sungyub/dapo-math-17k-verl huggingface dataset:
user: Right triangle \(ACD\) with a right angle at \(C\) is constructed outwards on the hypotenuse \(\overline{AC}\) of isosceles right triangle \(ABC\) with leg length \(1\), as shown, so that the two triangles have equal perimeters. Find the value of \(\sin(2\angle BAD)\). The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.

Answer: 16
user: The figure below is used to fold into a pyramid and consists of four equilateral triangles erected around a square with an area of 9. What is the length of the dashed path shown?

Answer: 24
user: A square with an area of $40$ is inscribed in a semicircle. What is the area of a square that could be inscribed in the entire circle with the same radius? Provide your answer as a number.

Answer: 100
user: How many ordered triples $(x, y, z)$ of positive integers satisfy the following conditions?
- $\text{lcm}(x, y) = 72$
- $\text{lcm}(x, z) = 600$
- $\text{lcm}(y, z) = 900$

Find the number of such ordered triples.

Answer: 15
user: Line \(\ell_1\) has equation \(3x-2y=1\) and goes through \(A=(-1,-2)\). Line \(\ell_2\) has equation \(y=1\) and meets line \(\ell_1\) at point \(B\). Line \(\ell_3\) has positive slope, goes through point \(A\), and meets \(\ell_2\) at point \(C\). The area of \(\triangle ABC\) is \(3\). The slope of \(\ell_3\) is in the form \(\frac{k}{m}\), where \(\frac{k}{m}\) is a simplified fraction. Please find the value of \(k + m\).

Answer: 7
user: A point \((x, y)\) is to be chosen in the coordinate plane so that it is equally distant from the x-axis, the y-axis, and the line \(x+y = 2\). The original answer is in the form \(k - \sqrt{m}\). Please find the value of \(k + m\).

Answer: 4
user: With $400$ members voting, the House of Representatives defeated a bill. A re-vote, with the same members voting, resulted in passage of the bill by twice the margin by which it was originally defeated. The number voting for the bill on the re-vote was $\frac{12}{11}$ of the number voting against it originally. How many more members voted for the bill the second time than voted for it the first time? Provide your answer as an integer.

Answer: 60
user: Let $S$ be a square of side length $1$. Two points are chosen independently at random on the sides of $S$. The probability that the straight-line distance between the points is at least $\frac{1}{2}$ is $\frac{a-b\pi}{c}$, where $a$, $b$, and $c$ are positive integers and $\gcd(a,b,c)=1$. Find the value of $a+b+c$.

Answer: 59
user: What is the sum of the roots of \(z^{12} = 64\) that have a positive real part? Express your answer in the form \(k\sqrt{m} + n\sqrt{p}\), where k, m, n, and p are integers. Please provide the value of k + m + n + p.

Answer: 11
user: $A$, $B$, and $C$ 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. Find the greatest possible integer value for the mean in pounds of the rocks in the combined piles $B$ and $C$.

Answer: 59
user: Let \(ABCDA'B'C'D'\) be a rectangular prism with \(|AB|=2|BC|\). \(E\) is a point on the edge \([BB']\) satisfying \(|EB'|=6|EB|\). Let \(F\) and \(F'\) be the feet of the perpendiculars from \(E\) at \(\triangle AEC\) and \(\triangle A'EC'\), respectively. If \(m(\widehat{FEF'})=60^{\circ}\), find the value of \(|BC|/|BE|\). The original answer is in the form \(\frac{k}{m}\sqrt{n}\). Please give the value of k + m + n.

Answer: 20
user: If $yz:zx:xy = 1:2:3$, find the ratio $\frac{x}{yz}:\frac{y}{zx}$. The original answer is in the format m:n, please give the value of m + n.

Answer: 5
user: A right rectangular prism has edge lengths \(\log_2 x\), \(\log_3 x\), and \(\log_4 x\). The surface area and volume of the prism are numerically equal. What is the value of \( x \)?

Answer: 576
user: What is the probability that a random arrangement of the letters in the word 'ARROW' will have both R's next to each other? Express your answer in the form of a simplified fraction \(\frac{k}{m}\). Please give the value of k + m.

