meetforever/GRPO_CL huggingface dataset:
user: A circle has an inscribed triangle whose sides are $5\sqrt{3}$, $10\sqrt{3}$, and $15$. If the measure of the angle subtended at the center of the circle by the shortest side is $x^\circ$, what is the value of $x$?
Answer: 60
user: A point \(P\) lies in the same plane as a given square with side length \(1\). The vertices of the square, taken counterclockwise, are labeled as \(A\), \(B\), \(C\), and \(D\). The distances from point \(P\) to the vertices \(A\), \(B\), and \(C\) are \(u\), \(v\), and \(w\), respectively. Determine the greatest distance that \(P\) can be from vertex \(D\), given that \(u^2 + v^2 = w^2\). The original answer is in the form \(k + \sqrt{m}\). Please provide the value of \(k + m\).
Answer: 4
user: A spider has one sock and one shoe for each of its eight legs. In how many different orders can the spider put on its socks and shoes, assuming that, on each leg, the sock must be put on before the shoe? Provide your answer in the form of \(\frac{m}{n}\) and find the value of m + n.
Answer: 20922789888256
user: Suppose that \(\cos(3x) + 3\cos(x) = -2\). Find the value of \(\cos(2x)\). The original answer is in the form \(\sqrt[3]{a} - b\). Please provide the value of a + b.
Answer: 3
user: At summer camp, there are $20$ campers in each of the swimming class, the archery class, and the rock climbing class. Each camper is in at least one of these classes. If $4$ campers are in all three classes, and $24$ campers are in exactly one of the classes, find the number of campers who are in exactly two classes.
Answer: 12
user: If Scott rolls four fair six-sided dice, find the probability that he rolls more 2's than 1's. The probability is given in the form \(\frac{m}{n}\). Please calculate the value of m + n.
Answer: 1717
user: If \(\frac{x}{x-1} = \frac{y^2+2y-1}{y^2-2y-2}\), find the value of \(x\). The original answer can be expressed in the form \(ay^2 + by + c\). Please find the value of \(a + b + c\).
Answer: 2
user: How many real roots of the equation \[x^2 - 18\lfloor x \rfloor + 77 = 0\] are not integer, where \(\lfloor x \rfloor\) denotes the greatest integer not exceeding the real number \(x\)?
Answer: 3
user: If \( |x - \log y| = x + \log y \) where \( x \) and \( \log y \) are real, find the expression that represents the relationship between \( x \) and \( y \). The original answer is in the form \( mx(y-n)=p \). Please provide the value of \( m+n+p \).
Answer: 2
user: The addition below is incorrect. The display can be made correct by changing one digit $d$, wherever it occurs, to another digit $e$. What is the sum of $d$ and $e$?

\[
\begin{array}{ccccccc}
& 7 & 4 & 2 & 5 & 8 & 6 \\
+ & 8 & 2 & 9 & 4 & 3 & 0 \\
\hline
1 & 2 & 1 & 2 & 0 & 1 & 6 \\
\end{array}
\]
Answer: 8
user: Find the first step in finding the product \((3x + 2)(x - 5)\) by use of the distributive property in the form \(a(b + c) = ab + ac\). The original answer is in the form \((a+b)x + (a+b)(c)\). Please give the value of a + b + c.
Answer: 0
user: The first $2007$ positive integers are each written in base $3$. How many of these base-$3$ representations are palindromes? (A palindrome is a number that reads the same forward and backward.)
Answer: 100
user: Triangle $ABC$ has a right angle at $C$, $AC = 3$ and $BC = 4$. Triangle $ABD$ has a right angle at $A$ and $AD = 12$. Points $C$ and $D$ are on opposite sides of $\overline{AB}$. The line through $D$ parallel to $\overline{AC}$ meets $\overline{CB}$ extended at $E$. If $\frac{DE}{DB} = \frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, then find the value of $m + n$.
Answer: 128
user: In \(\triangle ABC\), point \(D\) is on \(AC\) and point \(F\) is on \(BC\). It is given that \(AB \perp AC\), \(AF \perp BC\), and \(BD = DC = FC = 1\). Find the length of \(AC\). The original answer is in \(\sqrt[k]{m}\) format, please give the value of k + m.
Answer: 5
user: For how many ordered pairs $(a,b)$ of integers does the polynomial $x^3 + ax^2 + bx + 6$ have 3 distinct integer roots?
