ChenDRAG/VeRL_math_validation huggingface dataset:
Problem: An urn contains one red ball and one blue ball. A box of extra red and blue balls lies nearby. George performs the following operation four times: he draws a ball from the urn at random and then takes a ball of the same color from the box and returns those two matching balls to the urn. After the four iterations, the urn contains six balls. What is the probability that the urn contains three balls of each color? Express your answer in the form \(\frac{k}{m}\), where k and m are integers. Then find the value of k + m.
Answer: 6
Problem: Define \[P(x) =(x-1^2)(x-2^2)\cdots(x-100^2).\] Find the number of integers $n$ such that $P(n)\leq 0$.
Answer: 5100
Problem: The vertices of a quadrilateral lie on the graph of $y=\ln{x}$, and the $x$-coordinates of these vertices are consecutive positive integers. The area of the quadrilateral is $\ln{\frac{91}{90}}$. What is the $x$-coordinate of the leftmost vertex?
Answer: 12
Problem: How many positive integer factors of $2020$ have more than $3$ factors? (As an example, $12$ has $6$ factors, namely $1,2,3,4,6,$ and $12.$) Provide your answer as an integer.
Answer: 7
Problem: Let $(a,b,c,d)$ be an ordered quadruple of not necessarily distinct integers, each one of them in the set $\{0,1,2,3\}.$ Find the number of such quadruples for which $a\cdot d-b\cdot c$ is odd. (For example, $(0,3,1,1)$ is one such quadruple, because $0\cdot 1-3\cdot 1 = -3$ is odd.)
Answer: 96
Problem: Quadrilateral $ABCD$ satisfies $\angle ABC = \angle ACD = 90^{\circ}$, $AC=20$, and $CD=30$. Diagonals $\overline{AC}$ and $\overline{BD}$ intersect at point $E$, and $AE=5$. Find the area of quadrilateral $ABCD$.
Answer: 360
Problem: 已知 $F_{1}, F_{2}$ 为椭圆 $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)$ 的左、右焦点，该椭圆上存在两点 $A, B$ ，使得 $\overrightarrow{F_{1} A}=3 \overrightarrow{F_{2} B}$ 。求该椭圆的离心率的取值范围。答案的形式为\left(\frac{c}{d}, 1\right)，请给出c+d的值。
Answer: 3
Problem: 设有三个函数，第一个是 $y=\varphi(x)$ ，它的反函数是第二个函数，而第三个函数的图象与第二个函数的图象关于 $x+y=0$ 对称。第三个函数的表达式为 $y=ax+b\varphi(cx)$，请给出a+b+c的值。
Answer: -2
Problem: Rectangle $ABCD$ is inscribed in a semicircle with diameter $\overline{FE},$ as shown in the figure. Let $DA=16,$ and let $FD=AE=9.$ What is the area of $ABCD$?
Answer: 240
Problem: In a certain card game, a player is dealt a hand of $10$ cards from a deck of $52$ distinct cards. The number of distinct (unordered) hands that can be dealt to the player can be written as $158A00A4AA0$. Find the digit $A$.
Answer: 2
Problem: Quadrilateral $ABCD$ satisfies $\angle ABC = \angle ACD = 90^{\circ}, AC=20,$ and $CD=30.$ Diagonals $\overline{AC}$ and $\overline{BD}$ intersect at point $E,$ and $AE=5.$ What is the area of quadrilateral $ABCD$?
Answer: 360
Problem: Let $B$ be a right rectangular prism (box) with edge lengths $1,$ $3,$ and $4$, together with its interior. For real $r\geq0$, let $S(r)$ be the set of points in $3$-dimensional space that lie within a distance $r$ of some point in $B$. The volume of $S(r)$ can be expressed as $ar^{3} + br^{2} + cr +d$, where $a,$ $b,$ $c,$ and $d$ are positive real numbers. Find the value of $\frac{bc}{ad}$.
