meetforever/GRPO_CL huggingface dataset:
user: If the distinct non-zero numbers $x(y - z)$, $y(z - x)$, $z(x - y)$ form a geometric progression with common ratio $r$, find the equation that $r$ satisfies. The equation is in the form $ax^2 + bx + c = 0$. Please give the value of $a + b + c$.
Answer: 3
user: How many pairs of real numbers $(x, y)$ are there such that \( x^{4} - 2^{-y^{2}} x^{2} - \lfloor x^{2} \rfloor + 1 = 0 \), where \( \lfloor a \rfloor \) denotes the greatest integer not exceeding \( a \)?
Answer: 2
user: How many ordered triples $(a, b, c)$ of non-zero real numbers have the property that each number is the product of the other two?
Answer: 4
user: A laser is placed at the point \((3,5)\). The laser beam travels in a straight line. Larry wants the beam to hit and bounce off the \(y\)-axis, then hit and bounce off the \(x\)-axis, and finally hit the point \((7,5)\). Find the total distance the beam will travel along this path. The original answer is in the form \(k\sqrt{m}\), please give the value of \(k + m\).
Answer: 12
user: In $\triangle ABC$ with a right angle at $C$, the altitude $CH$ and the median $CM$ trisect the right angle. If the area of $\triangle CHM$ is $K$, find the area of $\triangle ABC$ in terms of $K$. The answer is in the form of $aK$, please provide the value of $a$.
Answer: 4
user: A fair standard six-sided die is tossed three times. Given that the sum of the first two tosses equals the third, find the probability that at least one $2$ is tossed. The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
Answer: 23
user: How many positive integers $n$ are there such that $n!(2n+1)$ and $221$ are relatively prime? Provide your answer as an integer.
Answer: 10
user: A house worth $9000 is sold by Mr. A to Mr. B at a 10\% loss. Mr. B sells the house back to Mr. A at a 10\% gain. How much money does Mr. A lose in total from these two transactions?
Answer: 810
user: How many polynomial functions $f$ of degree $\ge 1$ satisfy the following conditions?
\[
f(x^2) = [f(x)]^2 = f(f(x))
\]
Provide your answer as an integer.
Answer: 1
user: A circle with a radius of $10$ inches has its center at the vertex $C$ of an equilateral triangle $\triangle ABC$ and passes through the other two vertices. The side $AC$ extended through $C$ intersects the circle at $D$. Determine the measure of angle $\angle ADB$ in degrees. If the answer is $x^\circ$, what is the value of $x$?
Answer: 30
user: After rationalizing the numerator of \(\frac{\sqrt{3}-\sqrt{2}}{\sqrt{3}}\), the denominator can be expressed in the form \( k + \sqrt{m} \). Please find the value of \( k + m \).
Answer: 9
user: Call a fraction $\frac{a}{b}$, not necessarily in the simplest form, *special* if $a$ and $b$ are positive integers whose sum is 15. Find how many distinct integers can be written as the sum of two, not necessarily different, special fractions.
Answer: 11
user: Brenda and Sally run in opposite directions on a circular track, starting at diametrically opposite points. They first meet after Brenda has run $100$ meters. They next meet after Sally has run $150$ meters past their first meeting point. Each girl runs at a constant speed. What is the length of the track in meters?
Answer: 350
user: For how many ordered pairs of positive integers \((m,n)\), does \(m \cdot n\) divide \(2008 \cdot 2009 \cdot 2010\)? Express your answer as a product of powers of prime numbers in the form \(2^k \cdot 3^m \cdot 5^n\). Find the value of \(k + m + n\).