Answer: 5
user: A telephone number has the form $ABC - DEF - GHIJ$, where each letter represents a different digit. The digits in each part of the numbers are in decreasing order; that is, $A > B > C$, $D > E > F$, and $G > H > I > J$. Furthermore, $D$, $E$, and $F$ are consecutive even digits; $G$, $H$, $I$, and $J$ are consecutive odd digits; and $A + B + C = 9$. Find the value of $A$.
Answer: 8
user: Halfway through a $100$-shot archery tournament, Chelsea leads by $50$ points. For each shot, a bullseye scores $10$ points, with other possible scores being $8$, $4$, $2$, and $0$ points. Chelsea always scores at least $4$ points on each shot. If Chelsea's next $n$ shots are bullseyes, she will be guaranteed victory. What is the minimum value for $n$?
Answer: 42
user: Equilateral $\triangle ABC$ is inscribed in a circle. A second circle is tangent internally to the circumcircle at $T$ and tangent to sides $AB$ and $AC$ at points $P$ and $Q$. If side $BC$ has length $12$, find the length of segment $PQ$.
Answer: 8
user: A rectangular box measures $a \times b \times c$, where $a,$ $b,$ and $c$ are integers and $1 \leq a \leq b \leq c$. The volume and surface area of the box are numerically equal. Find the number of ordered triples $(a,b,c)$ that are possible.
Answer: 10
user: At the beginning of the school year, $50\%$ of all students in Mr. Well's math class answered "Yes" to the question "Do you love math", and $50\%$ answered "No." At the end of the school year, $70\%$ answered "Yes" and $30\%$ answered "No." Altogether, $x\%$ of the students gave a different answer at the beginning and end of the school year. What is the difference between the maximum and the minimum possible values of $x$? If the answer is $y\%$, what is the value of $y$?
Answer: 60
user: Let $f$ be a real-valued function defined over ordered pairs of integers such that \[f(x+3m-2n, y-4m+5n) = f(x,y)\] for every integers $x,y,m,n$. Find the maximum number of elements in the range set of $f$.
Answer: 7
user: After simple interest for two months at $5\%$ per annum was credited, a Boy Scout Troop had a total of $\textdollar 255.31$ in the Council Treasury. The interest credited was a number of dollars plus the following number of cents. If the answer is \( x \) cents, what is the value of \( x \)?
Answer: 11
user: From a two-digit number $N$, we subtract the number with the digits reversed and find that the result is a positive perfect cube. How many values of $N$ satisfy this condition?
Answer: 7
user: Let $r$ be the number that results when both the base and the exponent of $a^b$ are tripled, where $a, b>0$. If $r$ equals the product of $a^b$ and $x^b$ where $x>0$, find the value of $x$. The original answer is in the form of $k \cdot m^n$, where $k$, $m$, and $n$ are constants. Please find the value of $k + m + n$.
Answer: 30
user: If $\frac{8}{19}$ of the product of the largest two elements of a positive integer set is not greater than the sum of the other elements, what is the minimum possible value of the largest number in the set?
Answer: 13
user: How many positive perfect squares less than $2023$ are divisible by $5$? Fill in the blank with the number of such perfect squares.
Answer: 8
user: Let \(ABCD\) be a regular tetrahedron and let \(E\) be a point inside the face \(ABC\). Denote by \(s\) the sum of the distances from \(E\) to the faces \(DAB\), \(DBC\), \(DCA\), and by \(S\) the sum of the distances from \(E\) to the edges \(AB\), \(BC\), \(CA\). The original answer is in \(\frac{k\sqrt{m}}{n}\) format. Please give the value of \(k + m + n\).
Answer: 7
user: If five boys and three girls are randomly divided into two four-person teams, find the probability that all three girls will end up on the same team. Express your answer in the form of \(\frac{k}{m}\) where the fraction is simplified, and provide the value of k + m.
Answer: 8
user: If $ab \neq 0$ and $|a| \neq |b|$, find the number of distinct values of $x$ satisfying the equation
\[ \frac{x-a}{b} + \frac{x-b}{a} = \frac{b}{x-a} + \frac{a}{x-b} \].
Answer: 3
user: Let $ABCD$ be a convex quadrilateral such that $m(\widehat{ABD})=40^\circ$, $m(\widehat{DBC})=70^\circ$, $m(\widehat{BDA})=80^\circ$, and $m(\widehat{BDC})=50^\circ$. If the answer is $x^\circ$, what is the value of $x$ for $m(\widehat{CAD})$?