Answer: 19
Problem: Let $T$ be the triangle in the coordinate plane with vertices $(0,0), (4,0),$ and $(0,3).$ Consider the following five isometries (rigid transformations) of the plane: rotations of $90^{\circ}, 180^{\circ},$ and $270^{\circ}$ counterclockwise around the origin, reflection across the $x$-axis, and reflection across the $y$-axis. Find the number of the $125$ sequences of three of these transformations (not necessarily distinct) that will return $T$ to its original position.
Answer: 12
Problem: Two different cubes of the same size are to be painted, with the color of each face being chosen independently and at random to be either black or white. Find the probability that after they are painted, the cubes can be rotated to be identical in appearance. The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
Answer: 1171
Problem: In square \(ABCD\), points \(E\) and \(H\) lie on \(\overline{AB}\) and \(\overline{DA}\), respectively, so that \(AE=AH.\) Points \(F\) and \(G\) lie on \(\overline{BC}\) and \(\overline{CD}\), respectively, and points \(I\) and \(J\) lie on \(\overline{EH}\) so that \(\overline{FI} \perp \overline{EH}\) and \(\overline{GJ} \perp \overline{EH}\). See the figure below. Triangle \(AEH\), quadrilateral \(BFIE\), quadrilateral \(DHJG\), and pentagon \(FCGJI\) each has area \(1.\) The original answer is in the form \(k - m\sqrt{n}\), please give the value of \(k + m + n\).
Answer: 14
Problem: 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 $n$?
Answer: 48
Problem: How many positive integers $n$ satisfy 
$\dfrac{n+1000}{70} = \lfloor \sqrt{n} \rfloor?$
(Recall that $\lfloor x\rfloor$ is the greatest integer not exceeding $x$.)
Answer: 6
Problem: For how many positive integers $n \le 1000$ is
$\left\lfloor \dfrac{998}{n} \right\rfloor+\left\lfloor \dfrac{999}{n} \right\rfloor+\left\lfloor \dfrac{1000}{n}\right \rfloor$
not divisible by $3$?
Answer: 22
Problem: What is the remainder when $2^{202} +202$ is divided by $2^{101}+2^{51}+1$?
Answer: 201
Problem: Let $(a_n)$ and $(b_n)$ be the sequences of real numbers such that
\[ (2 + i)^n = a_n + b_ni \] for all integers $n\geq 0$, where $i = \sqrt{-1}$. Find the value of \[\sum_{n=0}^\infty\frac{a_nb_n}{7^n}\,.\] The original answer is in the form \(\frac{k}{m}\). Please give the value of \(k + m\).
Answer: 23
Problem: What is the maximum value of \(\frac{(2^t-3t)t}{4^t}\) for real values of \(t?\) The original answer is in \(\frac{k}{m}\) format, where \(\frac{k}{m}\) is a simplified fraction. Please provide the value of k + m.
Answer: 13
Problem: When a positive integer $N$ is fed into a machine, the output is a number calculated according to the rule shown below.

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For example, starting with an input of $N=7,$ the machine will output $3 \cdot 7 +1 = 22.$ Then if the output is repeatedly inserted into the machine five more times, the final output is $26.$\[7 \to 22 \to 11 \to 34 \to 17 \to 52 \to 26\]When the same $6$-step process is applied to a different starting value of $N,$ the final output is $1.$ What is the sum of all such integers $N?$\[N \to \rule{0.5cm}{0.15mm} \to \rule{0.5cm}{0.15mm} \to \rule{0.5cm}{0.15mm} \to \rule{0.5cm}{0.15mm} \to \rule{0.5cm}{0.15mm} \to 1\]
Answer: 83
Problem: Square \(ABCD\) in the coordinate plane has vertices at the points \(A(1,1), B(-1,1), C(-1,-1),\) and \(D(1,-1).\) Consider the following four transformations:

\(\quad\bullet\qquad\) \(L,\) a rotation of \(90^{\circ}\) counterclockwise around the origin;

\(\quad\bullet\qquad\) \(R,\) a rotation of \(90^{\circ}\) clockwise around the origin;

\(\quad\bullet\qquad\) \(H,\) a reflection across the \(x\)-axis; and

\(\quad\bullet\qquad\) \(V,\) a reflection across the \(y\)-axis.

