ChenDRAG/VeRL_math_validation huggingface dataset:
Problem: There are three flies of negligible size that start at the same position on a circular track with a circumference of 1000 meters. They fly clockwise at speeds of 2, 6, and $k$ meters per second, respectively, where $k$ is some positive integer with $7 \le k \le 2013$. Suppose that at some point in time, all three flies meet at a location different from their starting point. How many possible values of $k$ are there?
Answer: 501
Problem: For what natural number $x$ will the value of the polynomial $x^3 + 7x^2 + 6x + 1$ be the cube of a natural number?
Answer: 7
Problem: Suppose that $x^2 + px + q$ has two distinct roots $x = a$ and $x = b$. Furthermore, suppose that the positive difference between the roots of $x^2 + ax + b$, the positive difference between the roots of $x^2 + bx + a$, and twice the positive difference between the roots of $x^2 + px + q$ are all equal. Given that $q$ can be expressed in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, compute $m+n$.
Answer: 21
Problem: A convex polyhedron has $n$ faces, all of which are congruent triangles with angles $36^{\circ}$, $72^{\circ}$, and $72^{\circ}$. Determine, with proof, the maximum possible value of $n$.
Answer: 20
Problem: Julia adds up the numbers from $1$ to $2016$ in a calculator. However, every time she inputs a $2$, the calculator malfunctions and inputs a $3$ instead. For example, when Julia inputs $202$, the calculator inputs $303$ instead. How much larger is the total sum returned by the broken calculator?
Answer: 39202
Problem: Of the $9$ people who reached the final stage of the competition, only $4$ should receive a prize. The candidates were renumbered and lined up in a circle. Then a certain number $m$ (possibly greater than $9$) and the direction of reference were determined. People began to be counted, starting from the first. Each one became a winner and was eliminated from the drawing, and counting, starting from the next, continued until four winners were identified. The first three prizes were awarded to three people who had numbers $2$, $7$, and $5$ in the original lineup (they were eliminated in that order). What number did the fourth winner of the competition have in the initial lineup?
Answer: 8
Problem: Centuries ago, the pirate Captain Blackboard buried a vast amount of treasure in a single cell of a $2 \times 4$ grid-structured island. The treasure was buried in a single cell of an $M \times N$ ($2 \le M$, $N$) grid. You and your crew have reached the island and have brought special treasure detectors to find the cell with the treasure. 

For each detector, you can set it up to scan a specific subgrid $[a,b] \times [c,d]$ with $1 \le a \le b \le 2$ and $1 \le c \le d \le 4$. Running the detector will tell you whether the treasure is in the region or not, though it cannot say where in the region the treasure was detected. 

You plan on setting up $Q$ detectors, which may only be run simultaneously after all $Q$ detectors are ready. What is the minimum $Q$ required to guarantee to determine the location of the Blackboard’s legendary treasure?
Answer: 4
Problem: The set $X$ has $1983$ members. There exists a family of subsets $\{S_1, S_2, \ldots , S_k \}$ such that:

1. The union of any three of these subsets is the entire set $X$.
2. The union of any two of them contains at most $1979$ members.

What is the largest possible value of $k$?
Answer: 31
Problem: Define a function $g : \mathbb{N} \rightarrow \mathbb{R}$ such that:
\[
g(x) = \sqrt{4^x + \sqrt{4^{x+1} + \sqrt{4^{x+2} + \cdots}}}
\]
Find the last two digits in the decimal representation of $g(2021)$.
Answer: 53
Problem: Let $S = \{ 1, 2, \ldots, 15 \}$. Let $A_1, A_2, \ldots, A_n$ be $n$ subsets of $S$ which satisfy the following conditions:

1. $|A_i| = 7$, for $i = 1, 2, \ldots, n$;

2. $|A_i \cap A_j| \leq 3$, for $1 \leq i < j \leq n$;

