ChenDRAG/VeRL_math_validation huggingface dataset:
Problem: The grid below contains six rows with six points in each row. Points that are adjacent either horizontally or vertically are a distance of 2 units apart. Find the area of the irregularly shaped ten-sided figure shown.

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Answer: 66
Problem: Jesse has ten squares, labeled $1, 2, \dots, 10$. In how many ways can he color each square either red, green, yellow, or blue such that for all $1 \le i < j \le 10$, if $i$ divides $j$, then the $i$-th and $j$-th squares have different colors?
Answer: 2304
Problem: It is given a polygon with $2013$ sides labeled as $A_{1}A_{2}\ldots A_{2013}$. The vertices are marked with numbers such that the sum of the numbers on any $9$ consecutive vertices is constant and equal to $300$. Given that $A_{13}$ is marked with $13$ and $A_{20}$ is marked with $20$, determine the number with which $A_{2013}$ is marked.
Answer: 67
Problem: Bob, having little else to do, rolls a fair $6$-sided die until the sum of his rolls is greater than or equal to $700$. What is the expected number of rolls needed? Any answer within $0.0001$ of the correct answer will be accepted.
Answer: 200
Problem: A man rows at a speed of $2$ mph in still water. He sets out on a trip towards a spot $2$ miles downstream. He rows with the current until he is halfway there, then turns back and rows against the current for $15$ minutes. Then, he turns around again and rows with the current until he reaches his destination. The entire trip takes him $70$ minutes. The speed of the current can be represented as $\frac{p}{q}$ mph, where $p$ and $q$ are relatively prime positive integers. Find $10p + q$.
Answer: 47
Problem: Triangle $ABC$ has sidelengths $AB=1$, $BC=\sqrt{3}$, and $AC=2$. Points $D$, $E$, and $F$ are chosen on $AB$, $BC$, and $AC$ respectively, such that $\angle EDF = \angle DFA = 90^{\circ}$. Given that the maximum possible value of $[DEF]^2$ can be expressed as $\frac{a}{b}$ for positive integers $a, b$ with $\gcd(a, b) = 1$, find $a + b$. (Here $[DEF]$ denotes the area of triangle $DEF$.)
Answer: 67
Problem: Triangle $ABC$ satisfies $AB=28$, $BC=32$, and $CA=36$. Let $M$ and $N$ be the midpoints of $\overline{AB}$ and $\overline{AC}$ respectively. Point $P$ is the unique point in the plane of $\triangle ABC$ such that $\triangle PBM \sim \triangle PNC$. What is $AP$?
Answer: 27
Problem: It is known that odd prime numbers $x, y, z$ satisfy the following conditions:

- $x \mid (y^5 + 1)$
- $y \mid (z^5 + 1)$
- $z \mid (x^5 + 1)$

Find the minimum value of the product $xyz$.
Answer: 2013
Problem: How many ordered triples \((a, b, c)\) of positive integers are there such that none of \(a, b, c\) exceeds 2010 and each of \(a, b, c\) divides \(a + b + c\)?
Answer: 9045
Problem: Find the highest degree $k$ of $1991$ for which $1991^k$ divides the number 
\[ 1990^{1991^{1992}} + 1992^{1991^{1990}}. \]
Answer: 1991
Problem: Let $a$, $b$, and $c$ be real numbers satisfying the equations:
\[
a^3 + abc = 26 \\
b^3 + abc = 78 \\
c^3 - abc = 104
\]
Find $a^3 + b^3 + c^3$.
Answer: 184
Problem: Simon expands factored polynomials with his favorite AI, ChatSFFT. However, he has not paid for a premium ChatSFFT account, so when he goes to expand \((m - a)(n - b)\), where \(a, b, m, n\) are integers, ChatSFFT returns the sum of the two factors instead of the product. However, when Simon plugs in certain pairs of integer values for \(m\) and \(n\), he realizes that the value of ChatSFFT’s result is the same as the real result in terms of \(a\) and \(b\). How many such pairs are there?
Answer: 2
Problem: At CMU, markers come in two colors: blue and orange. Zachary fills a hat randomly with three markers such that each color is chosen with equal probability. Then, Chase shuffles an additional orange marker into the hat. If Zachary chooses one of the markers in the hat at random and it turns out to be orange, the probability that there is a second orange marker in the hat can be expressed as a simplified fraction $\frac{m}{n}$. Find $m+n$.
Answer: 39
Problem: Let \( n \) be a set of integers. \( S(n) \) is defined as the sum of the elements of \( n \). Consider the set \( T = \{1, 2, 3, 4, 5, 6, 7, 8, 9\} \). Let \( A \) and \( B \) be subsets of \( T \) such that \( A \cup B = T \) and \( A \cap B = \varnothing \). The probability that \( S(A) \geq 4S(B) \) can be expressed as \( \frac{p}{q} \). Compute \( p + q \).
Answer: 545
Problem: Find the smallest positive integer $n$ such that the 73 fractions $\frac{19}{n+21}, \frac{20}{n+22}, \frac{21}{n+23}, \ldots, \frac{91}{n+93}$ are all irreducible.
Answer: 95
Problem: What is the tens digit of the sum \[ \left(1!\right)^2 + \left(2!\right)^2 + \left(3!\right)^2 + \ldots + \left(2018!\right)^2 \] ?
Answer: 1
Problem: Peter is randomly filling boxes with candy. If he has 10 pieces of candy and 5 boxes in a row labeled $A$, $B$, $C$, $D$, and $E$, how many ways can he distribute the candy so that no two adjacent boxes are empty?
Answer: 771
Problem: Given the following matrix:

$$\begin{pmatrix}
11 & 17 & 25 & 19 & 16 \\
24 & 10 & 13 & 15 & 3 \\
12 & 5 & 14 & 2 & 18 \\
23 & 4 & 1 & 8 & 22 \\
6 & 20 & 7 & 21 & 9
\end{pmatrix},$$

choose five of these elements, no two from the same row or column, in such a way that the minimum of these elements is as large as possible.
Answer: 15
Problem: Four students from Harvard, including one named Jack, and five students from MIT, including one named Jill, are planning to attend a Boston Celtics game. However, they discovered that only $5$ tickets are available, meaning $4$ of them must return. Given that at least one student from each school must attend the game, and at least one of Jack or Jill must attend, how many ways are there to select the $5$ people who can attend the game?
Answer: 104
Problem: Find the sum of all integers $n$ for which $n - 3$ and $n^2 + 4$ are both perfect cubes.
Answer: 13
Problem: Let $ATHEM$ be a convex pentagon with $AT = 14$, $TH = MA = 20$, $HE = EM = 15$, and $\angle THE = \angle EMA = 90^{\circ}$. Find the area of $ATHEM$.
Answer: 468
Problem: Let $f(N) = N \left( \frac{9}{10} \right)^N$, and let $\frac{m}{n}$ denote the maximum value of $f(N)$, as $N$ ranges over the positive integers. If $m$ and $n$ are relatively prime positive integers, find the remainder when $m + n$ is divided by $1000$.
Answer: 401
Problem: Determine the number of all positive integers which cannot be written in the form $80k + 3m$, where $k, m \in \mathbb{N} = \{0, 1, 2, \ldots\}$.
Answer: 79
Problem: Find the sum of all integers $0 \le a \le 124$ such that $a^3 - 2$ is a multiple of $125$.
Answer: 53
Problem: Find the least possible value of $\frac{(x^2+1)(4y^2+1)(9z^2+1)}{6xyz}$ if $x$, $y$, $z$ are not necessarily distinct positive real numbers.
Answer: 8
Problem: I have $8$ unit cubes of different colors, which I want to glue together into a $2 \times 2 \times 2$ cube. How many distinct $2 \times 2 \times 2$ cubes can I make? Rotations of the same cube are not considered distinct, but reflections are.
Answer: 1680
Problem: There are $2016$ customers who entered a shop on a particular day. Each customer entered the shop exactly once (i.e., the customer entered the shop, stayed there for some time, and then left the shop without returning).