Answer: 7
user: In $\triangle ABC$, $\angle A = 100^\circ$, $\angle B = 50^\circ$, $\angle C = 30^\circ$, $\overline{AH}$ is an altitude, and $\overline{BM}$ is a median. If the answer is $x^\circ$, what is the value of $x$ for $\angle MHC$?

Answer: 30
user: Find the number of triples \((a,b,c)\) of positive integers which satisfy the simultaneous equations:
\[ ab + bc = 44, \]
\[ ac + bc = 23. \]

Answer: 2
user: $f(x) = \frac{x^5}{5x^4 - 10x^3 + 10x^2 - 5x + 1}$. 

Find the value of:

$$\sum_{i=1}^{2009} f\left(\frac{i}{2009}\right).$$

Answer: 1005
user: Find the number of distinct real roots of the following equation:

$$x^2 + \frac{9x^2}{(x + 3)^2} = 40$$

Provide your answer as an integer.

Answer: 2
user: What is the smallest positive integer $n$ such that $\sqrt{n} - \sqrt{n-1} < 0.01$?

Answer: 2501
user: Jerry has a four-sided die, a six-sided die, and an eight-sided die. Each die is numbered starting at one. Jerry rolls the three dice simultaneously. Find the probability that they all show different numbers. The answer is in the form \frac{k}{m}, where the fraction is in its simplest form. Please find the value of k + m.

Answer: 13
user: The sides of triangle $BAC$ are in the ratio $2:3:4$. $BD$ is the angle-bisector drawn to the shortest side $AC$, dividing it into segments $AD$ and $CD$. If the length of $AC$ is $10$, find the length of the longer segment of $AC$ in the form of a fraction $\frac{r}{s}$, where $r$ and $s$ are integers. Please provide the value of r + s.

Answer: 47
user: How many subsets of the set $\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}$ are there that do not contain 4 consecutive integers?

Answer: 773
user: The length of rectangle $ABCD$ is $5$ inches and its width is $3$ inches. Diagonal $AC$ is divided into three equal segments by points $E$ and $F$. Find the area of triangle $BEF$, expressed in square inches. The original answer is in \frac{k}{m} format, please give the value of k + m.

Answer: 7
user: How many primes $p$ are there such that the number of positive divisors of $p^2+23$ is equal to $14$?

Answer: 1
user: What is the sum of $3 + 3^2 + 3^{2^2} + 3^{2^3} + \dots + 3^{2^{2006}}$ in $\mod 11$?

Answer: 10
user: There are $20$ people in a certain community. $10$ of them speak English, $10$ of them speak German, and $10$ of them speak French. We call a committee a $3$-subset of this community if there is at least one who speaks English, at least one who speaks German, and at least one who speaks French in this subset. At most, how many committees are there in this community?

Answer: 1020
user: After finding the average of $35$ scores, a student carelessly included the average with the $35$ scores and found the average of these $36$ numbers. The original answer is in the format k:m, where k and m are integers. Please find the value of k + m.

Answer: 2
user: The roots of the polynomial $10x^3 - 39x^2 + 29x - 6$ are the height, length, and width of a rectangular box (right rectangular prism). A new rectangular box is formed by lengthening each edge of the original box by 2 units. What is the volume of the new box?

Answer: 30
user: A student is asked to measure $30.0 \, \text{g}$ of methanol $(d=0.7914 \, \text{g/mL \text{ at } 25^{\circ}\text{C}})$ but has only a graduated cylinder with which to measure it. Calculate the volume of methanol, in mL, that the student should use to obtain the required $30.0 \, \text{g}$, and then multiply this volume by 10. What is the resulting value?

Answer: 379
user: Tasha and Amy both pick a number, and they notice that Tasha's number is greater than Amy's number by $12$. They each square their numbers to get a new number and see that the sum of these new numbers is half of $169$. Finally, they square their new numbers and note that Tasha's latest number is now greater than Amy's by $5070$. What was the sum of their original numbers?

Answer: 5
user: How many prime numbers less than $100$ can be represented as the sum of squares of consecutive positive integers?

Answer: 5
user: Find the number of pairs $(m, n)$ of integers which satisfy the equation $m^3 + 6m^2 + 5m = 27n^3 + 9n^2 + 9n + 1$. Provide your answer as an integer.