3. For any 3-element subset $M$ of $S$, there exists $A_k$ such that $M \subset A_k$.

Find the smallest possible value of $n$.
Answer: 15
Problem: Let $ABC$ be a triangle with $AB = 3$, $BC = 4$, and $CA = 5$. The line through $A$ perpendicular to $AC$ intersects line $BC$ at $D$, and the line through $C$ perpendicular to $AC$ intersects line $AB$ at $E$. Compute the area of triangle $BDE$.
Answer: 6
Problem: A natural number $n$ is at least two digits long. If we write a certain digit between the tens digit and the units digit of this number, we obtain six times the number $n$. Find all numbers $n$ with this property.
Answer: 18
Problem: Let $p$, $q$, and $r$ be prime numbers such that $2pqr + p + q + r = 2020$. Find $pq + qr + rp$.
Answer: 585
Problem: The Houson Association of Mathematics Educators decides to hold a grand forum on mathematics education and invites a number of politicians from around the United States to participate. Around lunch time, the politicians decide to play a game. In this game, players can score $19$ points for pegging the coordinator of the gathering with a spitball, $9$ points for downing an entire cup of the forum’s interpretation of coffee, or $8$ points for quoting more than three consecutive words from the speech Senator Bobbo delivered before lunch.

What is the product of the two greatest scores that a player cannot score in this game?
Answer: 1209
Problem: Let $P$ be the probability that the product of $2020$ real numbers chosen independently and uniformly at random from the interval $[-1, 2]$ is positive. The value of $2P - 1$ can be written in the form $\left(\frac{m}{n}\right)^b$, where $m$, $n$, and $b$ are positive integers such that $m$ and $n$ are relatively prime and $b$ is as large as possible. Compute $m + n + b$.
Answer: 2024
Problem: The equation $AB \times CD = EFGH$, where each letter $A$, $B$, $C$, $D$, $E$, $F$, $G$, and $H$ represents a different digit, and the values of $A$, $C$, and $E$ are all nonzero, has many solutions. For example, $46 \times 85 = 3910$. Find the smallest value of the four-digit number $EFGH$ for which there is a solution.
Answer: 1058
Problem: Ted flips seven fair coins. There are relatively prime positive integers $m$ and $n$ such that $\frac{m}{n}$ is the probability that Ted flips at least two heads given that he flips at least three tails. Find $m+n$.
Answer: 190
Problem: The first $510$ positive integers are written on a blackboard: $1, 2, 3, \ldots, 510$. An operation consists of erasing two numbers whose sum is a prime number. What is the maximum number of operations that can be performed in a row? Show how it is accomplished and explain why it cannot be done in more operations.
Answer: 255
Problem: The Garfield Super Winners play $100$ games of foosball, where teams score a non-negative integer number of points, and the team with more points after ten minutes wins (if both teams have the same number of points, it is a draw). Suppose that the Garfield Super Winners score an average of $7$ points per game but allow an average of $8$ points per game. Given that the Garfield Super Winners never won or lost by more than $10$, what is the largest possible number of games that they could win?
Answer: 81
Problem: There is a pile with $15$ coins on a table. At each step, Pedro chooses one of the piles on the table with $a > 1$ coins and divides it into two piles with $b \geq 1$ and $c \geq 1$ coins. He writes the product $abc$ on the board. He continues this process until there are $15$ piles, each with $1$ coin. Determine all possible values that the final sum of the numbers on the board can have.
Answer: 1120
Problem: A clueless ant makes the following route: starting at point $A$, it goes $1$ cm north, then $2$ cm east, then $3$ cm south, then $4$ cm west, immediately $5$ cm north, continues $6$ cm east, and so on, finally $41$ cm north and ends at point $B$. Calculate the distance between $A$ and $B$ (in a straight line).
Answer: 29
Problem: A semicircular paper is folded along a chord such that the folded circular arc is tangent to the diameter of the semicircle. The radius of the semicircle is $4$ units, and the point of tangency divides the diameter in the ratio $7:1$. If the length of the crease (the dotted line segment in the figure) is $\ell$, then determine $\ell^2$.
Answer: 39
Problem: Compute the number of geometric sequences of length $3$ where each number is a positive integer no larger than $10$.
Answer: 18
Problem: A positive integer $n$ is called \textit{good} if $2 \mid \tau(n)$ and if its divisors are $1 = d_1 < d_2 < \ldots < d_{2k-1} < d_{2k} = n$, then $d_{k+1} - d_k = 2$ and $d_{k+2} - d_{k-1} = 65$. Find the smallest \textit{good} number.
Answer: 2024
Problem: Let right $\triangle ABC$ have $AC = 3$, $BC = 4$, and a right angle at $C$. Let $D$ be the projection from $C$ to $\overline{AB}$. Let $\omega$ be a circle with center $D$ and radius $\overline{CD}$, and let $E$ be a variable point on the circumference of $\omega$. Let $F$ be the reflection of $E$ over point $D$, and let $O$ be the center of the circumcircle of $\triangle ABE$. Let $H$ be the intersection of the altitudes of $\triangle EFO$. As $E$ varies, the path of $H$ traces a region $\mathcal R$. The area of $\mathcal R$ can be written as $\tfrac{m\pi}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $\sqrt{m}+\sqrt{n}$.
Answer: 179
Problem: Define $a_k = 2^{2^{k-2013}} + k$ for all integers $k$. Simplify the expression:

$$(a_0 + a_1)(a_1 - a_0)(a_2 - a_1)\cdots(a_{2013} - a_{2012}).$$
Answer: 7
Problem: Define a permutation of the set $\{1,2,3,...,n\}$ to be \textit{sortable} if, upon cancelling an appropriate term of such permutation, the remaining $n-1$ terms are in increasing order. If $f(n)$ is the number of sortable permutations of $\{1,2,3,...,n\}$, find the remainder when \(\sum\limits_{i=1}^{2018} (-1)^i \cdot f(i)\) is divided by $1000$. Note that the empty set is considered sortable.
Answer: 153
Problem: A number is called "cool" if the sum of its digits is a multiple of $17$, and the sum of the digits of its successor is also a multiple of $17$. What is the smallest cool number?
Answer: 8899
Problem: For distinct positive integers $a, b < 2012$, define $f(a, b)$ to be the number of integers $k$ with $1 \leq k < 2012$ such that the remainder when $ak$ is divided by $2012$ is greater than that of $bk$ divided by $2012$. Let $S$ be the minimum value of $f(a, b)$, where $a$ and $b$ range over all pairs of distinct positive integers less than $2012$. Determine $S$.
Answer: 502
Problem: A pair of positive integers $(m,n)$ is called *compatible* if $m \ge \frac{1}{2} n + 7$ and $n \ge \frac{1}{2} m + 7$. A positive integer $k \ge 1$ is called *lonely* if $(k,\ell)$ is not compatible for any integer $\ell \ge 1$. Find the sum of all lonely integers.
Answer: 91
Problem: Find the sum of the non-repeated roots of the polynomial $P(x) = x^6 - 5x^5 - 4x^4 - 5x^3 + 8x^2 + 7x + 7$.
Answer: 7
Problem: Find the least natural number $n$, such that the following inequality holds:
\[
\sqrt{\frac{n-2011}{2012}} - \sqrt{\frac{n-2012}{2011}} < \sqrt[3]{\frac{n-2013}{2011}} - \sqrt[3]{\frac{n-2011}{2013}}
\]
Answer: 4024
Problem: Let $ABCD$ be a trapezoid such that $AB \parallel CD$ and let $P = AC \cap BD$, with $AB = 21$, $CD = 7$, $AD = 13$, and $[ABCD] = 168$. Let the line parallel to $AB$ through $P$ intersect the circumcircle of $\triangle BCP$ at $X$. The circumcircles of $\triangle BCP$ and $\triangle APD$ intersect at $P$ and $Y$. Let $XY \cap BC = Z$. If $\angle ADC$ is obtuse, then $BZ = \frac{a}{b}$, where $a$ and $b$ are coprime positive integers. Compute $a + b$.
Answer: 17
Problem: For a prime $q$, let $\Phi_q(x) = x^{q-1} + x^{q-2} + \cdots + x + 1$. 
Find the sum of all primes $p$ such that $3 \le p \le 100$ and there exists an odd prime $q$ and a positive integer $N$ satisfying
\[\binom{N}{\Phi_q(p)} \equiv \binom{2\Phi_q(p)}{N} \not\equiv 0 \pmod{p}.\]
Answer: 420
Problem: In a volleyball tournament for the Euro-African cup, there were nine more teams from Europe than from Africa. Each pair of teams played exactly once, and the European teams won precisely nine times as many matches as the African teams, overall. What is the maximum number of matches that a single African team might have won?
Answer: 11
Problem: There are exactly two four-digit numbers that are multiples of three. These numbers satisfy the following conditions:
1. The first digit is double the second digit.
2. The third digit is three more than the fourth digit.
3. The second digit is $2$ less than the fourth digit.