Each of these transformations maps the squares onto itself, but the positions of the labeled vertices will change. For example, applying \(R\) and then \(V\) would send the vertex \(A\) at \((1,1)\) to \((-1,-1)\) and would send the vertex \(B\) at \((-1,1)\) to itself. Find how many sequences of \(20\) transformations chosen from \(\{L, R, H, V\}\) will send all of the labeled vertices back to their original positions. The original answer is in the form \(x * 2^y\), please give the value of \(x + y\).
Answer: 39
Problem: Jason rolls three fair standard six-sided dice. Then he looks at the rolls and chooses a subset of the dice (possibly empty, possibly all three dice) to reroll. After rerolling, he wins if and only if the sum of the numbers face up on the three dice is exactly \(7\). Jason always plays to optimize his chances of winning. What is the probability that he chooses to reroll exactly two of the dice? Express your answer in the form \(\frac{k}{m}\), where the fraction is in simplest form. Please give the value of \(k + m\).
Answer: 43
Problem: Let \( n \) be the least positive integer greater than 1000 for which
\(\gcd(63, n+120) = 21\) and \(\gcd(n+63, 120) = 60.\)
Find the sum of the digits of \( n \).
Answer: 18
Problem: How many positive integers $n$ satisfy 
$$\dfrac{n+1000}{70} = \lfloor \sqrt{n} \rfloor?$$
Answer: 6
Problem: Let $D(n)$ denote the number of ways of writing the positive integer $n$ as a product\[n = f_1\cdot f_2\cdots f_k,\]where $k\ge1$, the $f_i$ are integers strictly greater than $1$, and the order in which the factors are listed matters (that is, two representations that differ only in the order of the factors are counted as distinct). For example, the number $6$ can be written as $6$, $2\cdot 3$, and $3\cdot2$, so $D(6) = 3$. What is $D(96)$?
Answer: 112
Problem: Jason rolls three fair standard six-sided dice. Then he looks at the rolls and chooses a subset of the dice (possibly empty, possibly all three dice) to reroll. After rerolling, he wins if and only if the sum of the numbers face up on the three dice is exactly \(7\). Jason always plays to optimize his chances of winning. What is the probability that he chooses to reroll exactly two of the dice? Express your answer in the form \(\frac{k}{m}\), and find the value of k + m.
Answer: 43
Problem: The number $a=\frac{p}{q}$, where $p$ and $q$ are relatively prime positive integers, has the property that the sum of all real numbers $x$ satisfying
\[\lfloor x \rfloor \cdot \{x\} = a \cdot x^2\]
is $420$, where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$ and $\{x\}=x- \lfloor x \rfloor$ denotes the fractional part of $x$. What is $p+q$?
Answer: 929
Problem: For each real number \(a\) with \(0 \leq a \leq 1\), let numbers \(x\) and \(y\) be chosen independently at random from the intervals \([0, a]\) and \([0, 1]\), respectively, and let \(P(a)\) be the probability that
\(\sin^2{(\pi x)} + \sin^2{(\pi y)} > 1\).
Find the maximum value of \(P(a)\). The original answer is in the form \(k - \sqrt{m}\), please give the value of \(k + m\).
Answer: 4
Problem: At a math contest, $57$ students are wearing blue shirts, and another $75$ students are wearing yellow shirts. The $132$ students are assigned into $66$ pairs. In exactly $23$ of these pairs, both students are wearing blue shirts. In how many pairs are both students wearing yellow shirts?
Answer: 32
Problem: Menkara has a $4 \times 6$ index card. If she shortens the length of one side of this card by $1$ inch, the card would have area $18$ square inches. What would the area of the card be in square inches if instead she shortens the length of the other side by $1$ inch?
Answer: 20
Problem: What is the area of the shaded figure shown below?