Find the maximal $k$ such that the following condition holds:

There are $k$ customers such that either all of them were in the shop at a specific time instance, or no two of them were both in the shop at any time instance.
Answer: 45
Problem: Let $x$ be a number such that $x + \frac{1}{x} = -1$. Determine the value of $x^{1994} + \frac{1}{x^{1994}}$.
Answer: -1
Problem: Let $r_1, r_2, \ldots, r_{47}$ be the roots of $x^{47} - 1 = 0$. Compute \( \sum_{i=1}^{47} r_i^{2020} \).
Answer: 0
Problem: To each pair of nonzero real numbers $a$ and $b$, a real number $a*b$ is assigned such that:

1. $a*(b*c) = (a*b)*c$ for all $a, b, c$.
2. $a*a = 1$ for all $a$.

Solve the equation $x*36 = 216$.
Answer: 7776
Problem: While working with some data for the Iowa City Hospital, James got up to get a drink of water. When he returned, his computer displayed the "blue screen of death" (it had crashed). While rebooting his computer, James remembered that he was nearly done with his calculations since the last time he saved his data. He also kicked himself for not saving before he got up from his desk.

He had computed three positive integers $a$, $b$, and $c$, and recalled that their product is $24$, but he didn’t remember the values of the three integers themselves. What he really needed was their sum. He knows that the sum is an even two-digit integer less than $25$ with fewer than $6$ divisors. Help James by computing $a+b+c$.
Answer: 10
Problem: Let $\Gamma$ be a circle with center $A$, radius $1$, and diameter $BX$. Let $\Omega$ be a circle with center $C$, radius $1$, and diameter $DY$, where $X$ and $Y$ are on the same side of $AC$. $\Gamma$ meets $\Omega$ at two points, one of which is $Z$. The lines tangent to $\Gamma$ and $\Omega$ that pass through $Z$ cut out a sector of the plane containing no part of either circle and with angle $60^\circ$. If $\angle XY C = \angle CAB$ and $\angle XCD = 90^\circ$, then the length of $XY$ can be written in the form $\tfrac{\sqrt a+\sqrt b}{c}$ for integers $a, b, c$ where $\gcd(a, b, c) = 1$. Find $a + b + c$.
Answer: 16
Problem: Consider the set $E = \{5, 6, 7, 8, 9\}$. For any partition $\{A, B\}$ of $E$, with both $A$ and $B$ non-empty, consider the number obtained by adding the product of elements of $A$ to the product of elements of $B$. Let $N$ be the largest prime number among these numbers. Find the sum of the digits of $N$.
Answer: 17
Problem: How many subsets of the set $\{1, 2, 3, \ldots, 12\}$ contain exactly one or two prime numbers?
Answer: 1920
Problem: Let $\mathcal{P}_1$ and $\mathcal{P}_2$ be two parabolas with distinct directrices $\ell_1$ and $\ell_2$ and distinct foci $F_1$ and $F_2$ respectively. It is known that $F_1F_2 \parallel \ell_1 \parallel \ell_2$, $F_1$ lies on $\mathcal{P}_2$, and $F_2$ lies on $\mathcal{P}_1$. The two parabolas intersect at distinct points $A$ and $B$. Given that $F_1F_2=1$, the value of $AB^2$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Find $100m+n$.
Answer: 1504
Problem: Compute the sum of all positive integers $n$ for which the expression \[ \frac{n+7}{\sqrt{n-1}} \] is an integer.
Answer: 89
Problem: Let $N$ be a positive integer whose digits add up to $23$. What is the greatest possible product the digits of $N$ can have?
Answer: 4374
Problem: Let $g(x) = 1 + 2x + 3x^2 + 4x^3 + \ldots$. Find the coefficient of $x^{2015}$ in $f(x) = \frac{g(x)}{1-x}$.
Answer: 2033136
Problem: Define a sequence $\{a_n\}_{n \geq 1}$ recursively by $a_1=1$, $a_2=2$, and for all integers $n \geq 2$, $a_{n+1}=(n+1)^{a_n}$. Determine the number of integers $k$ between $2$ and $2020$, inclusive, such that $k+1$ divides $a_k - 1$.
Answer: 1009
Problem: How many ordered triples $(a, b, c)$, where $a$, $b$, and $c$ are from the set $\{ 1, 2, 3, \dots, 17 \}$, satisfy the equation:
\[
  a^3 + b^3 + c^3 + 2abc = a^2b + a^2c + b^2c + ab^2 + ac^2 + bc^2
\]
Answer: 408
Problem: In the triangle $ABC$, we have $\angle ABC = 100^\circ$, $\angle ACB = 65^\circ$, $M \in AB$, $N \in AC$, and $\angle MCB = 55^\circ$, $\angle NBC = 80^\circ$. Find $\angle NMC$.
Answer: 25
Problem: In quadrilateral $ABCD$, $AB \parallel CD$ and $BC \perp AB$. Lines $AC$ and $BD$ intersect at $E$. If $AB = 20$, $BC = 2016$, and $CD = 16$, find the area of $\triangle BCE$.
Answer: 8960
Problem: Purple College keeps a careful count of its students as they progress each year from the freshman class to the sophomore class to the junior class and, finally, to the senior class. Each year at the college, one-third of the freshman class drops out of school, 40 students in the sophomore class drop out of school, and one-tenth of the junior class drops out of school. Given that the college only admits new freshman students and wants to begin each school year with 3400 students enrolled, how many students does it need to admit into the freshman class each year?
Answer: 1185
Problem: Given that
\[\sum_{x=1}^{70} \sum_{y=1}^{70} \frac{x^{y}}{y} = \frac{m}{67!}\] 
for some positive integer $m$, find $m \pmod{71}$.
Answer: 12
Problem: On a $3 \times 3$ board, the numbers from $1$ to $9$ are written in some order without repeating. We say that the arrangement obtained is Isthmian if the numbers in any two adjacent squares have different parity. Determine the number of different Isthmian arrangements.