Find the difference between these two numbers.
Answer: 6333
Problem: For a particular value of the angle $\theta$, we can take the product of the two complex numbers $(8+i)\sin\theta + (7+4i)\cos\theta$ and $(1+8i)\sin\theta + (4+7i)\cos\theta$ to get a complex number in the form $a+bi$, where $a$ and $b$ are real numbers. Find the largest value for $a+b$.
Answer: 125
Problem: $ABC$ is an equilateral triangle with side length $11$ units. Consider the points $P_1, P_2, \dots, P_{10}$ dividing segment $BC$ into $11$ parts of unit length. Similarly, define $Q_1, Q_2, \dots, Q_{10}$ for the side $CA$ and $R_1, R_2, \dots, R_{10}$ for the side $AB$. Find the number of triples $(i,j,k)$ with $i,j,k \in \{1,2,\dots,10\}$ such that the centroids of triangles $ABC$ and $P_iQ_jR_k$ coincide.
Answer: 10
Problem: Elbert and Yaiza each draw $10$ cards from a $20$-card deck with cards numbered $1, 2, 3, \ldots, 20$. Then, starting with the player with the card numbered $1$, the players take turns placing down the lowest-numbered card from their hand that is greater than every card previously placed. When a player cannot place a card, they lose and the game ends.

Given that Yaiza lost and $5$ cards were placed in total, compute the number of ways the cards could have been initially distributed. (The order of cards in a player’s hand does not matter.)
Answer: 324
Problem: A "chaisri" figure is a triangle where the three vertices are vertices of a regular $2019$-gon. Two different chaisri figures may be formed by different regular $2019$-gons.

A "thubkaew" figure is a convex polygon that can be dissected into multiple chaisri figures, where each vertex of a dissected chaisri figure does not necessarily lie on the border of the convex polygon.

Determine the maximum number of vertices that a thubkaew figure may have.
Answer: 4038
Problem: What is the sum (in base $10$) of all the natural numbers less than $64$ which have exactly three ones in their base $2$ representation?
Answer: 630
Problem: In the figure, $ABCD$ is a convex quadrilateral. Points $K$, $L$, $M$, and $N$ are the midpoints of the sides $AB$, $BC$, $CD$, and $DA$, respectively. The quadrilateral $PQRS$ is formed by the intersections of lines $AK$, $BL$, $CM$, and $DN$. Determine the area of quadrilateral $PQRS$ given the following:

- The area of quadrilateral $ABCD$ is $3000$.
- The area of quadrilateral $AMQP$ is $513$.
- The area of quadrilateral $CKSR$ is $388$.
Answer: 599
Problem: For a positive integer $n$, define $f(n)$ as the largest integer $k$ such that $k! \leq n$. Define $g(n)$ as $n - (f(n))!$. For $j \geq 1$, define the function $g^j(n)$ as:
\[
g^j(n) = \underbrace{g(\dots(g(n))\dots)}_{\text{$j$ times}}.
\]
Find the smallest positive integer $n$ such that $g^j(n) > 0$ for all $j < 30$ and $g^{30}(n) = 0$.
Answer: 120959
Problem: A rectangle with sides $a$ and $b$ has an area of $24$ and a diagonal of length $11$. Find the perimeter of this rectangle.
Answer: 26
Problem: Suppose $z^3 = 2 + 2i$, where $i = \sqrt{-1}$. The product of all possible values of the real part of $z$ can be written in the form $\frac{p}{q}$ where $p$ and $q$ are relatively prime positive integers. Find $p + q$.
Answer: 3
Problem: Two bags contain some numbers, and the total number of numbers is prime.

When we transfer the number 170 from bag 1 to bag 2, the average in both bags increases by one.