Answer: 0
user: If the following instructions are carried out by a computer, what is the value of $X$ that will be printed because of instruction 5?

Start $X$ at $3$ and $S$ at $0$.
Increase the value of $X$ by $2$.
Increase the value of $S$ by the value of $X$.
If $S$ is at least $10000$, then go to instruction 5; otherwise, go to instruction 2 and proceed from there.
Print the value of $X$.
Stop.

Answer: 201
user: Rachel和Robert在一个圆形跑道上跑步。Rachel逆时针跑，每\(90\)秒完成一圈，Robert顺时针跑，每\(80\)秒完成一圈。两人同时从起点线出发。在他们开始跑步后的\(10\)分钟到\(11\)分钟之间的某个随机时刻，一位站在跑道内的摄影师拍了一张照片，照片显示了以起点线为中心的四分之一跑道。请问Rachel和Robert都在照片中的概率是多少？原始答案为\(\frac{k}{m}\)的形式，请给出k+m的值。

Answer: 19
user: A game is played with tokens according to the following rules. In each round, the player with the most tokens gives one token to each of the other players and also places one token into a discard pile. The game ends when some player runs out of tokens. Players $A$, $B$, and $C$ start with $15$, $14$, and $13$ tokens, respectively. How many rounds will there be in the game?

Answer: 37
user: Professor Gamble buys a lottery ticket, which requires that he pick six different integers from $1$ through $46$, inclusive. He chooses his numbers so that the sum of the base-ten logarithms of his six numbers is an integer. It so happens that the integers on the winning ticket have the same property—the sum of the base-ten logarithms is an integer. What is the probability that Professor Gamble holds the winning ticket? Provide your answer in the form of a simplified fraction $\frac{k}{m}$, and find the value of $k + m$.

Answer: 5
user: In a small pond, there are eleven lily pads in a row labeled $0$ through $10$. A frog is sitting on pad $1$. When the frog is on pad $N$, where $0<N<10$, it will jump to pad $N-1$ with probability $\frac{N}{10}$ and to pad $N+1$ with probability $1-\frac{N}{10}$. Each jump is independent of the previous jumps. If the frog reaches pad $0$, it will be eaten by a patiently waiting snake. If the frog reaches pad $10$, it will exit the pond, never to return. Find the probability that the frog will escape being eaten by the snake. The original answer is in \frac{m}{n} format, please give the value of m + n.

Answer: 209
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. Find the volume of this cube. The original answer is in the form \(k\sqrt{2} + m\), please give the value of k + m.

Answer: -2
user: The solutions to the equation \((z+6)^8=81\) are connected in the complex plane to form a convex regular polygon, three of whose vertices are labeled \(A, B,\) and \(C\). What is the least possible area of \(\triangle ABC\)? The original answer is in the form \(\frac{k}{m}\sqrt{n}-\frac{k}{m}\), please give the value of \(k + m + n\).

Answer: 7
user: Let $A$ be an even number but not divisible by $10$. What are the last two digits of $A^{20}$? Please provide the answer as a two-digit integer.

Answer: 76
user: Pentagon \(ABCDE\) is inscribed in a circle such that \(ACDE\) is a square with an area of \(12\). What is the largest possible area of pentagon \(ABCDE\)? The original answer is in the form of \(k + m\sqrt{2}\). Please provide the value of \(k + m\).

Answer: 12
user: The circumradius of an acute triangle \(ABC\) is twice the distance from its circumcenter to \(AB\). Given that \(|AC|=2\) and \(|BC|=3\), what is the altitude passing through \(C\)? The original answer is in the form \(\frac{k}{m}\sqrt{n}\), where \(\frac{k}{m}\) is in its simplest form. Please provide the value of \(k + m + n\).

Answer: 31
user: Jerry and Aaron both pick two integers from $1$ to $6$, inclusive, and independently and secretly tell their numbers to Dennis.

Dennis then announces, "Aaron's number is at least three times Jerry's number."

Aaron says, "I still don't know Jerry's number."

Jerry then replies, "Oh, now I know Aaron's number."

What is the sum of their numbers?

