ChenDRAG/VeRL_math_validation huggingface dataset:
Problem: Parallelogram $ABCD$ has area $1,000,000$. Vertex $A$ is at $(0,0)$ and all other vertices are in the first quadrant. Vertices $B$ and $D$ are lattice points on the lines $y = x$ and $y = kx$ for some integer $k > 1$, respectively. How many such parallelograms are there?
Answer: 784
Problem: Find the number of whole numbers between 1 and 1000 that do not contain the digit 1.
Answer: 728
Problem: Functions $f$ and $g$ are quadratic, $g(x) = - f(100 - x)$, and the graph of $g$ contains the vertex of the graph of $f$. The four $x$-intercepts on the two graphs have $x$-coordinates $x_1$, $x_2$, $x_3$, and $x_4$, in increasing order, and $x_3 - x_2 = 150$. Then $x_4 - x_1 = m + n\sqrt p$, where $m$, $n$, and $p$ are positive integers, and $p$ is not divisible by the square of any prime. Find the value of $m + n + p$.
Answer: 752
Problem: Convex quadrilateral $ABCD$ has $AB = 9$ and $CD = 12$. Diagonals $AC$ and $BD$ intersect at $E$, $AC = 14$, and $\triangle AED$ and $\triangle BEC$ have equal areas. What is the length of $AE$?
Answer: 6
Problem: Rachel and Robert run on a circular track. Rachel runs counterclockwise and completes a lap every \(90\) seconds, and Robert runs clockwise and completes a lap every \(80\) seconds. Both start from the same line at the same time. At some random time between \(10\) minutes and \(11\) minutes after they begin to run, a photographer standing inside the track takes a picture that shows one-fourth of the track, centered on the starting line. The original answer is in \(\frac{k}{m}\) format, please give the value of k + m, where k and m are integers in the simplest form of the fraction.
Answer: 19
Problem: A region \( S \) in the complex plane is defined by  
\( S = \{x + iy: -1 \le x \le 1, -1 \le y \le 1\} \).  
A complex number \( z = x + iy \) is chosen uniformly at random from \( S \).  
Find the probability that \(\left(\frac{3}{4} + \frac{3}{4}i\right)z\) is also in \( S \). The original answer is in \(\frac{k}{m}\) format, please give the value of \(k + m\).
Answer: 16
Problem: Three distinct vertices of a cube are chosen at random. What is the probability, in simplest form, that the plane determined by these three vertices contains points inside the cube? The original answer is in \(\frac{k}{m}\) format, where \(\frac{k}{m}\) is the simplest form of the probability. Please give the value of \(k + m\).
Answer: 11
Problem: The keystone arch is an ancient architectural feature. It is composed of congruent isosceles trapezoids fitted together along the non-parallel sides, as shown. The bottom sides of the two end trapezoids are horizontal. In an arch made with $9$ trapezoids, let $x$ be the angle measure in degrees of the larger interior angle of the trapezoid. If the answer is $x^\circ$, what is the value of $x$?
Answer: 100
Problem: The letters $A$, $B$, $C$ and $D$ represent digits. If $\begin{array}{rrr}&\mathrm{A}&\mathrm{B}\\+&\mathrm{C}&\mathrm{A}\\\hline&\mathrm{D}&\mathrm{A}\\\end{array}$ and $\begin{array}{ccc}&\mathrm{A}&\mathrm{B}\\-&\mathrm{C}&\mathrm{A}\\\hline&&\mathrm{A}\\\end{array}$, what digit does $D$ represent?
Answer: 9
Problem: Each face of a cube is given a single narrow stripe painted from the center of one edge to the center of the opposite edge. The choice of the edge pairing is made at random and independently for each face. What is the probability that there is a continuous stripe encircling the cube? Provide your answer as a simplified fraction in the form \frac{k}{m}, and find the value of k + m.
Answer: 19
Problem: A ticket to a school play costs $x$ dollars, where $x$ is a whole number. A group of $9$th graders buys tickets costing a total of $＄48$, and a group of $10$th graders buys tickets costing a total of $＄64$. How many values for $x$ are possible?
Answer: 5
Problem: A month with $31$ days has the same number of Mondays and Wednesdays. How many of the seven days of the week could be the first day of this month?
Answer: 3
Problem: Lucky Larry's teacher asked him to substitute numbers for $a$, $b$, $c$, $d$, and $e$ in the expression $a-(b-(c-(d+e)))$ and evaluate the result. Larry ignored the parentheses but added and subtracted correctly and obtained the correct result by coincidence. The number Larry substituted for $a$, $b$, $c$, and $d$ were $1$, $2$, $3$, and $4$, respectively. What number did Larry substitute for $e$?
Answer: 3
Problem: A month with 31 days has the same number of Mondays and Wednesdays. How many of the seven days of the week could be the first day of this month?
Answer: 3
Problem: 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
Problem: A palindrome, such as 83438, is a number that remains the same when its digits are reversed. The numbers \( x \) and \( x+32 \) are three-digit and four-digit palindromes, respectively. What is the sum of the digits of \( x \)?
Answer: 24
Problem: At the beginning of the school year, $50\%$ of all students in Mr. Well's 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 $x$?
Answer: 60
Problem: Every high school in the city of Euclid sent a team of $3$ students to a math contest. Each participant in the contest received a different score. Andrea's score was the median among all students, and hers was the highest score on her team. Andrea's teammates Beth and Carla placed $37$th and $64$th, respectively. How many schools are in the city? Fill in the blank with the number of schools.
Answer: 23
Problem: A ${palindrome}$, such as $83438$, is a number that remains the same when its digits are reversed. The numbers $x$ and $x+32$ are three-digit and four-digit palindromes, respectively. What is the sum of the digits of $x$?
Answer: 24
Problem: Let \( n \) be the smallest positive integer such that \( n \) is divisible by 20, \( n^2 \) is a perfect cube, and \( n^3 \) is a perfect square. Find the number of digits of \( n \).
Answer: 7
Problem: The average of the numbers \(1, 2, 3, \ldots, 98, 99,\) and \(x\) is \(100x\). The answer is in the form of \(\frac{k}{m}\), where \(k\) and \(m\) are integers. Please find the value of \(k + m\).
Answer: 151
Problem: The solution of the equation \(7^{x+7} = 8^x\) can be expressed in the form \(x = \log_b 7^7\). The original answer is in the format \(\frac{k}{m}\). Please give the value of \(k + m\).
Answer: 15
Problem: A palindrome between $1000$ and $10,000$ is chosen at random. Find the probability that it is divisible by $7$. The original answer is in \frac{k}{m} format, please give the value of k + m.
Answer: 6
Problem: In a magical swamp there are two species of talking amphibians: toads, whose statements are always true, and frogs, whose statements are always false. Four amphibians, Brian, Chris, LeRoy, and Mike live together in this swamp, and they make the following statements.  
  