If the total sum of all numbers is 2004, find the total number of numbers.
Answer: 167
Problem: There are $2022$ grids in a row. Two people, A and B, play a game with these grids. Initially, each odd-numbered grid from the left is marked with A's name, and each even-numbered grid from the left is marked with B's name. The players take turns performing the following action, starting with player A:

- One should select two grids marked with their own name, such that these two grids are not adjacent, and all grids between them are marked with the opponent's name. Then, change the name in all the grids between those two grids to one's own name.

The game continues until one player can no longer make a move. Find the largest positive integer $m$ such that:

- No matter how player B acts, player A can always ensure that there are at least $m$ grids marked with A's name.
Answer: 1011
Problem: The side lengths $a, b, c$ of a triangle $ABC$ are positive integers. Let:

$$T_{n} = (a+b+c)^{2n} - (a-b+c)^{2n} - (a+b-c)^{2n} + (a-b-c)^{2n}$$

for any positive integer $n$. If $\frac{T_{2}}{2T_{1}} = 2023$ and $a > b > c$, determine all possible perimeters of the triangle $ABC$.
Answer: 49
Problem: If the coefficient of the third term in the binomial expansion of $(1 - 3x)^{1/4}$ is $-\frac{a}{b}$, where $a$ and $b$ are relatively prime integers, find $a + b$.
Answer: 59
Problem: Let $A$ be the set of all sequences of 0's and 1's with length 4. What is the minimal number of sequences that can be chosen, such that any arbitrary sequence from $A$ differs in at most one position from one of the chosen sequences?
Answer: 4
Problem: Isosceles trapezoid $ABCD$ has side lengths $AB = 6$ and $CD = 12$, with $AD = BC$. It is given that $O$, the circumcenter of $ABCD$, lies in the interior of the trapezoid. The extensions of lines $AD$ and $BC$ intersect at $T$. Given that $OT = 18$, the area of $ABCD$ can be expressed as $a + b\sqrt{c}$ where $a$, $b$, and $c$ are positive integers, and $c$ is not divisible by the square of any prime. Compute $a+b+c$.
Answer: 84
Problem: A chess tournament is held with the participation of boys and girls. The number of girls is twice the number of boys. Each player plays against every other player exactly once. By the end of the tournament, there were no draws, and the ratio of girl winnings to boy winnings was $\frac{7}{9}$.  

How many players took part in the tournament?
Answer: 33
Problem: The boxes in the expression below are filled with the numbers $3$, $4$, $5$, $6$, $7$, and $8$, so that each number is used exactly once. What is the least possible value of the expression?

$$ \square \times \square + \square \times \square - \square \times \square $$
Answer: -18
Problem: The towns in one country are connected with bidirectional airlines, where at least one direction is paid. In a trip from town $A$ to town $B$, there are exactly 22 routes that are free. Find the least possible number of towns in the country.
Answer: 7
Problem: The side lengths of a scalene triangle are the roots of the polynomial \(x^3 - 20x^2 + 131x - 281.3\). Find the square of the area of the triangle.
Answer: 287
Problem: Let's say a positive integer $n$ is \textit{atresvido} if the set of its divisors (including 1 and $n$) can be split into 3 subsets such that the sum of the elements of each subset is the same. Determine the least number of divisors an \textit{atresvido} number can have.
Answer: 16
Problem: There are 2020 inhabitants in a town. Before Christmas, they are all happy. However, if an inhabitant does not receive any Christmas card from any other inhabitant, he or she will become sad. Unfortunately, there is only one post company that offers a single kind of service: before Christmas, each inhabitant may appoint two different other inhabitants, among which the company chooses one to whom to send a Christmas card on behalf of that inhabitant. It is known that the company makes the choices in such a way that as many inhabitants as possible will become sad. Find the least possible number of inhabitants who will become sad.
Answer: 674
Problem: $ABCD$ is a cyclic quadrilateral with circumcenter $O$ and circumradius $7$. The line $AB$ intersects $CD$ at $E$, and the line $DA$ intersects $CB$ at $F$. Given that $OE = 13$ and $OF = 14$, let $\cos \angle FOE = \frac{p}{q}$, where $p$ and $q$ are coprime integers. Find $p + q$.
Answer: 33
Problem: Let $f(n) = \sum_{d=1}^{n} \left\lfloor \frac{n}{d} \right\rfloor$ and $g(n) = f(n) - f(n - 1)$. For how many $n$ from $1$ to $100$ inclusive is $g(n)$ even?
Answer: 90
Problem: A "lame king" is a chess piece that can move from a cell to any cell that shares at least one vertex with it, except for the cells in the same column as the current cell. 