Answer: 30
user: For some positive integers $p$, there is a quadrilateral $ABCD$ with positive integer side lengths, perimeter $p$, right angles at $B$ and $C$, $AB=2$, and $CD=AD$. Find how many different values of $p < 2015$ are possible.
Answer: 31
user: Circles $A$, $B$, and $C$ each have a radius of $1$. 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: Let $x$ be a real number selected uniformly at random between 100 and 200. If $\lfloor \sqrt{x} \rfloor = 12$, find the probability that $\lfloor \sqrt{100x} \rfloor = 120$. ($\lfloor v \rfloor$ means the greatest integer less than or equal to $v$.) The original answer is in \frac{k}{m} format, please give the value of k + m.
Answer: 2741
user: If $x^4 + 4x^3 + 6px^2 + 4qx + r$ is exactly divisible by $x^3 + 3x^2 + 9x + 3$, find the value of $(p + q)r$.
Answer: 15
user: For how many integer values of $k$ do the graphs of $x^2 + y^2 = k^2$ and $xy = k$ \textit{not} intersect?
Answer: 2
user: Circles with centers \(P\), \(Q\), and \(R\), having radii \(1\), \(2\), and \(3\), respectively, lie on the same side of line \(l\) and are tangent to \(l\) at \(P'\), \(Q'\), and \(R'\), respectively, with \(Q'\) between \(P'\) and \(R'\). The circle with center \(Q\) is externally tangent to each of the other two circles. Find the area of triangle \(PQR\). The original answer is in the form \(\sqrt{a}-\sqrt{b}\), please provide the value of \(a+b\).
Answer: 8
user: Every cell of an $8\times8$ chessboard contains either $1$ or $-1$. It is known that there are at least four rows such that the sum of numbers inside the cells of those rows is positive. Find the maximum number of columns such that the sum of numbers inside the cells of those columns is less than $-3$.
Answer: 6
user: A 16-step path is to go from $( -4, -4)$ to $(4,4)$ with each step increasing either the $x$-coordinate or the $y$-coordinate by 1. Find the number of such paths that stay outside or on the boundary of the square $-2 \le x \le 2$, $-2 \le y \le 2$ at each step.
Answer: 1698
user: How many solutions does the equation $\tan{(2x)} = \cos{\left(\frac{x}{2}\right)}$ have on the interval $[0, 2\pi]$? Write your answer as an integer.
Answer: 5
user: In the adjoining figure, points $B$ and $C$ lie on line segment $AD$, and $AB$, $BC$, and $CD$ are diameters of circles $O$, $N$, and $P$, respectively. Circles $O$, $N$, and $P$ all have radius $15$, and the line $AG$ is tangent to circle $P$ at $G$. If $AG$ intersects circle $N$ at points $E$ and $F$, find the length of chord $EF$.
Answer: 24
user: A rectangle is inscribed in a triangle such that its base coincides with the base $b$ of the triangle. If the altitude of the triangle is $h$, and the altitude $x$ of the rectangle is half the base of the rectangle, find the expression for $x$ in terms of $b$ and $h$. The original answer is in the form \frac{kb}{mh+b}, please give the value of k + m.
Answer: 3
user: If \(x \in \left[-\frac{5\pi}{12}, -\frac{\pi}{3}\right]\), find the maximum value of \(y = \tan\left(x + \frac{2\pi}{3}\right) - \tan\left(x + \frac{\pi}{6}\right) + \cos\left(x + \frac{\pi}{6}\right)\). The original answer is in the form \(\frac{k}{m}\sqrt{n}\). Please provide the value of \(k + m + n\).
Answer: 20
user: If $n$ is any whole number, find the largest number by which $n^2(n^2 - 1)$ is always divisible.
Answer: 12
user: For every \( x, y \in \mathbb{R}^+ \), the function \( f: \mathbb{R}^+ \to \mathbb{R} \) satisfies the condition \( f(x) + f(y) = f(x)f(y) + 1 - \frac{1}{xy} \). If \( f(2) < 1 \), find the value of \( f(3) \). The original answer is in \(\frac{a}{b}\) format, please give the value of a + b.
Answer: 5
user: What is the sum of the real roots of the equation \(4x^4 - 3x^2 + 7x - 3 = 0\)? Please provide your answer in the form of an integer \(k\), where \(k\) is the sum of the real roots.