Note: Two arrangements are considered equal if one can be obtained from the other by rotating the board.
Answer: 720
Problem: 39 students participated in a math competition. The exam consisted of 6 problems, and each problem was worth 1 point for a correct solution and 0 points for an incorrect solution. For any 3 students, there is at most 1 problem that was not solved by any of the three. Let $B$ be the sum of all of the scores of the 39 students. Find the smallest possible value of $B$.
Answer: 165
Problem: Suppose $N$ is a $6$-digit number with base-$10$ representation $\underline{a}\,\underline{b}\,\underline{c}\,\underline{d}\,\underline{e}\,\underline{f}$. If $N$ is $\frac{6}{7}$ of the number with base-$10$ representation $\underline{d}\,\underline{e}\,\underline{f}\,\underline{a}\,\underline{b}\,\underline{c}$, find $N$.
Answer: 461538
Problem: Let $a, b$ be integers such that all the roots of the equation $(x^2+ax+20)(x^2+17x+b) = 0$ are negative integers. What is the smallest possible value of $a + b$?
Answer: 25
Problem: Let $S$ be a subset of the natural numbers such that $0 \in S$, and for all $n \in \mathbb{N}$, if $n$ is in $S$, then both $2n+1$ and $3n+2$ are in $S$. What is the smallest number of elements $S$ can have in the range $\{0,1,\ldots, 2019\}$?
Answer: 47
Problem: Call an integer $k$ debatable if the number of odd factors of $k$ is a power of two. What is the largest positive integer $n$ such that there exists $n$ consecutive debatable numbers? (Here, a power of two is defined to be any number of the form $2^m$, where $m$ is a nonnegative integer.)
Answer: 17
Problem: The number $2021$ leaves a remainder of $11$ when divided by a positive integer. Find the smallest such integer.
Answer: 15
Problem: For a permutation $\pi$ of the integers from 1 to 10, define
\[ S(\pi) = \sum_{i=1}^{9} (\pi(i) - \pi(i+1))\cdot (4 + \pi(i) + \pi(i+1)), \]
where $\pi (i)$ denotes the $i$th element of the permutation. Suppose that $M$ is the maximum possible value of $S(\pi)$ over all permutations $\pi$ of the integers from 1 to 10. Determine the number of permutations $\pi$ for which $S(\pi) = M$.
Answer: 40320
Problem: One hundred billion light years from Earth is planet Glorp. The inhabitants of Glorp are intelligent, uniform, amorphous beings with constant density which can modify their shape in any way, and reproduce by splitting. Suppose a Glorpian has somehow formed itself into a spinning cylinder in a frictionless environment. It then splits itself into two Glorpians of equal mass, which proceed to mold themselves into cylinders of the same height, but not the same radius, as the original Glorpian. If the new Glorpians' angular velocities after this are equal and the angular velocity of the original Glorpian is $\omega$, let the angular velocity of each of the new Glorpians be $\omega'$. Then, find $ \left( \frac {\omega'}{\omega} \right)^{10} $.
Answer: 1024
Problem: There is a rectangle $ABCD$ such that $AB = 12$ and $BC = 7$. Points $E$ and $F$ lie on sides $AB$ and $CD$, respectively, such that $\frac{AE}{EB} = 1$ and $\frac{CF}{FD} = \frac{1}{2}$. Let $X$ be the intersection of lines $AF$ and $DE$. What is the area of pentagon $BCFXE$?
Answer: 47
Problem: How many ways are there to choose (not necessarily distinct) integers $a, b, c$ from the set $\{1, 2, 3, 4\}$ such that $a^{(b^c)}$ is divisible by $4$?
Answer: 28
Problem: Find $a + b + c$, where $a$, $b$, and $c$ are the hundreds, tens, and units digits of the six-digit number $123abc$, which is a multiple of $990$.
Answer: 12
Problem: David, when submitting a problem for CMIMC, wrote his answer as $100\frac{x}{y}$, where $x$ and $y$ are two positive integers with $x < y$. Andrew interpreted the expression as a product of two rational numbers, while Patrick interpreted the answer as a mixed fraction. In this case, Patrick's number was exactly double Andrew's. What is the smallest possible value of $x + y$?
Answer: 299
Problem: The sweet shop named "Olympiad" sells boxes containing either 6, 9, or 20 chocolates. Groups of students from a nearby school collect money to buy chocolates so that each student receives one. They buy a box and distribute the chocolates accordingly. For example, students can form groups of 15 (using one box of 6 and one box of 9 chocolates) or 38 (using two boxes of 9 and one box of 20 chocolates).