Brian: "Mike and I are different species."  
  
Chris: "LeRoy is a frog."  
  
LeRoy: "Chris is a frog."  
  
Mike: "Of the four of us, at least two are toads."  
  
How many of these amphibians are frogs? Provide your answer as an integer.
Answer: 3
Problem: Angelina drove at an average rate of $80$ kmh and then stopped $20$ minutes for gas. After the stop, she drove at an average rate of $100$ kmh. Altogether she drove $250$ km in a total trip time of $3$ hours including the stop. Write the equation that could be used to solve for the time $t$ in hours that she drove before her stop. The original answer is in the form kt + m\left(\frac{n}{p} - t\right) = 250, please give the value of k + m + n + p.
Answer: 191
Problem: The average of the numbers $1, 2, 3, \cdots, 98, 99,$ and $x$ is $100x$. The original answer is in the form of \frac{k}{m}, where k and m are integers. Find the value of k + m.
Answer: 151
Problem: Nondegenerate $\triangle ABC$ has integer side lengths, $\overline{BD}$ is an angle bisector, $AD = 3$, and $DC=8$. What is the smallest possible value of the perimeter of $\triangle ABC$?
Answer: 33
Problem: Let $a$, $b$, $c$, $d$, and $e$ be positive integers with $a+b+c+d+e=2010$ and let $M$ be the largest of the sums $a+b$, $b+c$, $c+d$, and $d+e$. What is the smallest possible value of $M$?
Answer: 671
Problem: In a magical swamp there are two species of talking amphibians: toads, whose statements are always true, and frogs, whose statements are always false. Four amphibians, Brian, Chris, LeRoy, and Mike live together in this swamp, and they make the following statements.
Brian: "Mike and I are different species."
Chris: "LeRoy is a frog."
LeRoy: "Chris is a frog."
Mike: "Of the four of us, at least two are toads."
How many of these amphibians are frogs? Fill in the blank with the number of frogs.
Answer: 3
Problem: On a $50$-question multiple choice math contest, students receive $4$ points for a correct answer, $0$ points for an answer left blank, and $-1$ point for an incorrect answer. Jesse’s total score on the contest was $99$. What is the maximum number of questions that Jesse could have answered correctly?
Answer: 29
Problem: A coin is altered so that the probability that it lands on heads is less than \(\frac{1}{2}\) and when the coin is flipped four times, the probability of an equal number of heads and tails is \(\frac{1}{6}\). What is the probability that the coin lands on heads? Express your answer in the form \(\frac{k-\sqrt{m}}{n}\), where \(k\), \(m\), and \(n\) are integers, and give the value of \(k+m+n\).
Answer: 12
Problem: For how many ordered triples $(x,y,z)$ of nonnegative integers less than $20$ are there exactly two distinct elements in the set $\{i^x, (1+i)^y, z\}$, where $i=\sqrt{-1}$?
Answer: 225
Problem: A square of side length $1$ and a circle of radius $\dfrac{\sqrt{3}}{3}$ share the same center. Find the area inside the circle, but outside the square. The original answer is in the form \frac{k\pi}{m} - \frac{\sqrt{n}}{p}, please give the value of k + m + n + p.
Answer: 17
Problem: Bernardo randomly picks $3$ distinct numbers from the set \(\{1,2,3,\cdots,7,8,9\}\) and arranges them in descending order to form a $3$-digit number. Silvia randomly picks $3$ distinct numbers from the set \(\{1,2,3,\cdots,6,7,8\}\) and also arranges them in descending order to form a $3$-digit number. What is the probability that Bernardo's number is larger than Silvia's number? The answer should be in the form \(\frac{k}{m}\) in simplest terms. Please calculate the value of \(k + m\).
Answer: 93
Problem: Positive integers $a$, $b$, and $c$ are randomly and independently selected with replacement from the set $\{1, 2, 3,\dots, 2010\}$. Find the probability that $abc + ab + a$ is divisible by $3$. The original answer is in the form of a simplified fraction $\frac{m}{n}$, please give the value of $m + n$.
Answer: 40
Problem: A solid cube has side length $3$ 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
Problem: Every high school in the city of Euclid sent a team of $3$ students to a math contest. Each participant in the contest received a different score. Andrea's score was the median among all students, and hers was the highest score on her team. Andrea's teammates Beth and Carla placed $37$th and $64$th, respectively. How many schools are in the city?
