ChenDRAG/VeRL_math_validation huggingface dataset:
Problem: The diagram shows a semicircle with a diameter of $20$ and the largest circle that fits inside the semicircle. The area of the shaded region is $N\pi$, where $N$ is a positive integer. Find $N$.
Answer: 25
Problem: The sides of a triangle are $x$, $2x+1$, and $x+2$, where $x$ is a positive rational number. One of the angles of the triangle is $60$ degrees. Find the perimeter of the triangle.
Answer: 9
Problem: Compute the number of ordered quadruples of positive integers $(a, b, c, d)$ such that 
\[ a! \cdot b! \cdot c! \cdot d! = 24! \]
Answer: 28
Problem: Given any two positive real numbers $x$ and $y$, the operation $x \Diamond y$ is defined by some fixed rule. It satisfies the following equations for all $x, y > 0$:

1. $(x \cdot y) \Diamond y = x(y \Diamond y)$
2. $(x \Diamond 1) \Diamond x = x \Diamond 1$

Additionally, given that $1 \Diamond 1 = 1$, find the value of $19 \Diamond 98$.
Answer: 19
Problem: The squares of two positive integers differ by 2016. Find the maximum possible sum of the two integers.
Answer: 1008
Problem: Triangle $ABC$ is equilateral. Points $D$ and $E$ are the midpoints of segments $BC$ and $AC$, respectively. $F$ is the point on segment $AB$ such that $2BF = AF$. Let $P$ denote the intersection of $AD$ and $EF$. The value of $\frac{EP}{PF}$ can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.
Answer: 7
Problem: Congruent circles $\Gamma_1$ and $\Gamma_2$ have radius $2012$, and the center of $\Gamma_1$ lies on $\Gamma_2$. Suppose that $\Gamma_1$ and $\Gamma_2$ intersect at $A$ and $B$. The line through $A$ perpendicular to $AB$ meets $\Gamma_1$ and $\Gamma_2$ again at $C$ and $D$, respectively. Find the length of $CD$.
Answer: 4024
Problem: For some positive integer $n$, there exists $n$ different positive integers $a_1, a_2, \ldots, a_n$ such that:

1. $a_1 = 1, \quad a_n = 2000$
2. $\forall i \in \mathbb{Z}$ such that $2 \le i \le n, \quad a_i - a_{i-1} \in \{-3, 5\}$

Determine the maximum value of $n$.
Answer: 1996
Problem: For $n \in \mathbb{N}$, let $P(n)$ denote the product of the digits in $n$ and $S(n)$ denote the sum of the digits in $n$. Consider the set \(A=\{n \in \mathbb{N}: P(n)\) is non-zero, square-free, and \(S(n)\) is a proper divisor of \(P(n)\}\). Find the maximum possible number of digits of the numbers in \(A\).
Answer: 92
Problem: Quadrilateral $ABCD$ satisfies $AB = 8$, $BC = 5$, $CD = 17$, and $DA = 10$. Let $E$ be the intersection of $AC$ and $BD$. Suppose the ratio $BE : ED = 1 : 2$. Find the area of quadrilateral $ABCD$.
Answer: 60
Problem: Given are 100 different positive integers. We call a pair of numbers "good" if the ratio of these numbers is either 2 or 3. What is the maximum number of good pairs that these 100 numbers can form? (A number can be used in several pairs.)
Answer: 180
Problem: Let $n$ be an integer. We consider $s(n)$, the sum of the 2001 powers of $n$ with the exponents from 0 to 2000. Thus, $s(n) = \sum_{k=0}^{2000} n^k$. What is the unit digit of $s(n)$ in the decimal system?
Answer: 1
Problem: The number $104,060,465$ is divisible by a five-digit prime number. What is that prime number?
Answer: 10613
Problem: Let $a_1, a_2, a_3, \ldots$ be an infinite sequence where for all positive integers $i$, $a_i$ is chosen to be a random positive integer between $1$ and $2016$, inclusive. Let $S$ be the set of all positive integers $k$ such that for all positive integers $j < k$, $a_j \neq a_k$. 