Answer: 7
user: Five unit squares are arranged in the coordinate plane with the lower left corner at the origin. The slanted line, extending from \((a,0)\) to \((3,3)\), divides the entire region into two regions of equal area. The original answer is in the form \(\frac{k}{m}\). Please give the value of k + m.

Answer: 5
user: LeRoy and Bernardo went on a week-long trip together and agreed to share the costs equally. Over the week, each of them paid for various joint expenses such as gasoline and car rental. At the end of the trip, it turned out that LeRoy had paid \(A\) dollars and Bernardo had paid \(B\) dollars, where \(A < B\). How many dollars must LeRoy give to Bernardo so that they share the costs equally? Provide your answer in the form of \(\frac{k(B-A)}{m}\), and then find the value of k + m.

Answer: 3
user: How many values of $\theta$ in the interval $0 < \theta \le 2\pi$ satisfy $1 - 3\sin\theta + 5\cos3\theta = 0$? Provide your answer as an integer.

Answer: 6
user: Circles with centers $A$, $B$, and $C$ each have radius $r$, where $1 < r < 2$. The distance between each pair of centers is $2$. If $B'$ is the point of intersection of circle $A$ and circle $C$ which is outside circle $B$, and if $C'$ is the point of intersection of circle $A$ and circle $B$ which is outside circle $C$, find the length $B'C'$. The original answer is in the form $k+\sqrt{m(r^2-1)}$, please give the value of $k+m$.

Answer: 4
user: The length of a rectangle is \(5\) inches and its width is less than \(4\) inches. The rectangle is folded so that two diagonally opposite vertices coincide. If the length of the crease is \(\sqrt{6}\), what is the width of the rectangle in the form \(\sqrt{a}\)? Please provide the value of \(a\).

Answer: 5
user: The average (arithmetic mean) age of a group consisting of doctors and lawyers is 40. If the doctors average 35 years old and the lawyers average 50 years old, what is the ratio of the number of doctors to the number of lawyers? Express your answer as a simplified ratio in the form $m: n$, and then calculate the value of $m + n$.

Answer: 3
user: If $x + \sqrt{x^{2} - 1} + \frac{1}{x - \sqrt{x^{2} - 1}} = 20$, then find the value of $1000 \times \left( x^{2} + \sqrt{x^{4} - 1} + \frac{1}{x^{2} + \sqrt{x^{4} - 1}} \right)$.

Answer: 51005
user: If the radius of a circle is increased $100\%$, by what percentage is the area increased?

Answer: 300
user: \(y\) varies inversely as the square of \(x\). When \(y=16\), \(x=1\). Find the value of \(y\) when \(x=8\). The original answer is in the form \(\frac{k}{m}\), please give the value of \(k + m\).

Answer: 5
user: A parking lot has $16$ spaces in a row. Twelve cars arrive, each of which requires one parking space, and their drivers choose spaces at random from among the available spaces. Auntie Em then arrives in her SUV, which requires $2$ adjacent spaces. What is the probability that she is able to park? Express your answer in the form \(\frac{k}{m}\), where the fraction is simplified, and provide the value of \(k + m\).

Answer: 45
user: $ABCD$ is a cyclic quadrilateral. If $\angle B = \angle D$, $AC \cap BD = \{E\}$, $\angle BCD = 150^\circ$, $|BE| = x$, $|AC| = z$, then find $|ED|$ in terms of $x$ and $z$. The original answer is in the form \(\frac{kx + mz}{n}\), where n cannot be divisible by 2. Please provide the value of \(k + m + n\).

Answer: 1
user: The sequence $(a_n)$ is defined recursively by $a_0=1$, $a_1=\sqrt[19]{2}$, and $a_n=a_{n-1}a_{n-2}^2$ for $n \ge 2$. Find the smallest positive integer $k$ such that the product $a_1a_2 \cdots a_k$ is an integer.

Answer: 17
user: Let $D$ be a point on the side $[BC]$ of $\triangle ABC$ such that $|BD|=2$ and $|DC|=6$. If $|AB|=4$ and $m(\widehat{ACB})=20^\circ$, then what is the value of $m(\widehat{BAD})$ in degrees?