[asy] size(200); defaultpen(linewidth(0.4)+fontsize(12)); pen s = linewidth(0.8)+fontsize(8);  pair O,X,Y; O = origin; X = (6,0); Y = (0,5); fill((1,0)--(3,5)--(5,0)--(3,2)--cycle, palegray+opacity(0.2)); for (int i=1; i<7; ++i) { draw((i,0)--(i,5), gray+dashed); label("${"+string(i)+"}$", (i,0), 2*S); if (i<6) { draw((0,i)--(6,i), gray+dashed); label("${"+string(i)+"}$", (0,i), 2*W); } } label("$0$", O, 2*SW); draw(O--X+(0.35,0), black+1.5, EndArrow(10)); draw(O--Y+(0,0.35), black+1.5, EndArrow(10)); draw((1,0)--(3,5)--(5,0)--(3,2)--(1,0), black+1.5); [/asy]

Find the area of the shaded figure.
Answer: 6
Problem: Menkara has a $4 \times 6$ index card. If she shortens the length of one side of this card by $1$ inch, the card would have an area of $18$ square inches. What would the area of the card be in square inches if instead she shortens the length of the other side by $1$ inch?
Answer: 20
Problem: What is the maximum number of balls of clay of radius $2$ that can completely fit inside a cube of side length $6$ assuming the balls can be reshaped but not compressed before they are packed in the cube?
Answer: 6
Problem: The six-digit number $\underline{2}\,\underline{0}\,\underline{2}\,\underline{1}\,\underline{0}\,\underline{A}$ is prime for only one digit $A.$ What is the value of $A$?
Answer: 9
Problem: When a student multiplied the number $66$ by the repeating decimal,  
$\underline{1}.\underline{a} \ \underline{b} \ \underline{a}  \underline{b}\cdots=\underline{1}.\overline{\underline{a} \ \underline{b}},$  
where $a$ and $b$ are digits, he did not notice the notation and just multiplied $66~ times$ $\underline{1}.\underline{a} \ \underline{b}.$ Later he found that his answer is $0.5$ less than the correct answer. Find the $2$-digit number $\underline{a} \ \underline{b}.$
Answer: 75
Problem: The point $P(a,b)$ in the $xy$-plane is first rotated counterclockwise by $90^\circ$ around the point $(1,5)$ and then reflected about the line $y = -x$. The image of $P$ after these two transformations is at $(-6,3)$. What is the value of $b - a$?
Answer: 7
Problem: The six-digit number $\underline{2}\,\underline{0}\,\underline{2}\,\underline{1}\,\underline{0}\,\underline{A}$ is prime for only one digit $A.$ What is the value of $A?$
Answer: 9
Problem: 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 the number of distinct integers that can be written as the sum of two, not necessarily different, special fractions.
Answer: 11
Problem: Chantal and Jean start hiking from a trailhead toward a fire tower. Jean is wearing a heavy backpack and walks slower. Chantal starts walking at \(4\) miles per hour. Halfway to the tower, the trail becomes really steep, and Chantal slows down to \(2\) miles per hour. After reaching the tower, she immediately turns around and descends the steep part of the trail at \(3\) miles per hour. She meets Jean at the halfway point. What was Jean's average speed, in miles per hour, until they meet? The original answer is in \(\frac{k}{m}\) format, please give the value of \(k + m\).
Answer: 25
Problem: The least positive integer with exactly $2021$ distinct positive divisors can be written in the form $m \cdot 6^k$, where $m$ and $k$ are integers and $6$ is not a divisor of $m$. What is the value of $m+k$?
Answer: 58
Problem: The greatest prime number that is a divisor of $16{,}384$ is $2$ because $16{,}384 = 2^{14}$. Find the sum of the digits of the greatest prime number that is a divisor of $16{,}383$.
Answer: 10
Problem: As shown in the figure below, point $E$ lies on the opposite half-plane determined by line $CD$ from point $A$ so that $\angle CDE = 110^\circ$. Point $F$ lies on $\overline{AD}$ so that $DE=DF$, and $ABCD$ is a square. If the answer is $x^\circ$, what is the degree measure of $\angle AFE$?