Answer: -1
user: Four girls — Mary, Alina, Tina, and Hanna — sang songs in a concert as trios, with one girl sitting out each time. Hanna sang $7$ songs, which was more than any other girl, and Mary sang $4$ songs, which was fewer than any other girl. How many songs did these trios sing?
Answer: 7
user: For each $x$ in $[0,1]$, define
\[ f(x) = \begin{cases} 2x, & \text{if } 0 \leq x \leq \frac{1}{2}; \\
2 - 2x, & \text{if } \frac{1}{2} < x \leq 1. \end{cases} \]
Let $f^{[2]}(x) = f(f(x))$, and $f^{[n + 1]}(x) = f^{[n]}(f(x))$ for each integer $n \geq 2$. Find the number of values of $x$ in $[0,1]$ for which $f^{[2005]}(x) = \frac{1}{2}$. The original answer is in the form of $k^m$, please provide the value of $k + m$.
Answer: 2007
user: How many $6$-digit positive integers whose digits are different from $0$ are there such that each number generated by rearranging the digits of the original number is always divisible by $7$? Find the total number of such integers.
Answer: 133
user: If $n$ is a real number, then the simultaneous system of equations:
\[
\begin{align*}
  nx + y &= 1 \\
  ny + z &= 1 \\
  x + nz &= 1
\end{align*}
\]
has no solution if and only if $n$ is equal to $x$. Please provide the absolute value of $x$ as your final answer.
Answer: 1
user: The five solutions to the equation $$(z-1)(z^2+2z+4)(z^2+4z+6)=0$$ may be written in the form $x_k+y_ki$ for $1 \le k \le 5,$ where $x_k$ and $y_k$ are real. Let $\mathcal E$ be the unique ellipse that passes through the points $(x_1,y_1),(x_2,y_2),(x_3,y_3),(x_4,y_4),$ and $(x_5,y_5)$. The eccentricity of $\mathcal E$ can be written in the form $\sqrt{\frac mn}$ where $m$ and $n$ are relatively prime positive integers. What is $m+n$?
Answer: 7
user: In this diagram, $AB$ and $AC$ are the equal sides of an isosceles triangle $ABC$, in which an equilateral triangle $DEF$ is inscribed. Designate angle $BFD$ by $a$, angle $ADE$ by $b$, and angle $FEC$ by $c$. The relationship between $a$, $b$, and $c$ is such that $k(b + c) = 2a$. Please find the value of $k + m + n$, where $m$ and $n$ are the coefficients of $b$ and $c$, respectively.
Answer: 3
user: On a standard die, one of the dots is removed at random, with each dot equally likely to be chosen. The die is then rolled. What is the probability that the top face has an odd number of dots? Express your answer as a fraction in simplest form, \frac{k}{m}, and provide the value of k + m.
Answer: 32
user: The ratio of the interior angles of two regular polygons with sides of unit length is $3:2$. How many such pairs of polygons are there?
Answer: 3
user: In checking the petty cash, a clerk counts $q$ quarters, $d$ dimes, $n$ nickels, and $c$ cents. Later, he discovers that $x$ of the nickels were counted as quarters and $x$ of the dimes were counted as cents. To correct the total obtained, by how many cents must the clerk adjust the total? The original answer is in the form $kx$, please give the value of $k$.
Answer: 11
user: Given that $\angle BAC = 80^\circ$, $|AB| = |AC|$, $K \in [AB]$, $L \in [AB]$, $|AB|^2 = |AK| \cdot |AL|$, and $|BL| = |BC|$, find $\angle KCB$. If the answer is $x^\circ$, what is the value of $x$?
Answer: 25
user: Given the polynomial $a_0x^n + a_1x^{n-1} + \cdots + a_{n-1}x + a_n$, where $n$ is a non-negative integer, and $a_0$ is a positive integer. The remaining coefficients $a_1, a_2, \ldots, a_n$ are integers or zero. Define $h = n + a_0 + |a_1| + |a_2| + \cdots + |a_n|$. Find the number of polynomials for which $h = 3$.
Answer: 5
user: Let $f(x) = |x-2| + |x-4| - |2x-6|$ for $2 \leq x \leq 8$. Find the sum of the largest and smallest values of $f(x)$.
Answer: 2
user: A planet orbits around a star $S$, as shown in the figure. The semi-major axis of the orbit is $a$. The perigee, namely the shortest distance between the planet and the star, is $0.5a$. When the planet passes point $P$ (on the line through the star and perpendicular to the major axis), its speed is $v_1$. What is the value of $v_2$ in terms of $v_1$ when it passes the perigee? The original answer is in the form \(\frac{k}{\sqrt{m}}v_1\). Please calculate the value of \(k + m\).