Answer: 23
Problem: Equiangular hexagon $ABCDEF$ has side lengths $AB=CD=EF=1$ and $BC=DE=FA=r$. The area of $\triangle ACE$ is $70\%$ of the area of the hexagon. What is the sum of all possible values of $r$?
Answer: 6
Problem: The diagram shows an octagon consisting of $10$ unit squares. The portion below $\overline{PQ}$ is a unit square and a triangle with base $5$. If $\overline{PQ}$ bisects the area of the octagon, find the ratio \(\frac{XQ}{QY}\). The original answer is in the form \(\frac{k}{m}\), please provide the value of \(k + m\).
Answer: 5
Problem: Bernardo randomly picks $3$ distinct numbers from the set $\{1,2,3,...,7,8,9\}$ and arranges them in descending order to form a 3-digit number. Silvia randomly picks $3$ distinct numbers from the set $\{1,2,3,...,6,7,8\}$ and also arranges them in descending order to form a $3$-digit number. Find the probability that Bernardo's number is larger than Silvia's number. The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
Answer: 93
Problem: Positive integers \(a\), \(b\), and \(c\) are randomly and independently selected with replacement from the set \(\{1, 2, 3, \cdots, 2010\}\). Find the probability that \(abc + ab + a\) is divisible by \(3\). The original answer is in \(\frac{k}{m}\) format, please give the value of \(k + m\).
Answer: 40
Problem: 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 how many such paths stay outside or on the boundary of the square $-2 \le x \le 2$, $-2 \le y \le 2$ at each step.
Answer: 1698
Problem: A frog makes $3$ jumps, each exactly $1$ meter long. The directions of the jumps are chosen independently at random. Find the probability that the frog's final position is no more than $1$ meter from its starting position. The original answer is in the form \(\frac{k}{m}\). Please give the value of \(k + m\).
Answer: 5
Problem: Each of $2010$ boxes in a line contains a single red marble, and for $1 \le k \le 2010$, the box in the $k{th}$ position also contains $k$ white marbles. Isabella begins at the first box and successively draws a single marble at random from each box, in order. She stops when she first draws a red marble. Let $P(n)$ be the probability that Isabella stops after drawing exactly $n$ marbles. Find the smallest value of $n$ for which $P(n) < \frac{1}{2010}$.
Answer: 45
Problem: A high school basketball game between the Raiders and Wildcats was tied at the end of the first quarter. The number of points scored by the Raiders in each of the four quarters formed an increasing geometric sequence, and the number of points scored by the Wildcats in each of the four quarters formed an increasing arithmetic sequence. At the end of the fourth quarter, the Raiders had won by one point. Neither team scored more than $100$ points. What was the total number of points scored by the two teams in the first half?
Answer: 34
Problem: A fly trapped inside a cubical box with side length \(1\) meter decides to relieve its boredom by visiting each corner of the box. It will begin and end in the same corner and visit each of the other corners exactly once. To get from a corner to any other corner, it will either fly or crawl in a straight line. The maximum possible length of its path is in the form of \(k\sqrt{2} + m\sqrt{3}\). Please find the value of \(k + m\).
Answer: 8
Problem: Arithmetic sequences $\left(a_n\right)$ and $\left(b_n\right)$ have integer terms with $a_1=b_1=1<a_2 \le b_2$ and $a_n b_n = 2010$ for some $n$. What is the largest possible value of $n$?
Answer: 8
Problem: A geometric sequence $(a_n)$ has $a_1=\sin x$, $a_2=\cos x$, and $a_3= \tan x$ for some real number $x$. Find the value of $n$ such that $a_n=1+\cos x$.
Answer: 8
Problem: The polynomial $x^3-ax^2+bx-2010$ has three positive integer roots. What is the smallest possible value of $a$?
Answer: 78
Problem: 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 these values of $x$?
Answer: 4
Problem: Let $a > 0$, and let $P(x)$ be a polynomial with integer coefficients such that  
  