Answer: 9
user: In the mobile below, the two cross beams and the seven supporting strings are all massless. The hanging objects are $M_1 = 400 \, \text{g}$, $M_2 = 200 \, \text{g}$, and $M_4 = 500 \, \text{g}$. What is the value of $M_3$ for the system to be in static equilibrium? If the answer is $x \, \text{g}$, what is the value of $x$?
Answer: 700
user: There are two positive numbers that may be inserted between $3$ and $9$ such that the first three numbers form a geometric progression, while the last three numbers form an arithmetic progression. If the sum of those two positive numbers is $x$, find the value of $100x$.
Answer: 1125
user: A solid cube has a side length of $3$ inches. A $2$-inch by $2$-inch square hole is cut into the center of each face. The edges of each cut are parallel to the edges of the cube, and each hole goes all the way through the cube. What is the volume, in cubic inches, of the remaining solid?
Answer: 7
user: Let $f(x) = x^2 + 3x + 2$ and let $S$ be the set of integers $\{0, 1, 2, \dots, 25\}$. Find the number of members $s$ of $S$ such that $f(s)$ has remainder zero when divided by 6.
Answer: 17
user: A point in the plane, both of whose rectangular coordinates are integers with absolute values less than or equal to four, is chosen at random, with all such points having an equal probability of being chosen. Find the probability that the distance from the point to the origin is at most two units. Express your answer in the form \frac{k}{m} where k and m are coprime, and give the value of k + m.
Answer: 94
user: How many positive integers $n$ are there such that $n$ is a multiple of $5$, and the least common multiple of $5!$ and $n$ equals $5$ times the greatest common divisor of $10!$ and $n$? Provide your answer as an integer.
Answer: 48
user: Right triangle $ABC$ has side lengths $BC=6$, $AC=8$, and $AB=10$. A circle centered at $O$ is tangent to line $BC$ at $B$ and passes through $A$. A circle centered at $P$ is tangent to line $AC$ at $A$ and passes through $B$. The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
Answer: 47
user: A rock is dropped off a cliff of height \(h\). As it falls, a camera takes several photographs at random intervals. In each picture, the distance the rock has fallen is measured. Let the average (expected value) of all these distances be \(kh\). If the number of photographs taken is huge, find \(k\). The original answer is in the form \(\frac{m}{n}\), where m and n are integers. Please give the value of m + n.
Answer: 4
user: Find the number of all positive integers $n$ such that $n + s(n) = 2016$, where $s(n)$ is the sum of all digits of $n$.
Answer: 2
user: The graph of $y = x^6 - 10x^5 + 29x^4 - 4x^3 + ax^2$ lies above the line $y = bx + c$ except at three values of $x$, where the graph and the line intersect. What is the largest of those values?
Answer: 4
user: Let \[ S_1 = \{ (x,y)\ | \ \log_{10} (1 + x^2 + y^2) \le 1 + \log_{10}(x + y)\} \] and \[ S_2 = \{ (x,y)\ | \ \log_{10} (2 + x^2 + y^2) \le 2 + \log_{10}(x + y)\}. \] Find the ratio of the area of $S_2$ to the area of $S_1$.
Answer: 102
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 third red light and the 21st red light? If the answer is x feet, what is the value of 10x?

