{n}\) where the fraction is in its simplest form. Please give the value of \(p + n\).

Answer: 8
user: Rabbits Peter and Pauline have three offspring—Flopsie, Mopsie, and Cotton-tail. These five rabbits are to be distributed to four different pet stores so that no store gets both a parent and a child. It is not required that every store gets a rabbit. In how many different ways can this be done?

Answer: 204
user: In triangle $ABC$, medians $AD$ and $CE$ intersect at $P$, $PE=1.5$, $PD=2$, and $DE=2.5$. If the area of $AEDC$ is $x$, what is the value of $10x$?

Answer: 135
user: $A$, $B$, $C$ are three piles of rocks. The mean weight of the rocks in $A$ is $40$ pounds, the mean weight of the rocks in $B$ is $50$ pounds, the mean weight of the rocks in the combined piles $A$ and $B$ is $43$ pounds, and the mean weight of the rocks in the combined piles $A$ and $C$ is $44$ pounds. What is the greatest possible integer value for the mean in pounds of the rocks in the combined piles $B$ and $C$?

Answer: 59
user: Daphne is visited periodically by her three best friends: Alice, Beatrix, and Claire. Alice visits every third day, Beatrix visits every fourth day, and Claire visits every fifth day. All three friends visited Daphne yesterday. How many days of the next $365$-day period will exactly two friends visit her?

Answer: 54
user: Let \(a, b,\) and \(c\) be real numbers such that 
 
\(a+b+c=2,\) and 
\(a^2+b^2+c^2=12\) 
 
Find the difference between the maximum and minimum possible values of \(c\). The original answer is in \(\frac{m}{n}\) format, please give the value of \(m+n\).

Answer: 19
user: Let points $A = (0, 0)$, $B = (1, 2)$, $C=(3, 3)$, and $D = (4, 0)$. Quadrilateral $ABCD$ is cut into equal area pieces by a line passing through $A$. This line intersects $\overline{CD}$ at point $\left(\frac{p}{q}, \frac{r}{s}\right)$, where these fractions are in lowest terms. Find the value of $p+q+r+s$.

Answer: 58
user: The number $2013$ has the property that its units digit is the sum of its other digits, that is $2+0+1=3$. Find how many integers less than $2013$ but greater than $1000$ have this property.

Answer: 46
user: Six spheres of radius \(1\) are positioned so that their centers are at the vertices of a regular hexagon of side length \(2\). The six spheres are internally tangent to a larger sphere whose center is the center of the hexagon. An eighth sphere is externally tangent to the six smaller spheres and internally tangent to the larger sphere. The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.

Answer: 5
user: Isabella uses one-foot cubical blocks to build a rectangular fort that is $12$ feet long, $10$ feet wide, and $5$ feet high. The floor and the four walls are all one foot thick. How many blocks does the fort contain?

Answer: 280
user: In base $10$, the number $2013$ ends in the digit $3$. In base $9$, on the other hand, the same number is written as $(2676)_{9}$ and ends in the digit $6$. For how many positive integers $b$ does the base-$b$-representation of $2013$ end in the digit $3$?

Answer: 13
user: The real numbers \(c, b, a\) form an arithmetic sequence with \(a \geq b \geq c \geq 0\). The quadratic \(ax^2+bx+c\) has exactly one root. The original answer is in the form \(-k+\sqrt{m}\). Please give the value of \(k + m\).

Answer: 5
user: In $\bigtriangleup ABC$, $AB = 86$, and $AC = 97$. A circle with center $A$ and radius $AB$ intersects $\overline{BC}$ at points $B$ and $X$. Moreover, $\overline{BX}$ and $\overline{CX}$ have integer lengths. What is the length of $BC$?

Answer: 61
user: In triangle $ABC$, $AB=13$, $BC=14$, and $CA=15$. Distinct points $D$, $E$, and $F$ lie on segments $\overline{BC}$, $\overline{CA}$, and $\overline{DE}$, respectively, such that $\overline{AD}\perp\overline{BC}$, $\overline{DE}\perp\overline{AC}$, and $\overline{AF}\perp\overline{BF}$. The length of segment $\overline{DF}$ can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Answer: 21
user: The number $2013$ is expressed in the form 
$2013 = \frac {a_1!a_2!...a_m!}{b_1!b_2!...b_n!}$, 
where $a_1 \ge a_2 \ge \cdots \ge a_m$ and $b_1 \ge b_2 \ge \cdots \ge b_n$ are positive integers and $a_1 + b_1$ is as small as possible. Find the value of $|a_1 - b_1|$.