A lame king is placed in the top-left cell of a $7 \times 7$ grid. Compute the maximum number of cells it can visit without visiting the same cell twice (including its starting cell).
Answer: 25
Problem: Find all real numbers $a$ for which the following equation has a unique real solution:
$$\sum_{i=1}^{99} |x-i| = a.$$
Answer: 2450
Problem: A square grid $100 \times 100$ is tiled in two ways: only with dominoes and only with $2 \times 2$ squares. What is the minimum number of dominoes that are entirely inside some $2 \times 2$ square?
Answer: 100
Problem: Find the smallest prime number that cannot be expressed as the difference of a power of $2$ and a power of $3$, regardless of the order.
Answer: 41
Problem: Given that \[A=\sum_{n=1}^\infty\frac{\sin(n)}{n},\] determine $\lfloor 100A \rfloor$.
Answer: 107
Problem: What is the least possible sum of two positive integers $a$ and $b$ where $a \cdot b = 10!$?
Answer: 3810
Problem: There are $n$ schools in a city. The $i$th school dispatches $C_i$ students to watch a football match, where $1 \leq C_i \leq 39$ and $1 \leq i \leq n$. The total number of students is $\sum_{i=1}^{n} C_{i} = 1990$. Each line contains 199 seats, but students from the same school must sit in the same line. How many lines of seats are needed to ensure all students have a seat?
Answer: 11
Problem: Let $x, y, z$ be complex numbers satisfying the following system of equations:  
\[  
\begin{align*}  
z^2 + 5x &= 10z, \\  
y^2 + 5z &= 10y, \\  
x^2 + 5y &= 10x.  
\end{align*}  
\]  
Find the sum of all possible values of $z$.
Answer: 40
Problem: Let $ABC$ be an acute triangle with orthocenter $H$. Let $M$, $N$, and $R$ be the midpoints of $AB$, $BC$, and $AH$, respectively. If $\angle ABC = 70^\circ$, compute $\angle MNR$.
Answer: 20
Problem: For how many integers $n \neq 1$ does $(n-1)^3$ divide $n^{2018(n-1)} - 1$?
Answer: 5
Problem: In an old script found in the ruins of Persepolis, it is written:

"This script has been finished in a year whose 13th power is 258145266804692077858261512663. If you are skilled in Arithmetic, you will know the year this script is finished easily."

Find the year the script is finished. Provide a reason for your answer.
Answer: 183
Problem: How many values of $x$ satisfy the equation $$(x^2 - 9x + 19)^{x^2 + 16x + 60} = 1?$$
Answer: 5
Problem: 12 distinct points are equally spaced around a circle. How many ways can Bryan choose 3 points out of these 12 points such that they form an acute triangle? Rotations of a set of points are considered distinct.
Answer: 40
Problem: Let triangle $ABC$ with incenter $I$ and circumcircle $\Gamma$ satisfy $AB = 6\sqrt{3}$, $BC = 14$, and $CA = 22$. Construct points $P$ and $Q$ on rays $BA$ and $CA$ such that $BP = CQ = 14$. Lines $PI$ and $QI$ meet the tangents from $B$ and $C$ to $\Gamma$, respectively, at points $X$ and $Y$. If $XY$ can be expressed as $a\sqrt{b} - c$ for positive integers $a, b, c$ with $c$ squarefree, find $a + b + c$.
Answer: 31
Problem: Hamilton Avenue has eight houses. On one side of the street are the houses numbered $1, 3, 5, 7$ and directly opposite are houses $2, 4, 6, 8$ respectively. An eccentric postman starts deliveries at house $1$ and delivers letters to each of the houses, finally returning to house $1$ for a cup of tea. Throughout the entire journey he must observe the following rules:  
1. The numbers of the houses delivered to must follow an odd-even-odd-even pattern throughout.  
2. Each house except house $1$ is visited exactly once (house $1$ is visited twice).  
3. The postman at no time is allowed to cross the road to the house directly opposite.  

How many different delivery sequences are possible?
Answer: 12