ChenDRAG/VeRL_math_validation huggingface dataset:
Problem: Alison is eating $2401$ grains of rice for lunch. She eats the rice in a peculiar manner: 

1. If she has only one grain of rice remaining, she eats it.
2. Otherwise, she finds the smallest positive integer $d > 1$ for which she can group the rice into equal groups of size $d$ with none left over. 
3. She then groups the rice into groups of size $d$, eats one grain from each group, and puts the rice back into a single pile.

How many steps does it take her to finish all her rice?
Answer: 17
Problem: Nils has a telephone number with eight different digits. He has made $28$ cards with statements of the type "The digit $a$ occurs earlier than the digit $b$ in my telephone number" – one for each pair of digits appearing in his number. How many cards can Nils show you without revealing his number?
Answer: 27
Problem: For a math tournament, each person is assigned an ID which consists of two uppercase letters followed by two digits. All IDs have the property that either the letters are the same, the digits are the same, or both the letters are the same and the digits are the same. Compute the number of possible IDs that the tournament can generate.
Answer: 9100
Problem: Compute the area of the resulting shape, drawn in red above. 

\[ \text{[asy]} 
\text{size(5cm);} 
\text{defaultpen(fontsize(6pt));} 
\text{draw((0,0)--(4,0)--(4,4)--(0,4)--cycle);} 
\text{draw((0,0)--(-4,0)--(-4,-4)--(0,-4)--cycle);} 
\text{draw((1,-1)--(1,3)--(-3,3)--(-3,-1)--cycle);} 
\text{draw((-1,1)--(-1,-3)--(3,-3)--(3,1)--cycle);} 
\text{draw((-4,-4)--(0,-4)--(0,-3)--(3,-3)--(3,0)--(4,0)--(4,4)--(0,4)--(0,3)--(-3,3)--(-3,0)--(-4,0)--cycle, red+1.2);} 
\text{label("1", (-3.5,0), S);} 
\text{label("2", (-2,0), S);} 
\text{label("1", (-0.5,0), S);} 
\text{label("1", (3.5,0), S);} 
\text{label("2", (2,0), S);} 
\text{label("1", (0.5,0), S);} 
\text{label("1", (0,3.5), E);} 
\text{label("2", (0,2), E);} 
\text{label("1", (0,0.5), E);} 
\text{label("1", (0,-3.5), E);} 
\text{label("2", (0,-2), E);} 
\text{label("1", (0,-0.5), E);} 
\text{[/asy]} \]
Answer: 50
Problem: When Meena turned 16 years old, her parents gave her a cake with \( n \) candles, where \( n \) has exactly 16 different positive integer divisors. What is the smallest possible value of \( n \)?
Answer: 120
Problem: You have 5000 distinct finite sets. Their intersection is empty. However, the intersection of any two sets is nonempty. What is the smallest possible number of elements contained in their union?
Answer: 14
Problem: The largest factor of \( n \) not equal to \( n \) is 35. Compute the largest possible value of \( n \).
Answer: 175
Problem: Suppose a biased coin gives head with probability $\frac{2}{3}$. The coin is tossed repeatedly. If it shows heads, then player $A$ rolls a fair die; otherwise, player $B$ rolls the same die. The process ends when one of the players gets a $6$, and that player is declared the winner.\n\nIf the probability that $A$ will win is given by $\frac{m}{n}$, where $m$ and $n$ are coprime, then what is the value of $m^2n$?
Answer: 12
Problem: On the party, every boy gave $1$ candy to every girl, and every girl gave $1$ candy to every boy. Then, every boy ate $2$ candies, and every girl ate $3$ candies. It is known that $\frac{1}{4}$ of all candies were eaten. Find the greatest possible number of children at the party.
Answer: 35
Problem: A page of an exercise book is painted with $23$ colors, arranged in squares. A pair of colors is called \textit{good} if there are neighboring squares painted with these colors. What is the minimum number of good pairs?
Answer: 22
Problem: In the parallelogram $ABCD$, a line through $C$ intersects the diagonal $BD$ at $E$ and $AB$ at $F$. If $F$ is the midpoint of $AB$ and the area of $\triangle BEC$ is $100$, find the area of the quadrilateral $AFED$.
Answer: 250
Problem: Let $a$ and $b$ be positive integers such that $a + ab = 1443$ and $ab + b = 1444$. Find $10a + b$.
Answer: 408
Problem: Define $f(n) = \text{LCM}(1, 2, \ldots, n)$. Determine the smallest positive integer $a$ such that $f(a) = f(a+2)$.
Answer: 13
Problem: George and two of his friends go to a famous jiaozi restaurant, which serves only two kinds of jiaozi: pork jiaozi and vegetable jiaozi. Each person orders exactly $15$ jiaozi. How many different ways could the three of them order? Two ways of ordering are different if one person orders a different number of pork jiaozi in both orders.
Answer: 4096
Problem: A positive integer is called \textit{oneic} if it consists of only $1$'s. For example, the smallest three oneic numbers are $1$, $11$, and $111$. Find the number of $1$'s in the smallest oneic number that is divisible by $63$.
Answer: 18
Problem: $A, B, C$, and $D$ are all on a circle, and $ABCD$ is a convex quadrilateral. If $AB = 13$, $BC = 13$, $CD = 37$, and $AD = 47$, what is the area of $ABCD$?
Answer: 504
Problem: The sequence of positive integers $a_1, a_2, \ldots, a_{2025}$ is defined as follows:
- $a_1 = 2^{2024} + 1$
- For $1 \leq n \leq 2024$, $a_{n+1}$ is the greatest prime factor of $a_n^2 - 1$.