Answer: 170
Problem: 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
Problem: When a student multiplied the number $66$ by the repeating decimal,  
$$\underline{1}.\underline{a} \ \underline{b} \ \underline{a} \ \underline{b}\ldots=\underline{1}.\overline{\underline{a} \ \underline{b}},$$  
where $a$ and $b$ are digits, he did not notice the notation and just multiplied $66$ times $\underline{1}.\underline{a} \ \underline{b}.$ Later he found that his answer is $0.5$ less than the correct answer. Find the $2$-digit number $\underline{a} \ \underline{b}.$
Answer: 75
Problem: Mr. Zhou places all the integers from $1$ to $225$ into a $15$ by $15$ grid. He places $1$ in the middle square (eighth row and eighth column) and places other numbers one by one clockwise, as shown in part in the diagram below. Find the sum of the greatest number and the least number that appear in the second row from the top.
Answer: 367
Problem: Three equally spaced parallel lines intersect a circle, creating three chords of lengths $38, 38,$ and $34$. What is the distance between two adjacent parallel lines? Provide your answer as an integer.
Answer: 6
Problem: A two-digit positive integer is said to be cuddly if it is equal to the sum of its nonzero tens digit and the square of its units digit. How many two-digit positive integers are cuddly?
Answer: 1
Problem: The greatest prime number that is a divisor of $16{,}384$ is $2$ because $16{,}384 = 2^{14}$. What is the sum of the digits of the greatest prime number that is a divisor of $16{,}383$? Provide your answer as a single integer.
Answer: 10
Problem: Let $M$ be the least common multiple of all the integers $10$ through $30,$ inclusive. Let $N$ be the least common multiple of $M, 32, 33, 34, 35, 36, 37, 38, 39,$ and $40.$ Find the value of $\frac{N}{M}.$
Answer: 74
Problem: The product of the lengths of the two congruent sides of an obtuse isosceles triangle is equal to the product of the base and twice the triangle's height to the base. What is the measure of the vertex angle of this triangle, in degrees?
Answer: 150
Problem: Which expression is equivalent to

$(2+3)(2^2+3^2)(2^4+3^4)(2^8+3^8)(2^{16}+3^{16})(2^{32}+3^{32})(2^{64}+3^{64})?$

Provide your answer in the form $a^{b} - c^{b}$, and then find the value of $a + b + c$.
Answer: 133
Problem: When a certain unfair die is rolled, an even number is $3$ times as likely to appear as an odd number. The die is rolled twice. Find the probability that the sum of the numbers rolled is even. The original answer is in \frac{k}{m} format, please give the value of k + m.
Answer: 13
Problem: A right rectangular prism whose surface area and volume are numerically equal has edge lengths $\log_{2}x, \log_{3}x,$ and $\log_{4}x.$ What is the integer value of $x$?
Answer: 576
Problem: Triangle $ABC$ is equilateral with side length $6$. Suppose that $O$ is the center of the inscribed circle of this triangle. If the area of the circle passing through $A$, $O$, and $C$ is $x\pi$, what is the value of $x$?
Answer: 12
Problem: Two distinct numbers are selected from the set $\{1,2,3,4,\dots,36,37\}$ so that the sum of the remaining $35$ numbers is the product of these two numbers. What is the difference of these two numbers?
Answer: 10
Problem: $A$ school has $100$ students and $5$ teachers. In the first period, each student is taking one class, and each teacher is teaching one class. The enrollments in the classes are $50, 20, 20, 5,$ and $5$. Let $t$ be the average value obtained if a teacher is picked at random and the number of students in their class is noted. Let $s$ be the average value obtained if a student was picked at random and the number of students in their class, including the student, is noted. What is $10(t-s)$?
Answer: -135
Problem: Forty slips of paper numbered $1$ to $40$ are placed in a hat. Alice and Bob each draw one number from the hat without replacement, keeping their numbers hidden from each other. Alice says, "I can't tell who has the larger number." Then Bob says, "I know who has the larger number." Alice says, "You do? Is your number prime?" Bob replies, "Yes." Alice says, "In that case, if I multiply your number by $100$ and add my number, the result is a perfect square. " What is the sum of the two numbers drawn from the hat?