- So $1 \in S$; 
- $2 \in S$ if and only if $a_1 \neq a_2$; 
- $3 \in S$ if and only if $a_1 \neq a_3$ and $a_2 \neq a_3$; and so on.

In simplest form, let $\frac{p}{q}$ be the expected number of positive integers $m$ such that $m$ and $m+1$ are in $S$. Compute $pq$.
Answer: 4030
Problem: Find the largest $n$ such that the last nonzero digit of $n!$ is $1$.
Answer: 1
Problem: Find the greatest prime number $p$ such that $p^3$ divides \[ \frac{122!}{121} + 123! \]
Answer: 61
Problem: Lev and Alex are racing on a number line. Alex is much faster, so he goes to sleep until Lev reaches $100$. Lev runs at $5$ integers per minute and Alex runs at $7$ integers per minute (in the same direction). How many minutes from the start of the race will it take Alex to catch up to Lev (who is still running after Alex wakes up)?
Answer: 70
Problem: A three-digit number $\overline{ABC}$ is called \textit{spicy} if it satisfies the equation $\overline{ABC} = A^3 + B^3 + C^3$. Determine the unique number $n$ for which both $n$ and $n+1$ are \textit{spicy} numbers.
Answer: 370
Problem: Find the largest positive integer $n < 30$ such that $\frac{1}{2}(n^8 + 3n^4 - 4)$ is not divisible by the square of any prime number.
Answer: 20
Problem: Triangle $ABC$ is scalene. Points $P$ and $Q$ are on segment $BC$ with $P$ between $B$ and $Q$ such that $BP=21$, $PQ=35$, and $QC=100$. If $AP$ and $AQ$ trisect $\angle A$, then $\frac{AB}{AC}$ can be written uniquely as $\frac{p\sqrt{q}}{r}$, where $p$ and $r$ are relatively prime positive integers and $q$ is a positive integer not divisible by the square of any prime. Determine $p+q+r$.
Answer: 92
Problem: Find all natural numbers $n$ such that the product of its decimal digits equals $n^2 - 10n - 22$.
Answer: 12
Problem: A pyramid has a base which is an equilateral triangle with side length $300$ centimeters. The vertex of the pyramid is $100$ centimeters above the center of the triangular base. A mouse starts at a corner of the base of the pyramid and walks up the edge of the pyramid toward the vertex at the top. When the mouse has walked a distance of $134$ centimeters, how many centimeters above the base of the pyramid is the mouse?
Answer: 67
Problem: Let $n \ge 3$. Suppose $a_1, a_2, \ldots, a_n$ are $n$ distinct pairwise real numbers. In terms of $n$, find the smallest possible number of different assumed values by the following $n$ numbers:

$$a_1 + a_2, a_2 + a_3, \ldots, a_{n-1} + a_n, a_n + a_1$$
Answer: 3
Problem: Find the value of the sum: 
\[ \sum_{i=1}^{2001} \lfloor \sqrt{i} \rfloor \]
where \( \lfloor x \rfloor \) denotes the greatest integer which does not exceed \( x \).
Answer: 58718
Problem: Find the smallest positive integer, $n$, which can be expressed as the sum of distinct positive integers $a, b, c$ such that $a + b$, $a + c$, and $b + c$ are perfect squares.
Answer: 55
Problem: Find the number of triples of real numbers \((a, b, c)\) such that \(a^{20} + b^{20} + c^{20} = a^{24} + b^{24} + c^{24} = 1\).
Answer: 6
Problem: Joe bikes $x$ miles East at $20$ mph to his friend’s house. He then turns South and bikes $x$ miles at $20$ mph to the store. Then, Joe turns East again and goes to his grandma’s house at $14$ mph. On this last leg, he has to carry flour he bought for her at the store. Her house is $2$ more miles from the store than Joe’s friend’s house is from the store. Joe spends a total of 1 hour on the bike to get to his grandma’s house. If Joe then rides straight home in his grandma’s helicopter at $78$ mph, how many minutes does it take Joe to get home from his grandma’s house?
Answer: 10
Problem: For each of the $9$ positive integers $n, 2n, 3n, \ldots, 9n$, Alice takes the first decimal digit (from the left) and writes it onto a blackboard. She selected $n$ such that among the nine digits on the blackboard, there is the least possible number of different digits. What is this number of different digits equal to?
Answer: 4
Problem: Let $P(x) = x^3 + ax^2 + bx + c$ be a polynomial where $a, b, c$ are integers and $c$ is odd. Let $p_i$ be the value of $P(x)$ at $x = i$. Given that $p_1^3 + p_2^3 + p_3^3 = 3p_1p_2p_3$, find the value of $p_2 + 2p_1 - 3p_0$.
Answer: 18
Problem: For $1 \leq n \leq 2016$, how many integers $n$ satisfy the condition that the remainder when $n$ is divided by $20$ is smaller than the remainder when $n$ is divided by $16$?
Answer: 600
Problem: Define $f(x) = x + \sqrt{x + \sqrt{x + \sqrt{x + \sqrt{x + \ldots}}}}$. Find the smallest positive integer $x$ such that $f(x) \ge 50\sqrt{x}$.
Answer: 2400
Problem: A triangle has sides of length $48$, $55$, and $73$. A square is inscribed in the triangle such that one side of the square lies on the longest side of the triangle, and the two vertices not on that side of the square touch the other two sides of the triangle. If $c$ and $d$ are relatively prime positive integers such that $\frac{c}{d}$ is the length of a side of the square, find the value of $c+d$.
Answer: 200689
Problem: How many 4-digit numbers have exactly two odd digits and two even digits?
Answer: 3375
Problem: Find the sum of all distinct prime factors of $2023^3 - 2000^3 - 23^3$.
Answer: 57
Problem: Let $P(x) = x^3 - 6x^2 - 5x + 4$. Suppose that $y$ and $z$ are real numbers such that \[ zP(y) = P(y - n) + P(y + n) \] for all real numbers $n$. Evaluate $P(y)$.
Answer: -22
Problem: Let $f_1(x) = x^2 - 3$ and $f_n(x) = f_1(f_{n-1}(x))$ for $n \ge 2$. Let $m_n$ be the smallest positive root of $f_n$, and $M_n$ be the largest positive root of $f_n$. If $x$ is the least number such that $M_n \le m_n \cdot x$ for all $n \ge 1$, compute $x^2$.
Answer: 4
Problem: Let $ABCD$ be a convex quadrilateral with $AC = 7$ and $BD = 17$. Let $M$, $P$, $N$, $Q$ be the midpoints of sides $AB$, $BC$, $CD$, and $DA$, respectively. Compute $MN^2 + PQ^2$.
Answer: 169
Problem: Let $a, b, c$ be integers, not all the same, with $a, b, c \geq 4$ that satisfy the equation:
\[ 4abc = (a + 3)(b + 3)(c + 3). \]
Find the numerical value of $a + b + c$.
Answer: 16
Problem: The battery life on a computer decreases at a rate proportional to the display brightness. Austin starts his day with both his battery life and brightness at $100\%$. Whenever his battery life (expressed as a percentage) reaches a multiple of $25$, he also decreases the brightness of his display to that multiple of $25$. If left at $100\%$ brightness, the computer runs out of battery in $1$ hour. Compute the amount of time, in minutes, it takes for Austin’s computer to reach $0\%$ battery using his modified scheme.
Answer: 125
Problem: Let $O$ be the set of odd numbers between 0 and 100. Let $T$ be the set of subsets of $O$ of size 25. For any finite subset of integers $S$, let $P(S)$ be the product of the elements of $S$. Define $n = \sum_{S \in T} P(S)$. If you divide $n$ by 17, what is the remainder?
Answer: 15
Problem: 15 ladies and 30 gentlemen attend a luxurious party. At the start of the party, each lady shakes hands with a random gentleman. At the end of the party, each lady shakes hands with another random gentleman. A lady may shake hands with the same gentleman twice (first at the start and then at the end of the party), and no two ladies shake hands with the same gentleman at the same time.