Find the value of $a_{2024} + a_{2025}$.
Answer: 5
Problem: A convex $2019$-gon $A_1A_2\ldots A_{2019}$ is divided into smaller pieces by cutting along its $2019$ diagonals of the form $A_iA_{i+3}$ for $1 \le i \le 2019$, where $A_{2020} = A_1$, $A_{2021} = A_2$, and $A_{2022} = A_3$. What is the least possible number of resulting pieces?
Answer: 2019
Problem: 5 people stand in a line facing one direction. In every round, the person at the front moves randomly to any position in the line, including the front or the end. Suppose that \( \frac{m}{n} \) is the expected number of rounds needed for the last person of the initial line to appear at the front of the line, where \( m \) and \( n \) are relatively prime positive integers. What is \( m + n \)?
Answer: 137
Problem: Let $k$ be a positive integer. A sequence $a_0, a_1, \ldots, a_n$, where $n > 0$, of positive integers satisfies the following conditions:

1. $a_0 = a_n = 1$;
2. $2 \leq a_i \leq k$ for each $i = 1, 2, \ldots, n-1$;
3. For each $j = 2, 3, \ldots, k$, the number $j$ appears $\phi(j)$ times in the sequence $a_0, a_1, \ldots, a_n$, where $\phi(j)$ is the number of positive integers that do not exceed $j$ and are coprime to $j$;
4. For any $i = 1, 2, \ldots, n-1$, $\gcd(a_i, a_{i-1}) = 1 = \gcd(a_i, a_{i+1})$, and $a_i$ divides $a_{i-1} + a_{i+1}$.