The shopkeeper has assured the students that he can fulfill the chocolate needs of any group. If a new box needs to be opened for any group size that doesn't fit perfectly (such as groups of 4, 7, or 10 students), then the shopkeeper will give all the chocolates for free to that group.

Is it possible to construct the largest group that receives free chocolates, and if so, how many students are there in this group?
Answer: 43
Problem: Let $a > b$ be positive integers. Compute the smallest possible integer value of \( \frac{a! + 1}{b! + 1} \).
Answer: 103
Problem: Let $n$ be a positive integer with $k$ digits. A number $m$ is called an \( alero \) of \( n \) if there exist distinct digits \( a_1, a_2, \dotsb, a_k \), all different from each other and from zero, such that \( m \) is obtained by adding the digit \( a_i \) to the \( i \)-th digit of \( n \), and no sum exceeds 9.

For example, if \( n = 2024 \) and we choose \( a_1 = 2 \), \( a_2 = 1 \), \( a_3 = 5 \), \( a_4 = 3 \), then \( m = 4177 \) is an alero of \( n \), but if we choose the digits \( a_1 = 2 \), \( a_2 = 1 \), \( a_3 = 5 \), \( a_4 = 6 \), then we don't obtain an alero of \( n \), because \( 4 + 6 \) exceeds 9.