**Note:** 1 foot is equal to 12 inches.
Answer: 225
user: In the given diagram, semi-circles are constructed on diameters $\overline{AB}$, $\overline{AC}$, and $\overline{CB}$, such that they are mutually tangent. If $\overline{CD} \bot \overline{AB}$, find the ratio of the shaded area to the area of a circle with $\overline{CD}$ as its radius. The original answer is in the form \frac{k}{m}, please give the value of k + m.
Answer: 5
user: Let \(a\) and \(b\) be positive reals such that:
\[ a = 1 + \frac{a}{b} \]
\[ b = 3 + \frac{4 + a}{b - 2} \]
The original answer is in the form \(\sqrt[3]{k}\). Please find the value of \(k\).
Answer: 2
user: For how many integers $n$, are there four distinct real numbers satisfying the equation $|x^2 - 4x - 7| = n$?
Answer: 10
user: If $x, y > 0$, $\log_y x + \log_x y = \frac{10}{3}$ and $xy = 144$, then find the value of $\frac{x+y}{2}$. The original answer is in the form of k\sqrt{m}, please give the value of k + m.
Answer: 16
user: $n$ integers are arranged along a circle such that each number is equal to the absolute value of the difference of the two numbers following it in the clockwise direction. If the sum of all numbers is $278$, find how many different values $n$ can take.
Answer: 2
user: The sum of four two-digit numbers is $221$. None of the eight digits is $0$ and no two of them are the same. Which digit is **not** included among the eight digits?
Answer: 4
user: Points \(A\), \(B\), and \(C\) are on a circle with radius \(r\). They are positioned such that \(AB = AC\), \(AB > r\), and the length of the minor arc \(BC\) is \(r\). If angles are measured in radians, find the value of \(\frac{AB}{BC}\). The original answer is in the form \(\frac{k}{m}\csc\left(\frac{n}{p}\right)\), where \(\frac{k}{m}\) is in simplest form. Please provide the value of \(k + m + n + p\).
Answer: 8
user: Suppose \(S\) cans of soda can be purchased from a vending machine for \(Q\) quarters. Find the expression that describes the number of cans of soda that can be purchased for \(D\) dollars, where \(1\) dollar is worth \(4\) quarters. The original answer is in \(\frac{kDS}{mQ}\) format, please give the value of k + m + n.
Answer: 5
user: Let $ABCD$ be a trapezoid with $AB$ and $CD$ parallel, $\angle D = 2 \angle B$, $AD = 5$, and $CD = 2$. Find the length of $AB$.
Answer: 7
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 \(k + m\sqrt{n}\), please give the value of \(k + m + n\).
Answer: 14
user: Each circle in an infinite sequence with decreasing radii is tangent externally to the one following it and to both sides of a given right angle. Find the ratio of the area of the first circle to the sum of areas of all other circles in the sequence in the form $(k+m\sqrt{2}):1$. What is the value of $k+m$?
Answer: 28
user: How many integers $n$ with $0 \leq n < 840$ are there such that $840$ divides $n^8-n^4+n-1$? Provide your answer as an integer.
Answer: 2
user: How many non-negative integer triples $(m,n,k)$ are there such that $m^3 - n^3 = 9^k + 123$?
Answer: 1
user: Consider sequences of positive real numbers of the form $x, 2000, y, \ldots,$ in which every term after the first is 1 less than the product of its two immediate neighbors. Find the number of different values of $x$ for which the term 2001 appears somewhere in the sequence.
Answer: 4
user: In the five-sided star shown, the letters $A,B,C,D,$ and $E$ are replaced by the numbers $3,5,6,7,$ and $9$, although not necessarily in this order. The sums of the numbers at the ends of the line segments $\overline{AB},\overline{BC},\overline{CD},\overline{DE},$ and $\overline{EA}$ form an arithmetic sequence, although not necessarily in this order. What is the middle term of the arithmetic sequence?
Answer: 12
user: The polynomial \((x+y)^9\) is expanded in decreasing powers of \(x\). The second and third terms have equal values when evaluated at \(x=p\) and \(y=q\), where \(p\) and \(q\) are positive numbers whose sum is one. The original answer is in \(\frac{k}{m}\) format, please give the value of \(k + m\).
Answer: 9
user: How many nonnegative integers can be written in the form $a_7\cdot3^7 + a_6\cdot3^6 + a_5\cdot3^5 + a_4\cdot3^4 + a_3\cdot3^3 + a_2\cdot3^2 + a_1\cdot3^1 + a_0\cdot3^0$, where $a_i \in \{-1,0,1\}$ for $0 \le i \le 7$?
Answer: 3281
user: The number $n$ is called a composite number if it can be written in the form $n = a \times b$, where $a, b$ are positive integers greater than $1$. Write the number $2013$ as a sum of $m$ composite numbers. What is the largest possible value of $m$? Please provide the value of m.
Answer: 502
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 $N$. Find the perimeter of $\triangle AMN$.
Answer: 30
user: Each of the small circles in the figure has radius one. The innermost circle is tangent to the six circles that surround it, and each of those circles is tangent to the large circle and to its small-circle neighbors. Find the area of the shaded region. If the area is $x\pi$, what is the value of $x$?
Answer: 2
user: What is the average length in letters of a word in this question? Provide your answer as an integer.
Answer: 4
user: A grasshopper jumps either $364$ or $715$ units on the real number line. If it starts from the point $0$, find the smallest distance that the grasshopper can be away from the point $2010$.
Answer: 5
user: The first 9 positive odd integers are placed in a magic square such that the sum of the numbers in each row, column, and diagonal is equal. Find the value of $A + E$.

\[
\begin{array}{|c|c|c|}
\hline
A & 1 & B \\
\hline
5 & C & 13 \\
\hline
D & E & 3 \\
\hline
\end{array}
\]
Answer: 32
user: In the following equation, each of the letters represents uniquely a different digit in base ten:

\[ (YE) \cdot (ME) = TTT \]

Find the sum $E+M+T+Y$.
Answer: 21
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?

1. Start $X$ at $3$ and $S$ at $0$.
2. Increase the value of $X$ by $2$.
3. Increase the value of $S$ by the value of $X$.
4. If $S$ is at least $10000$, then go to instruction 5; otherwise, go to instruction 2 and proceed from there.
5. Print the value of $X$.
6. 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