Suppose there is another sequence $b_0, b_1, \ldots, b_n$ of integers such that $\frac{b_{i+1}}{a_{i+1}} > \frac{b_i}{a_i}$ for all $i = 0, 1, \ldots, n-1$. Find the minimum value of $b_n - b_0$.
Answer: 1
Problem: \[ \int_{-10}^{10} \left| 4 - \left| 3 - \left| 2 - \left| 1 - |x| \right| \right| \right| \right| \, \mathrm{d}x \]
Answer: 44
Problem: Let $f_n(x) = n + x^2$. Evaluate the product \( \text{gcd}\{f_{2001}(2002), f_{2001}(2003)\} \times \text{gcd}\{f_{2011}(2012), f_{2011}(2013)\} \times \text{gcd}\{f_{2021}(2022), f_{2021}(2023)\} \), where \( \text{gcd}\{x, y\} \) is the greatest common divisor of \( x \) and \( y \).
Answer: 125
Problem: Gabriela found an encyclopedia with $2023$ pages, numbered from $1$ to $2023$. She noticed that the pages formed only by even digits have a blue mark, and that every three pages starting from page two have a red mark. How many pages of the encyclopedia have both colors?
Answer: 44
Problem: There are 300 children in a camp. Everyone has no more than $k-1$ friends. What is the smallest $k$ for which it might be impossible to create some new friendships so that everyone has exactly $k$ friends?
Answer: 151
Problem: Let $x_n$ be the smallest positive integer such that $7^n$ divides $x_n^2 - 2$. Find $x_1 + x_2 + x_3$.
Answer: 121
Problem: Suppose \((a,b)\) is an ordered pair of integers such that the three numbers \(a\), \(b\), and \(ab\) form an arithmetic progression, in that order. Find the sum of all possible values of \(a\).
Answer: 8
Problem: The area of a circle centered at the origin, which is inscribed in the parabola $y = x^2 - 25$, can be expressed as $\frac{a}{b}\pi$, where $a$ and $b$ are coprime positive integers. What is the value of $a + b$?
Answer: 103
Problem: The product \(\left(\frac{1}{2^3-1}+\frac{1}{2}\right)\left(\frac{1}{3^3-1}+\frac{1}{2}\right)\left(\frac{1}{4^3-1}+\frac{1}{2}\right)\cdots\left(\frac{1}{100^3-1}+\frac{1}{2}\right)\) can be written as \(\frac{r}{s2^t}\) where \(r\), \(s\), and \(t\) are positive integers and \(r\) and \(s\) are odd and relatively prime. Find \(r+s+t\).
Answer: 5990
Problem: Let $Q(x) = a_0 + a_1x + \dots + a_nx^n$ be a polynomial with integer coefficients, and $0 \le a_i < 3$ for all $0 \le i \le n$.

Given that $Q(\sqrt{3}) = 20 + 17\sqrt{3}$, compute $Q(2)$.
Answer: 86
Problem: A meeting is held at a round table. It is known that:
- 7 women have a woman on their right side.
- 12 women have a man on their right side.
- 75% of the men have a woman on their right side.

Determine the total number of people sitting at the round table.
Answer: 35
Problem: Set $M=\{1,2,\ldots,1000\}$. For any $X\subseteq M$ ($X\neq\varnothing$), define $a_X$ as the sum of the minimum and maximum numbers in $X$. Determine the arithmetic mean of all $a_X$.
Answer: 1001
Problem: Amy places positive integers in each of these cells so that each row and column contains each of $1, 2, 3, 4, 5$ exactly once. Find the sum of the numbers in the gray cells.