$P(1) = P(3) = P(5) = P(7) = a,$  and  
  
$P(2) = P(4) = P(6) = P(8) = -a.$ 
  
What is the smallest possible value of $a$?
Answer: 315
Problem: Let \(ABCD\) be a cyclic quadrilateral. The side lengths of \(ABCD\) are distinct integers less than \(15\) such that \(BC\cdot CD=AB\cdot DA\). Find the largest possible value of \(BD\) in the form \(\sqrt{\frac{x}{y}}\). Please provide the value of \(x + y\).
Answer: 427
Problem: Seven distinct pieces of candy are to be distributed among three bags. The red bag and the blue bag must each receive at least one piece of candy; the white bag may remain empty. How many arrangements are possible?
Answer: 1932
Problem: Each of $2010$ boxes in a line contains a single red marble, and for $1 \le k \le 2010$, the box in the $k{th}$ position also contains $k$ white marbles. Isabella begins at the first box and successively draws a single marble at random from each box, in order. She stops when she first draws a red marble. Let $P(n)$ be the probability that Isabella stops after drawing exactly $n$ marbles. What is the smallest value of $n$ for which $P(n) < \frac{1}{2010}$?
Answer: 45
Problem: The number obtained from the last two nonzero digits of $90!$ is equal to $n$. What is the value of $n$?
Answer: 12
Problem: Monic quadratic polynomial $P(x)$ and $Q(x)$ have the property that $P(Q(x))$ has zeros at $x=-23, -21, -17,$ and $-15$, and $Q(P(x))$ has zeros at $x=-59,-57,-51$ and $-49$. Find the sum of the minimum values of $P(x)$ and $Q(x)$.
Answer: -100
Problem: Semicircles $POQ$ and $ROS$ pass through the center $O$. What is the ratio of the combined areas of the two semicircles to the area of circle $O$? The answer is in the form of \frac{k}{m}, where k and m are integers. Please provide the value of k + m.
Answer: 3
Problem: $A$ high school basketball game between the Raiders and Wildcats was tied at the end of the first quarter. The number of points scored by the Raiders in each of the four quarters formed an increasing geometric sequence, and the number of points scored by the Wildcats in each of the four quarters formed an increasing arithmetic sequence. At the end of the fourth quarter, the Raiders had won by one point. Neither team scored more than $100$ points. What was the total number of points scored by the two teams in the first half?
Answer: 34
Problem: Let $f(x) = \log_{10} \left(\sin(\pi x) \cdot \sin(2 \pi x) \cdot \sin (3 \pi x) \cdots \sin(8 \pi x)\right)$. The intersection of the domain of $f(x)$ with the interval $[0,1]$ is a union of $n$ disjoint open intervals. What is the value of $n$?
Answer: 12
Problem: The set of real numbers $x$ for which  