Answer: 8
user: Regular hexagon $ABCDEF$ has vertices $A$ and $C$ at $(0,0)$ and $(7,1)$ respectively. What is the area of the hexagon? The answer should be in the form $k\sqrt{m}$, where k and m are integers. Please find the value of k + m.
Answer: 28
user: In \(\triangle ABC\), \(\angle C = 90^\circ\), \(AC = 6\), and \(BC = 8\). Points \(D\) and \(E\) are on \(\overline{AB}\) and \(\overline{BC}\), respectively, and \(\angle BED = 90^\circ\). If \(DE = 4\), find the value of \(BD\). The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
Answer: 23
user: A right triangle $ABC$ with hypotenuse $AB$ has side $AC = 15$. Altitude $CH$ divides $AB$ into segments $AH$ and $HB$, with $HB = 16$. Find the area of $\triangle ABC$.
Answer: 150
user: In the $xy$-plane, consider the L-shaped region bounded by horizontal and vertical segments with vertices at $(0,0)$, $(0,3)$, $(3,3)$, $(3,1)$, $(5,1)$, and $(5,0)$. Find the slope of the line through the origin that divides the area of this region exactly in half. The original answer is in \(\frac{k}{m}\) format, where \(\frac{k}{m}\) is a simplified fraction. Please give the value of \(k + m\).
Answer: 16
user: How many ordered integer pairs $(x,y)$ are there such that $xy = 4(y^2 + x)$?
Answer: 14
user: Let \(ABC\) be a triangle such that \(|AB| = 8\) and \(|AC| = 2|BC|\). What is the largest value of the altitude from side \([AB]\)? The original answer is in the form \(\frac{k}{m}\). Please provide the value of \(k + m\).
Answer: 19
user: A positive integer $n$ has 60 divisors and $7n$ has 80 divisors. What is the greatest integer $k$ such that $7^k$ divides $n$?
Answer: 2
user: How many ordered integer pairs $(x, y)$, where $0 \leq x, y < 31$, satisfy the equation $(x^2 - 18)^2 \equiv y^2 \pmod{31}$? Provide your answer as an integer.
Answer: 60
user: Lily has an unfair coin that has a \(\frac{2}{3}\) probability of showing heads and a \(\frac{1}{3}\) probability of showing tails. She flips the coin twice. What is the probability that the first flip is heads while the second is tails? Provide your answer in the form of \(\frac{k}{m}\), where the fraction is in simplest form, and find the value of k + m.
Answer: 11
user: Find the number of counterexamples to the statement: 
\[ \text{If } N \text{ is an odd positive integer the sum of whose digits is 4 and none of whose digits is 0, then } N \text{ is prime.} \]
Provide your answer as an integer.
Answer: 2
user: Jamal wants to store $30$ computer files on floppy disks, each of which has a capacity of $1.44$ megabytes (MB). Three of his files require $0.8$ MB of memory each, $12$ more require $0.7$ MB each, and the remaining $15$ require $0.4$ MB each. No file can be split between floppy disks. What is the minimal number of floppy disks that will hold all the files?
Answer: 13
user: Distinct points \(A\) and \(B\) are on a semicircle with diameter \(MN\) and center \(C\). The point \(P\) is on \(CN\) and \(\angle CAP = \angle CBP = 10^{\circ}\). If \(\stackrel{\frown}{MA} = 40^{\circ}\), then \(\stackrel{\frown}{BN}\) equals how many degrees? Please provide your answer as an integer representing the number of degrees.
Answer: 20
user: The base of a triangle is $15$ inches. Two lines are drawn parallel to the base, terminating in the other two sides, and dividing the triangle into three equal areas. Find the length of the parallel closer to the base in inches. The answer should be in the form $k\sqrt{m}$, where k and m are integers. Please calculate the value of k + m.
Answer: 11
user: Find the set of values of $m$ for which $x^2 + 3xy + x + my - m$ has two factors, with integer coefficients, which are linear in $x$ and $y$. The original answer is in the form of $\{p, q\}$, where $p$ and $q$ are integers. Please provide the value of $p + q$.
Answer: 12
user: Figure \(ABCD\) is a trapezoid with \(AB \parallel DC\), \(AB = 5\), \(BC = 3 \sqrt{2}\), \(\angle BCD = 45^\circ\), and \(\angle CDA = 60^\circ\). Find the length of \(DC\). The original answer is in the form \(m + \sqrt{n}\). Please give the value of \(m + n\).