\begin{asy}
import graph; size(4cm);
fill((4,0)--(10,0)--(10,6)--(4,6)--cycle, gray(.7));
draw((0,0)--(10,0));
draw((0,2)--(10,2));
draw((0,4)--(10,4));
draw((0,6)--(10,6));
draw((0,8)--(10,8));
draw((0,10)--(10,10));
draw((0,0)--(0,10));
draw((2,0)--(2,10));
draw((4,0)--(4,10));
draw((6,0)--(6,10));
draw((8,0)--(8,10));
draw((10,0)--(10,10));
label("1",(1,9));
label("2",(3,9));
label("3",(1,7));
label("4",(3,7));
\end{asy}
Answer: 25
Problem: Start with an angle of $60^\circ$ and bisect it, then bisect the lower $30^\circ$ angle, then the upper $15^\circ$ angle, and so on, always alternating between the upper and lower of the previous two angles constructed. This process approaches a limiting line that divides the original $60^\circ$ angle into two angles. Find the measure (degrees) of the smaller angle.
Answer: 20
Problem: Let $a_1, a_2, a_3, \dots, a_{20}$ be a permutation of the numbers $1, 2, \dots, 20$. How many different values can the expression $a_1 - a_2 + a_3 - \dots - a_{20}$ have?
Answer: 101
Problem: Given a parallelogram $ABCD$, let $\mathcal{P}$ be a plane such that the distance from vertex $A$ to $\mathcal{P}$ is $49$, the distance from vertex $B$ to $\mathcal{P}$ is $25$, and the distance from vertex $C$ to $\mathcal{P}$ is $36$. Find the sum of all possible distances from vertex $D$ to $\mathcal{P}$.
Answer: 220
Problem: Consider the base 27 number \[ n = ABCDEFGHIJKLMNOPQRSTUVWXYZ \] where each letter has the value of its position in the alphabet. What remainder do you get when you divide $n$ by 100? (The remainder is an integer between 0 and 99, inclusive.)
Answer: 25
Problem: Yang has a standard $6$-sided die, a standard $8$-sided die, and a standard $10$-sided die. He tosses these three dice simultaneously. The probability that the three numbers that show up form the side lengths of a right triangle can be expressed as $\frac{m}{n}$, for relatively prime positive integers $m$ and $n$. Find $100m+n$.
Answer: 1180
Problem: Alice's favorite number has the following properties:

- It has 8 distinct digits.
- The digits are decreasing when read from left to right.
- It is divisible by 180.

What is Alice's favorite number?
Answer: 97654320
Problem: An isosceles triangle has angles of $50^\circ$, $x^\circ$, and $y^\circ$. Find the maximum possible value of $x - y$.
Answer: 30
Problem: A point $P$ is chosen uniformly at random in the interior of triangle $ABC$ with side lengths $AB = 5$, $BC = 12$, $CA = 13$. The probability that a circle with radius $\frac{1}{3}$ centered at $P$ does not intersect the perimeter of $ABC$ can be written as $\frac{m}{n}$ where $m, n$ are relatively prime positive integers. Find $m + n$.
Answer: 61
Problem: How many integer pairs \((x, y)\) satisfy the equation \(x^2 + y^2 = 9999(x - y)\)?
Answer: 0
Problem: ABCD is a cyclic quadrilateral with circumcenter $O$ and circumradius 7. The line segments $AB$ and $CD$ intersect at point $E$, while $DA$ and $CB$ intersect at point $F$. Given that $OE = 13$ and $OF = 14$, let $\cos\angle FOE = \frac{p}{q}$, where $p$ and $q$ are coprime integers. Find the sum $p + q$.
Answer: 33
Problem: Let $P_0 = (3,1)$ and define $P_{n+1} = (x_n, y_n)$ for $n \ge 0$ by \[ x_{n+1} = - \frac{3x_n - y_n}{2}, \quad y_{n+1} = - \frac{x_n + y_n}{2} \] Find the area of the quadrilateral formed by the points $P_{96}, P_{97}, P_{98}, P_{99}$.
Answer: 8
Problem: Suppose that $ABCDE$ is a convex pentagon with $\angle A = 90^\circ$, $\angle B = 105^\circ$, $\angle C = 90^\circ$ and $AB = 2$, $BC = CD = DE = \sqrt{2}$. If the length of $AE$ is $\sqrt{a} - b$ where $a$ and $b$ are integers, what is the value of $a + b$?
Answer: 4
Problem: Let $d$ be the greatest common divisor of $2^{30^{10}} - 2$ and $2^{30^{45}} - 2$. Find the remainder when $d$ is divided by $2013$.
Answer: 2012
Problem: Let a positive integer $n$ be \textit{nice} if there exists a positive integer $m$ such that:
\[
n^3 < 5mn < n^3 + 100.
\]
Find the number of \textit{nice} positive integers.
Answer: 53
Problem: The Miami Heat and the San Antonio Spurs are playing a best-of-five series basketball championship, in which the team that first wins three games wins the whole series. Assume that the probability that the Heat wins a given game is $x$ (there are no ties). The expected value for the total number of games played can be written as $f(x)$, with $f$ a polynomial. Find $f(-1)$.
Answer: 21
Problem: In $\triangle ABC$, $AB = AC = 14 \sqrt{2}$, $D$ is the midpoint of $CA$, and $E$ is the midpoint of $BD$. Suppose $\triangle CDE$ is similar to $\triangle ABC$. Find the length of $BD$.
Answer: 14
Problem: An equiangular hexagon has side lengths $1, 1, a, 1, 1, a$ in that order. Given that there exists a circle that intersects the hexagon at $12$ distinct points, we have $M < a < N$ for some real numbers $M$ and $N$. Determine the minimum possible value of the ratio $\frac{N}{M}$.
Answer: 2
Problem: Let $\{a,b,c,d,e,f,g,h,i\}$ be a permutation of $\{1,2,3,4,5,6,7,8,9\}$ such that $\gcd(c,d)=\gcd(f,g)=1$ and
\[(10a+b)^{c/d}=e^{f/g}.\]
Given that $h>i$, evaluate $10h+i$.
Answer: 65
Problem: Suppose that $P(x, y, z)$ is a homogeneous degree 4 polynomial in three variables such that $P(a, b, c) = P(b, c, a)$ and $P(a, a, b) = 0$ for all real $a$, $b$, and $c$. If $P(1, 2, 3) = 1$, compute $P(2, 4, 8)$.