$$\dfrac{1}{x-2009}+\dfrac{1}{x-2010}+\dfrac{1}{x-2011}\ge1$$  

is the union of intervals of the form $a<x\le b$. What is the sum of the lengths of these intervals?
Answer: 3
Problem: Jim starts with a positive integer $n$ and creates a sequence of numbers. Each successive number is obtained by subtracting the largest possible integer square less than or equal to the current number until zero is reached. For example, if Jim starts with $n = 55$, then his sequence contains $5$ numbers:
$$\begin{array}{ccccc}
{}&{}&{}&{}&55\\
55&-&7^2&=&6\\
6&-&2^2&=&2\\
2&-&1^2&=&1\\
1&-&1^2&=&0\\
\end{array}$$
Let $N$ be the smallest number for which Jim’s sequence has $8$ numbers. What is the units digit of $N$?
Answer: 3
Problem: Let $a > 0$, and let $P(x)$ be a polynomial with integer coefficients such that  
$P(1) = P(3) = P(5) = P(7) = a$,  and  
$P(2) = P(4) = P(6) = P(8) = -a$.  
Find the smallest possible value of $a$.
Answer: 315
Problem: Two quadrilaterals are considered the same if one can be obtained from the other by a rotation and a translation. How many different convex cyclic quadrilaterals are there with integer sides and perimeter equal to $32$? Provide your answer as an integer.
Answer: 568
Problem: Suppose [$a$ $b$] denotes the average of $a$ and $b$, and {$a$ $b$ $c$} denotes the average of $a$, $b$, and $c$. Calculate the value of $\{\{{1 1 0}\} { [0 1] } 0\}$. The original answer is in \(\frac{k}{m}\) format, please give the value of k + m.
Answer: 25
Problem: Extend the square pattern of $8$ black and $17$ white square tiles by attaching a border of black tiles around the square. What is the ratio of black tiles to white tiles in the extended pattern? The original answer is in the form of a ratio, $a:b$. Please provide the value of $a + b$.
Answer: 49
Problem: Last summer $30\%$ of the birds living on Town Lake were geese, $25\%$ were swans, $10\%$ were herons, and $35\%$ were ducks. What is the percentage of the birds that were not swans and were geese? If the answer is x%, what is the value of x?
Answer: 40
Problem: Let $N$ be the second smallest positive integer that is divisible by every positive integer less than $7$. What is the sum of the digits of $N$?
Answer: 3
Problem: The players on a basketball team made some three-point shots, some two-point shots, and some one-point free throws. They scored as many points with two-point shots as with three-point shots. Their number of successful free throws was one more than their number of successful two-point shots. The team's total score was $61$ points. How many free throws did they make?
Answer: 13
Problem: Two tangents to a circle are drawn from a point $A$. The points of contact $B$ and $C$ divide the circle into arcs with lengths in the ratio $2 : 3$. What is the degree measure of $\angle{BAC}$? If the answer is $x^\circ$, what is the value of $x$?
Answer: 36
Problem: A majority of the $30$ students in Ms. Demeanor's class bought pencils at the school bookstore. Each of these students bought the same number of pencils, and this number was greater than $1$. The cost of a pencil in cents was greater than the number of pencils each student bought, and the total cost of all the pencils was $17.71$. What was the cost of a pencil in cents?
Answer: 11
Problem: Let \(x\) and \(y\) be two-digit positive integers with mean \(60\). Find the maximum value of the ratio \(\frac{x}{y}\). The answer is in the form \(\frac{m}{n}\), please provide the value of \(m + n\).
Answer: 40
Problem: In the eight term sequence $A$, $B$, $C$, $D$, $E$, $F$, $G$, $H$, the value of $C$ is $5$ and the sum of any three consecutive terms is $30$. Find the value of $A+H$.
Answer: 25
Problem: At a twins and triplets convention, there were $9$ sets of twins and $6$ sets of triplets, all from different families. Each twin shook hands with all the twins except his/her siblings and with half the triplets. Each triplet shook hands with all the triplets except his/her siblings and with half the twins. How many handshakes took place?
Answer: 441
Problem: Two real numbers are selected independently and at random from the interval \([-20,10]\). Find the probability that the product of those numbers is greater than zero. Provide your answer as a simplified fraction \(\frac{k}{m}\), and then give the value of \(k + m\).
Answer: 14
Problem: A pair of standard 6-sided dice is rolled once. The sum of the numbers rolled determines the diameter of a circle. What is the probability that the numerical value of the area of the circle is less than the numerical value of the circle's circumference? Provide your answer as a fraction in simplest form, \(\frac{k}{m}\), and find the value of k + m.
Answer: 13
Problem: Rectangle $ABCD$ has $AB=6$ and $BC=3$. Point $M$ is chosen on side $AB$ so that $\angle AMD=\angle CMD$. If the degree measure of $\angle AMD$ is $x^\circ$, what is the value of $x$?
Answer: 75
Problem: Circles $A, B,$ and $C$ each have radius $1$. 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