Answer: 27
Problem: The base-nine representation of the number $N$ is $27{,}006{,}000{,}052_{\text{nine}}.$ Find the remainder when $N$ is divided by $5$.
Answer: 3
Problem: Grandma has just finished baking a large rectangular pan of brownies. She is planning to make rectangular pieces of equal size and shape, with straight cuts parallel to the sides of the pan. Each cut must be made entirely across the pan. Grandma wants to make the same number of interior pieces as pieces along the perimeter of the pan. What is the greatest possible number of brownies she can produce?
Answer: 60
Problem: 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, then hit the point $(7,5)$. What is the total distance $d$ the beam will travel along this path? If the answer is $a\sqrt{b}$, where $a$ and $b$ are integers and $b$ is not divisible by the square of any prime, what is the value of $a$?
Answer: 10
Problem: Triangle \(ABC\) has \(AB=13, BC=14\) and \(AC=15\). Let \(P\) be the point on \(\overline{AC}\) such that \(PC=10\). There are exactly two points \(D\) and \(E\) on line \(BP\) such that quadrilaterals \(ABCD\) and \(ABCE\) are trapezoids. The distance \(DE\) can be expressed in the form \(k\sqrt{m}\). Please find the value of \(k + m\).
Answer: 14
Problem: Emily sees a ship traveling at a constant speed along a straight section of a river. She walks parallel to the riverbank at a uniform rate faster than the ship. She counts $210$ equal steps walking from the back of the ship to the front. Walking in the opposite direction, she counts $42$ steps of the same size from the front of the ship to the back. In terms of Emily's equal steps, what is the length of the ship?
Answer: 70
Problem: A regular hexagon of side length $1$ is inscribed in a circle. Each minor arc of the circle determined by a side of the hexagon is reflected over that side. Find the area of the region bounded by these $6$ reflected arcs. The original answer is in the form of k\sqrt{m} - n\pi, please provide the value of k + m + n.
Answer: 7
Problem: Consider two concentric circles of radius $17$ and $19.$ The larger circle has a chord, half of which lies inside the smaller circle. What is the length of the chord in the larger circle? The original answer is in the form of k\sqrt{m}, please give the value of k + m.
Answer: 14
Problem: Una rolls $6$ standard $6$-sided dice simultaneously and calculates the product of the $6$ numbers obtained. What is the probability that the product is divisible by $4$? Provide your answer in the form of \(\frac{k}{m}\) where the fraction is in simplest form. Please give the value of \(k + m\).
Answer: 123
Problem: Let $N = 34 \cdot 34 \cdot 63 \cdot 270$. What is the value of $y$ if the ratio of the sum of the odd divisors of $N$ to the sum of the even divisors of $N$ is $1 : y$?
Answer: 14
Problem: Suppose that $S$ is a finite set of positive integers. If the greatest integer in $S$ is removed from $S$, then the average value (arithmetic mean) of the integers remaining is $32$. If the least integer in $S$ is also removed, then the average value of the integers remaining is $35$. If the greatest integer is then returned to the set, the average value of the integers rises to $40$. The greatest integer in the original set $S$ is $72$ greater than the least integer in $S$. If the average value of all the integers in the set $S$ is $x$, what is the value of $10x$?
Answer: 368
Problem: For \(n\) a positive integer, let \(f(n)\) be the quotient obtained when the sum of all positive divisors of \(n\) is divided by \(n.\) For example, \[f(14)=(1+2+7+14)\div 14=\frac{12}{7}\]
Find the value of \(f(768)-f(384).\) The original answer is in \(\frac{k}{m}\) format, please give the value of \(k + m\).
Answer: 193
Problem: The angle bisector of the acute angle formed at the origin by the graphs of the lines \(y = x\) and \(y=3x\) has equation \(y=kx.\) The original answer is in the form \(\frac{k+\sqrt{m}}{n}\). Please find the value of \(k + m + n\).
Answer: 8
Problem: How many values of $\theta$ in the interval $0<\theta\le 2\pi$ satisfy 
$1-3\sin\theta+5\cos3\theta = 0?$
Answer: 6