Note: $P(x, y, z)$ is a homogeneous degree $4$ polynomial if it satisfies $P(ka, kb, kc) = k^4P(a, b, c)$ for all real $k, a, b, c$.
Answer: 56
Problem: Consider the arithmetic sequence of integers with first term $-7$ and second term $17$. What is the sum of the squares of the first three terms of the sequence?
Answer: 2019
Problem: Find the least odd positive integer that is the middle number of five consecutive integers, all of which are composite.
Answer: 93
Problem: Suppose that the polynomial $p(x) = x^{2010} \pm x^{2009} \pm \ldots \pm x \pm 1$ does not have a real root. What is the maximum number of coefficients that can be $-1$?
Answer: 1005
Problem: Let $x, y, z$ be positive real numbers whose sum is 2012. Find the maximum value of:
\[ \frac{(x^2 + y^2 + z^2)(x^3 + y^3 + z^3)}{x^4 + y^4 + z^4} \]
Answer: 2012
Problem: Find the number of pairs of integers $x$ and $y$ such that $x^2 + xy + y^2 = 28$.
Answer: 12
Problem: Consider the set $L$ of binary strings of length less than or equal to $9$. For a string $w$, define $w^{+}$ to be the set $\{w, w^2, w^3, \ldots\}$ where $w^k$ represents $w$ concatenated to itself $k$ times. How many ways are there to pick an ordered pair of (not necessarily distinct) elements $x, y \in L$ such that $x^{+} \cap y^{+} \neq \varnothing$?
Answer: 1250
Problem: Compute the maximum integer value of $k$ such that $2^k$ divides $3^{2n+3} + 40n - 27$ for any positive integer $n$.
Answer: 6
Problem: Let $V$ be a rectangular prism with integer side lengths. The largest face has area $240$ and the smallest face has area $48$. A third face has area $x$, where $x$ is not equal to $48$ or $240$. What is the sum of all possible values of $x$?
Answer: 260
Problem: Let $f: \mathbb{N} \to \mathbb{N}$ be a function satisfying the following conditions:

- $f(ab) = f(a)f(b)$ whenever the greatest common divisor (GCD) of $a$ and $b$ is $1$.
- $f(p + q) = f(p) + f(q)$ whenever $p$ and $q$ are prime numbers.