Find the smallest \( n \) which is a multiple of \( 2024 \) that has an alero which is also a multiple of \( 2024 \).
Answer: 10120
Problem: Consider a right-angled triangle $ABC$ with $\angle C = 90^\circ$. Suppose that the hypotenuse $AB$ is divided into four equal parts by the points $D, E, F$, such that $AD = DE = EF = FB$. If $CD^2 + CE^2 + CF^2 = 350$, find the length of $AB$.
Answer: 20
Problem: In a tournament, each player played exactly one game against each of the other players. In each game, the winner was awarded 1 point, the loser got 0 points, and each player earned \(\frac{1}{2}\) point if the game was a tie. After the completion of the tournament, it was found that exactly half of the points earned by each player were earned against the ten players with the least number of points. In particular, each of the ten lowest-scoring players earned half of their points against the other nine of the ten. What was the total number of players in the tournament?
Answer: 25
Problem: This sequence lists the perfect squares in increasing order: $0, 1, 4, 9, 16, \cdots, a, 10^8, b, \cdots$. Determine the value of $b-a$.
Answer: 40000
Problem: In a mathematical competition, $n = 10,000$ contestants participate. During the final party, the process of distributing a cake is as follows: 

1. The first contestant takes $\frac{1}{n}$ of the cake.
2. The second contestant takes $\frac{2}{n}$ of the remaining cake.
3. The third contestant takes $\frac{3}{n}$ of what remains after the first and second contestants have taken their portions.
4. This process continues until the last contestant, who takes all of the remaining cake.

Determine which competitor takes the largest piece of cake.
Answer: 100
Problem: An ant starts at the origin of the Cartesian coordinate plane. Each minute, it moves randomly one unit in one of the directions: up, down, left, or right, with all four directions being equally likely. Its direction each minute is independent of its direction in any previous minutes. It stops when it reaches a point $(x,y)$ such that $|x|+|y|=3$. The expected number of moves it makes before stopping can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $100m+n$.
Answer: 3907
Problem: A 10-digit arrangement $0,1,2,3,4,5,6,7,8,9$ is called beautiful if:
1. When read from left to right, $0,1,2,3,4$ form an increasing sequence.
2. $5,6,7,8,9$ form a decreasing sequence.
3. $0$ is not the leftmost digit.

For example, $9807123654$ is a beautiful arrangement.

Determine the number of beautiful arrangements.
Answer: 126
Problem: How many $6$-tuples $(a, b, c, d, e, f)$ of natural numbers are there for which $a > b > c > d > e > f$ and $a + f = b + e = c + d = 30$ are simultaneously true?
Answer: 364
Problem: $A_1, A_2, \cdots, A_8$ are fixed points on a circle. Determine the smallest positive integer $n$ such that among any $n$ triangles with these eight points as vertices, two of them will have a common side.
Answer: 9
Problem: In a country consisting of $2015$ cities, there is exactly one direct round flight between any two cities, operated by some air company. Find the minimal possible number of air companies required if direct flights between any three cities are operated by three different air companies.
Answer: 2015
Problem: A positive integer $n > 1$ is called beautiful if $n$ can be written in one and only one way as $n = a_1 + a_2 + \cdots + a_k = a_1 \cdot a_2 \cdots a_k$ for some positive integers $a_1, a_2, \ldots, a_k$, where $k > 1$ and $a_1 \geq a_2 \geq \cdots \geq a_k$. For example, 6 is beautiful since $6 = 3 \cdot 2 \cdot 1 = 3 + 2 + 1$, and this is unique. But 8 is not beautiful since $8 = 4 + 2 + 1 + 1 = 4 \cdot 2 \cdot 1 \cdot 1$ as well as $8 = 2 + 2 + 2 + 1 + 1 = 2 \cdot 2 \cdot 2 \cdot 1 \cdot 1$, so uniqueness is lost. Find the largest beautiful number less than 100.
Answer: 95