Answer: 11
user: Let \(Q(z)\) and \(R(z)\) be the unique polynomials such that:
\[
z^{2021}+1=(z^2+z+1)Q(z)+R(z)
\]
and the degree of \(R\) is less than \(2\). The original answer is in the form \(-a \cdot z + b\). Please find the value of \(a + b\).
Answer: 1
user: In a square $ABCD$ with side length $2$ units, $E$ is the midpoint of $AD$. A point $F$ is located on $BE$ such that $CF \perp BE$. If the area of the quadrilateral $CDEF$ is $x$ and $x$ is multiplied by 10, what is the value of $10x$?
Answer: 22
user: The buttons \( \{\times, +, \div\} \) on a calculator have their functions swapped. A button instead performs one of the other two functions; no two buttons have the same function. The calculator claims that \( 2 + 3 \div 4 = 10 \) and \( 4 \times 2 \div 3 = 5 \). What does \( 4 + 3 \times 2 \div 1 \) equal on this calculator? Find the value.
Answer: 7
user: On an auto trip, the distance read from the instrument panel was $450$ miles. With snow tires on for the return trip over the same route, the reading was $440$ miles. Find, to the nearest hundredth of an inch, the increase in radius of the wheels if the original radius was $15$ inches. If the answer is $x$ inches, what is the value of $100x$?
Answer: 34
user: Line $\frac{x}{4}+\frac{y}{3}=1$ and ellipse $\frac{x^2}{16}+\frac{y^2}{9}=1$ intersect at $A$ and $B$. A point on the ellipse $P$ satisfies that the area of $\triangle PAB$ is $3$. Find the number of such points.
Answer: 2
user: $P$ is the intersection point of diagonals of cyclic quadrilateral $ABCD$. The circumcenters of $\triangle APB$ and $\triangle CPD$ lie on the circumcircle of $ABCD$. If $AC + BD = 18$, find the area of $ABCD$. The original answer is in the form \(\frac{k\sqrt{m}}{n}\), please give the value of k + m + n.
Answer: 88
user: For all integers $n$ greater than $1$, define $a_n = \frac{1}{\log_n 2002}$. Let $b = a_2 + a_3 + a_4 + a_5$ and $c = a_{10} + a_{11} + a_{12} + a_{13} + a_{14}$. Find the value of $b - c$.
Answer: -1
user: Let $(a_n)_{n=1}^\infty$ be a real sequence such that \[a_n = (n-1)a_1 + (n-2)a_2 + \dots + 2a_{n-2} + a_{n-1}\] for every $n \geq 3$. If $a_{2011} = 2011$ and $a_{2012} = 2012$, find the value of $a_{2013}$.
Answer: 4025
user: The sides of $\triangle ABC$ form an arithmetic sequence of integers. The incircle $I$ is tangent to $AB$, $BC$, and $CA$ at points $D$, $E$, and $F$, respectively. Given that $DB = \frac{3}{2}$ and $FA = \frac{1}{2}$, find the radius of the incircle $I$. Express your answer in the form \frac{\sqrt{m}}{n}, where the fraction is in its simplest form. Calculate the value of m + n.
Answer: 21
user: How many ordered pairs of positive integers $(x,y)$ are there such that $y^2 - x^2 = 2y + 7x + 4$?
Answer: 1
user: A ball of radius \(R\) and mass \(m\) is magically put inside a thin shell of the same mass and radius \(2R\). The system is at rest on a horizontal frictionless surface initially. When the ball is, again magically, released inside the shell, it sloshes around in the shell and eventually stops at the bottom of the shell. How far does the shell move from its initial contact point with the surface? The original answer is in \(\frac{a}{b}\) format, please give the value of a + b.
Answer: 3
user: For how many integer values of $m$ do the lines $13x + 11y = 700$ and $y = mx - 1$ intersect at a point with integer-valued coordinates? Provide your answer as an integer.
Answer: 2
user: Find the value of the product \[ \frac{8}{4} \cdot \frac{12}{8} \cdot \frac{16}{12} \cdots \frac{4n + 4}{4n} \cdots \frac{2008}{2004}. \]
Answer: 502
user: A subset of the integers $1, 2, \ldots, 100$ has the property that none of its members is 3 times another. What is the largest number of members such a subset can have?
Answer: 76