Determine all possible values of $f(2002)$. Justify your answers.
Answer: 2002
Problem: Find the largest real number $a$ such that the system of equations:
\[
\begin{align*}
x - 4y &= 1 \\
ax + 3y &= 1
\end{align*}
\]
has an integer solution.
Answer: 1
Problem: In the addition problem:
\[
\begin{array}{cccccc}
& W & H & I & T & E \\
+ & W & A & T & E & R \\
\hline
P & I & C & N & I & C
\end{array}
\]
each distinct letter represents a different digit. Find the number represented by the answer \(\text{PICNIC}\).
Answer: 169069
Problem: Cédric needs to write down a phrase in French. He knows that there are $3$ words in the phrase and that the letters appear in the order: \[cesontoiseaux.\] He does not remember what the words are and which letters have what accents in the phrase. If $n$ is the number of possible phrases that he could write down, then determine the number of distinct primes in the prime factorization of $n$.
Answer: 4
Problem: Find the remainder when $333^{333}$ is divided by $33$.
Answer: 27
Problem: A positive integer $k$ is said to be visionary if there are integers $a > 0$ and $b \geq 0$ such that $a \cdot k + b \cdot (k + 1) = 2020.$ How many visionary integers are there?
Answer: 2020
Problem: A point \((a,b)\) in the plane is called sparkling if it also lies on the line \(ax+by=1\). Find the maximum possible distance between two sparkling points.
Answer: 2
Problem: Alice, Bob, Charlie, Diana, Emma, and Fred sit in a circle, in that order, and each roll a six-sided die. Each person looks at his or her own roll, and also looks at the roll of either the person to the right or to the left, deciding at random. Then, at the same time, Alice, Bob, Charlie, Diana, Emma, and Fred each state the expected sum of the dice rolls based on the information they have. All six people say different numbers; in particular, Alice, Bob, Charlie, and Diana say $19$, $22$, $21$, and $23$, respectively. Compute the product of the dice rolls.
Answer: 1440
Problem: In the rectangle $ABCD$, $BC = 5$, $EC = \frac{1}{3} CD$, and $F$ is the point where $AE$ and $BD$ intersect. The triangle $\triangle DFE$ has an area of $12$, and the triangle $\triangle ABF$ has an area of $27$. Find the area of the quadrilateral $BCEF$.
Answer: 33
Problem: We color the numbers $1, 2, 3, \ldots, 20$ with two colors: white and black, ensuring that both colors are used. Determine the number of ways to perform this coloring such that the greatest common divisor (GCD) of the product of the white numbers and the product of the black numbers is equal to $1$.
Answer: 62
Problem: How many two-digit factors does $555555$ have?
Answer: 12
Problem: 1 or -1 is written in 50 letters. These letters are placed into 50 envelopes. You can ask to learn the product of numbers written into any three letters. At least, how many questions are required to find the product of all of the 50 numbers?
Answer: 18
Problem: In the diagram below, $ABCD$ is a square and both $\triangle CFD$ and $\triangle CBE$ are equilateral. Find the degree measure of $\angle CEF$.
Answer: 15
Problem: Find the least positive integer $n$ greater than $1$ such that $n^3 - n^2$ is divisible by $7^2 \times 11$.
Answer: 56
Problem: Call a positive integer $n \geq 2$ "junk" if there exist two distinct $n$-digit binary strings $a_1a_2\cdots a_n$ and $b_1b_2\cdots b_n$ such that:
- $a_1 + a_2 = b_1 + b_2$,
- $a_{i-1} + a_i + a_{i+1} = b_{i-1} + b_i + b_{i+1}$ for all $2 \leq i \leq n-1$,
- $a_{n-1} + a_n = b_{n-1} + b_n$.