Answer: 2
user: A group of $12$ pirates agree to divide a treasure chest of gold coins among themselves as follows. The $k^{{th}}$ pirate to take a share takes $\frac{k}{12}$ of the coins that remain in the chest. The number of coins initially in the chest is the smallest number for which this arrangement will allow each pirate to receive a positive whole number of coins. How many coins does the $12^{{th}}$ pirate receive?

Answer: 1925
user: Samantha lives $2$ blocks west and $1$ block south of the southwest corner of City Park. Her school is $2$ blocks east and $2$ blocks north of the northeast corner of City Park. On school days she bikes on streets to the southwest corner of City Park, then takes a diagonal path through the park to the northeast corner, and then bikes on streets to school. If her route is as short as possible, how many different routes can she take?

Answer: 18
user: Six spheres of radius \(1\) are positioned so that their centers are at the vertices of a regular hexagon of side length \(2\). The six spheres are internally tangent to a larger sphere whose center is the center of the hexagon. An eighth sphere is externally tangent to the six smaller spheres and internally tangent to the larger sphere. The radius of this eighth sphere can be expressed in the form \(\frac{k}{m}\), where \(\frac{k}{m}\) is in simplest terms. Please find the value of \(k + m\).

Answer: 5
user: The regular octagon $ABCDEFGH$ has its center at $J$. Each of the vertices and the center are to be associated with one of the digits $1$ through $9$, with each digit used once, in such a way that the sums of the numbers on the lines $AJE$, $BJF$, $CJG$, and $DJH$ are all equal. In how many ways can this be done?

Answer: 1152
user: A palindrome is a nonnegative integer number that reads the same forwards and backwards when written in base $10$ with no leading zeros. A $6$-digit palindrome $n$ is chosen uniformly at random. Find the probability that \(\frac{n}{11}\) is also a palindrome. The original answer is in the form of \(\frac{k}{m}\), where \(\frac{k}{m}\) is a simplified fraction. Please give the value of \(k + m\).

Answer: 41
user: Toothpicks are used to make a grid that is $60$ toothpicks long and $32$ toothpicks wide. How many toothpicks are used altogether?

Answer: 3932
user: In $\triangle ABC$, $AB = 86$, and $AC=97$. A circle with center $A$ and radius $AB$ intersects $\overline{BC}$ at points $B$ and $X$. Moreover, $\overline{BX}$ and $\overline{CX}$ have integer lengths. What is the integer length of $BC$?

Answer: 61
user: In triangle $ABC$, $AB=13$, $BC=14$, and $CA=15$. Distinct points $D$, $E$, and $F$ lie on segments $\overline{BC}$, $\overline{CA}$, and $\overline{DE}$, respectively, such that $\overline{AD}\perp\overline{BC}$, $\overline{DE}\perp\overline{AC}$, and $\overline{AF}\perp\overline{BF}$. The length of segment $\overline{DF}$ can be written as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find the value of $m+n$.

Answer: 21
user: A positive integer $n$ is nice if there is a positive integer $m$ with exactly four positive divisors (including $1$ and $m$) such that the sum of the four divisors is equal to $n$. Find how many numbers in the set $\{ 2010,2011,2012,\cdots,2019 \}$ are nice.

Answer: 1
user: A ball with diameter 4 inches starts at point A to roll along the track shown. The track is comprised of 3 semicircular arcs whose radii are \(R_1 = 100\) inches, \(R_2 = 60\) inches, and \(R_3 = 80\) inches, respectively. The ball always remains in contact with the track and does not slip. If the answer is \(x\pi\), what is the value of \(x\) for the distance the center of the ball travels over the course from A to B?

Answer: 238
user: For certain real numbers \(a\), \(b\), and \(c\), the polynomial \[g(x) = x^3 + ax^2 + x + 10\] has three distinct roots, and each root of \(g(x)\) is also a root of the polynomial \[f(x) = x^4 + x^3 + bx^2 + 100x + c.\] Find the value of \(f(1)\). The original answer is in the form \(m - n\), please give the value of \(m + n\).

Answer: 7007
user: Three numbers $a, b, c$, none zero, form an arithmetic progression. Increasing $a$ by $1$ or increasing $c$ by $2$ results in a geometric progression. Find the value of $b$.

Answer: 12