Answer: 20
user: Call a positive integer *monotonous* if it is a one-digit number or its digits, when read from left to right, form either a strictly increasing or a strictly decreasing sequence. For example, 3, 23578, and 987620 are monotonous, but 88, 7434, and 23557 are not. Find the total number of monotonous positive integers.

Answer: 1524
user: Circle \(C_1\) and \(C_2\) each have radius \(1\), and the distance between their centers is \(\frac{1}{2}\). Circle \(C_3\) is the largest circle internally tangent to both \(C_1\) and \(C_2\). Circle \(C_4\) is internally tangent to both \(C_1\) and \(C_2\) and externally tangent to \(C_3\). The radius of \(C_4\) can be expressed in the form \(\frac{k}{m}\). Please find the value of \(k + m\).

Answer: 31
user: If \(q_1(x)\) and \(r_1\) are the quotient and remainder, respectively, when the polynomial \(x^8\) is divided by \(x + \frac{1}{2}\), and if \(q_2(x)\) and \(r_2\) are the quotient and remainder, respectively, when \(q_1(x)\) is divided by \(x + \frac{1}{2}\), find the value of \(r_2\). The original answer is in the form \(-\frac{k}{m}\). Please provide the value of \(k + m\).

Answer: 17
user: The circle with radius $4$ passing through points $A$ and $B$ is tangent to another circle with radius $3$ passing through points $A$ and $C$, where the distance $|AB| = 2$. If the line $BC$ is tangent to the second circle, what is the value of $|BC|^2$?

Answer: 7
user: Let $S$ be the set of points $(a,b)$ in the coordinate plane, where each of $a$ and $b$ may be $-1$, $0$, or $1$. How many distinct lines pass through at least two members of $S$?

Answer: 20
user: Find the ratio of the area of a square inscribed in a semicircle to the area of the square inscribed in the entire circle. Express your answer in the form of a ratio $a: b$, and find the value of $a + b$.

Answer: 7
user: Four circles, no two of which are congruent, have centers at $A$, $B$, $C$, and $D$, and points $P$ and $Q$ lie on all four circles. The radius of circle $A$ is $\frac{5}{8}$ times the radius of circle $B$, and the radius of circle $C$ is $\frac{5}{8}$ times the radius of circle $D$. Furthermore, $AB = CD = 39$ and $PQ = 48$. Let $R$ be the midpoint of $\overline{PQ}$. Find the value of $AR+BR+CR+DR$.

Answer: 192
user: A circle centered at \(A\) with a radius of 1 and a circle centered at \(B\) with a radius of 4 are externally tangent. A third circle is tangent to the first two and to one of their common external tangents as shown. The radius of the third circle is in the simplest fractional form \(\frac{k}{m}\). Please find the value of \(k + m\).

Answer: 13
user: Four coins are placed in a line. A passerby walks by and flips each coin, stopping if she ever obtains two adjacent heads. If the passerby manages to flip all four coins, find the number of possible head-tail combinations for her four flips.

Answer: 8
user: Let $a, b,$ and $c$ be positive integers with $a \geq b \geq c$ such that \[ \begin{align*} a^2 - b^2 - c^2 + ab &= 2011 \\ a^2 + 3b^2 + 3c^2 - 3ab - 2ac - 2bc &= -1997 \end{align*} \] What is the value of $a$?

Answer: 253
user: The roots of the equation \(2\sqrt{x} + 2x^{-\frac{1}{2}} = 5\) can be found by solving the quadratic equation \(kx^2 + mx + n = 0\). What are the values of \(k\), \(m\), and \(n\) in the equation? Please provide the sum of \(k\), \(m\), and \(n\).

Answer: -9
user: The circle $C_1$ with radius $6$ and the circle $C_2$ with radius $8$ are externally tangent to each other at $A$. The circle $C_3$, which is externally tangent to both $C_1$ and $C_2$, has a radius of $21$. The common tangent of $C_1$ and $C_2$ that passes through $A$ meets $C_3$ at points $B$ and $C$. The original answer is in the form $k\sqrt{m}$, please find the value of $k + m$.

Answer: 27
user: What is the area enclosed by the graph of $|3x| + |4y| = 12$?