Find the number of junk positive integers less than or equal to $2016$.
Answer: 672
Problem: Alex places a rook on any square of an empty $8 \times 8$ chessboard. Then he places additional rooks, one at a time, each attacking an odd number of rooks already on the board. A rook attacks to the left, to the right, above, and below, and only the first rook in each direction. What is the maximum number of rooks Alex can place on the chessboard?
Answer: 16
Problem: How many ordered triples $(a, b, c)$ of integers with $-15 \le a, b, c \le 15$ are there such that the three equations $ax + by = c$, $bx + cy = a$, and $cx + ay = b$ correspond to lines that are distinct and concurrent?
Answer: 720
Problem: Find the number of positive integer divisors of \( \sum_{k=0}^{50}(-3)^k\binom{100}{2k} \).
Answer: 100
Problem: Given a $4 \times 4$ squares table, how many ways can we fill the table with $\{0,1\}$ such that any two neighboring squares (sharing one common side) have a product equal to $0$?
Answer: 1234
Problem: Let $N$ be the number of functions $f:\{1,2,3,4,5,6,7,8,9,10\} \rightarrow \{1,2,3,4,5\}$ that have the property that for $1\leq x\leq 5$ it is true that $f(f(x))=x$. Given that $N$ can be written in the form $5^a\cdot b$ for positive integers $a$ and $b$ with $b$ not divisible by $5$, find $a+b$.
Answer: 31
Problem: We define the polynomial $P(x) = 2014x^{2013} + 2013x^{2012} + \cdots + 4x^3 + 3x^2 + 2x$. Find the largest prime divisor of $P(2)$.
Answer: 61
Problem: There were three times as many red candies as blue candies on a table. After Darrel took the same number of red candies and blue candies, there were four times as many red candies as blue candies left on the table. Then after Cloe took $12$ red candies and $12$ blue candies, there were five times as many red candies as blue candies left on the table. Find the total number of candies that Darrel took.
Answer: 48
Problem: There are $38$ people in the California Baseball League (CBL). The CBL cannot start playing games until people are split into teams of exactly $9$ people, with each person in exactly one team. Moreover, there must be an even number of teams. What is the fewest number of people who must join the CBL such that the CBL can start playing games? The CBL may not revoke membership of the $38$ people already in the CBL.
Answer: 16
Problem: Let $a$ and $b$ be two positive integers satisfying the equation  
\[ 20\sqrt{12} = a\sqrt{b}. \]  
Compute the sum of all possible distinct products $ab$.
Answer: 10800
Problem: How many numbers $\overline{abc}$ with $a, b, c > 0$ exist such that $$\overline{cba} \mid \overline{abc}$$? 

Note: The vertical line denotes that $\overline{cba}$ divides $\overline{abc}.$
Answer: 81
Problem: Given $x, y, z \in \mathbb{R}^+$, which are solutions to the system of equations:
\[
x^2 + xy + y^2 = 57
\]
\[
y^2 + yz + z^2 = 84
\]
\[
z^2 + zx + x^2 = 111
\]
What is the value of $xy + 3yz + 5zx$?
Answer: 270
Problem: Compute the sum of all prime numbers $p$ with $p \ge 5$ such that $p$ divides $(p + 3)^{p-3} + (p + 5)^{p-5}$.
Answer: 322
Problem: The product 
$$\left(\frac{1+1}{1^2+1}+\frac{1}{4}\right)\left(\frac{2+1}{2^2+1}+\frac{1}{4}\right)\left(\frac{3+1}{3^2+1}+\frac{1}{4}\right)\cdots\left(\frac{2022+1}{2022^2+1}+\frac{1}{4}\right)$$
can be written as $\frac{q}{2^r\cdot s}$, where $r$ is a positive integer, and $q$ and $s$ are relatively prime odd positive integers. Find $s$.
Answer: 1
Problem: Find the smallest integer $k \geq 2$ such that for every partition of the set $\{2, 3, \ldots, k\}$ into two parts, at least one of these parts contains (not necessarily distinct) numbers $a$, $b$, and $c$ with $ab = c$.
Answer: 32
Problem: Let $X = \cos\frac{2\pi}{7} + \cos\frac{4\pi}{7} + \cos\frac{6\pi}{7} + \cdots + \cos\frac{2006\pi}{7} + \cos\frac{2008\pi}{7}$. Compute $\Big|\lfloor 2008 X \rfloor \Big|$.
Answer: 1004
Problem: Using only once each of the digits $1, 2, 3, 4, 5, 6, 7$, and $8$, write the square and the cube of a positive integer. Determine what that number can be.
Answer: 24