Answer: 24
user: Integers \(a\), \(b\), \(c\), and \(d\), not necessarily distinct, are chosen independently and at random from \(0\) to \(2007\), inclusive. Find the probability that \(ad - bc\) is even. The original answer is in the form \(\frac{k}{m}\), where \(\frac{k}{m}\) is in simplest terms. Please calculate the value of \(k + m\).

Answer: 13
user: A class of $50$ students took an exam with $4$ questions. At least $1$ of any $40$ students gave exactly $3$ correct answers, at least $2$ of any $40$ gave exactly $2$ correct answers, and at least $3$ of any $40$ gave exactly $1$ correct answer. At least $4$ of any $40$ students gave exactly $4$ wrong answers. What is the least number of students who gave an odd number of correct answers?

Answer: 24
user: What is the greatest number of consecutive integers whose sum is $45$? Provide your answer as an integer.

Answer: 90
user: Three non-overlapping regular plane polygons, at least two of which are congruent, all have sides of length $1$. The polygons meet at a point $A$ such that the sum of the three interior angles at $A$ is $360^\circ$. Thus, the three polygons form a new polygon with $A$ as an interior point. What is the largest possible perimeter that this polygon can have?

Answer: 21
user: A regular polygon with $n$ sides is inscribed in a circle with radius $R$. The area of the polygon is $3R^2$. Determine the value of $n$.

Answer: 12
user: How many different sentences with two words can be written using all letters of the word $\text{YARI\c{S}MA}$?

(The Turkish word $\text{YARI\c{S}MA}$ means $\text{CONTEST}$. It will produce the same result.)

Find the total number of different sentences.

Answer: 15120
user: Let \(a_n\) denote the angle opposite to the side of length \(4n^2\) units in an integer right-angled triangle with side lengths \(4n^2\), \(4n^4+1\), and \(4n^4-1\), where \(n \in \mathbb{N}\). Find the value of:
\[
\lim_{p \to \infty} \sum_{n=1}^p a_n
\]
The original answer is in the format \(\frac{k\pi}{m}\). Please give the value of \(k + m\).

Answer: 3
user: To the nearest thousandth, $\log_{10}2$ is $0.301$ and $\log_{10}3$ is $0.477$. Find the best approximation of $\log_5 10$ in the form of a fraction \(\frac{k}{m}\). Please provide the value of k + m.

Answer: 17
user: How many pairs of positive integers \((a,b)\) with \(a \geq b\) are there such that \(a^2+b^2\) divides both \(a^3+b\) and \(a+b^3\)?

Answer: 1
user: How many ordered pairs of positive integers $(b,c)$ exist where both $x^2+bx+c=0$ and $x^2+cx+b=0$ do not have distinct, real solutions?

Answer: 6
user: The sides of a triangle are \(25, 39,\) and \(40\). What is the diameter of the circumscribed circle? The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.

Answer: 128
user: For $n$ a positive integer, let $R(n)$ be the sum of the remainders when $n$ is divided by $2$, $3$, $4$, $5$, $6$, $7$, $8$, $9$, and $10$. For example, $R(15) = 1+0+3+0+3+1+7+6+5=26$. Find the number of two-digit positive integers $n$ such that $R(n) = R(n+1)$.

Answer: 2
user: If the tangents of all interior angles of a triangle are integers, what is the sum of these integers?

Answer: 6
user: At a dance party, a group of boys and girls exchange dances as follows: one boy dances with $5$ girls, a second boy dances with $6$ girls, and so on, with the last boy dancing with all the girls. If $b$ represents the number of boys and $g$ the number of girls, find the relation between $b$ and $g$. The original answer is in the form $b = mg + n$. Please find the value of $m + n$.

Answer: -3
user: In triangle \(ABC\), the median from vertex \(A\) is perpendicular to the median from vertex \(B\). If the lengths of sides \(AC\) and \(BC\) are \(6\) and \(7\) respectively, find the length of side \(AB\). The original answer is in the form \(\sqrt{k}\). Please find the value of \(k\).

Answer: 17
user: What is the sum of cubes of real roots of the equation $x^3-2x^2-x+1=0$?

Answer: 11