Let $m$ and $n$ be relatively prime positive integers such that $\frac{m}{n}$ is the probability that the collection of ladies and gentlemen that shook hands at least once can be arranged in a single circle such that each lady is directly adjacent to someone if and only if she shook hands with that person. Find the remainder when $m$ is divided by 10000.
Answer: 1
Problem: For each positive integer $n$, let $r_n$ be the smallest positive root of the equation $x^n = 7x - 4$. There are positive real numbers $a$, $b$, and $c$ such that \[\lim_{n \to \infty} a^n (r_n - b) = c.\] If $100a + 10b + c = \frac{p}{7}$ for some integer $p$, find $p$.
Answer: 1266
Problem: In a table with $88$ rows and $253$ columns, each cell is colored either purple or yellow. Suppose that for each yellow cell $c$, the following condition holds:  
$$x(c) \cdot y(c) \geq 184,$$  
where $x(c)$ is the number of purple cells in the same row as $c$, and $y(c)$ is the number of purple cells in the same column as $c$.  
Find the least possible number of cells that are colored purple.
Answer: 2024
Problem: A semicircle with diameter length $16$ contains a circle with radius $3$ that is tangent to both the inside of the semicircle and its diameter. A second larger circle is tangent to the inside of the semicircle, the outside of the smaller circle, and the diameter of the semicircle. The diameter of the second circle can be expressed as $\frac{n + k\sqrt{2}}{m}$, where $m$, $n$, and $k$ are positive integers, and $m$ and $n$ have no common factors. Find $m + n + k$.
Answer: 361
Problem: Define $\lfloor x \rfloor$ as the largest integer less than or equal to $x$. Define $\{x \} = x - \lfloor x \rfloor$. For example, $\{ 3 \} = 3-3 = 0$, $\{ \pi \} = \pi - 3$, and $\{ - \pi \} = 4-\pi$. If $\{n\} + \{ 3n\} = 1.4$, then find the sum of all possible values of $100\{n\}$.
Answer: 145
Problem: Let $a, b, c, p, q, r > 0$ such that $(a,b,c)$ is a geometric progression and $(p, q, r)$ is an arithmetic progression. If 
\[a^p b^q c^r = 6 \quad \text{and} \quad a^q b^r c^p = 29\]
then compute $\lfloor a^r b^p c^q \rfloor$.
Answer: 29
Problem: Consider the sum $$S = \sum^{2021}_{j=1} \left|\sin \frac{2\pi j}{2021}\right|.$$ The value of $S$ can be written as $\tan \left( \frac{c\pi}{d} \right)$ for some relatively prime positive integers $c, d$, satisfying $2c < d$. Find the value of $c + d$.
Answer: 3031
Problem: Let $T = \frac{1}{4}x^{2} - \frac{1}{5}y^{2} + \frac{1}{6}z^{2}$ where $x, y, z$ are real numbers such that $1 \leq x, y, z \leq 4$ and $x - y + z = 4$. Find the smallest value of $10 \times T$.
Answer: 23
Problem: An angle is considered as a point and two rays coming out of it. Find the largest value of $n$ such that it is possible to place $n$ angles of $60^\circ$ on a plane in such a way that any pair of these angles has exactly 4 intersection points.
Answer: 2
Problem: For how many positive integers $n$ less than $2018$ does $n^2$ have the same remainder when divided by $7$, $11$, and $13$?
Answer: 36
Problem: There are some balls, each marked with a positive integer not exceeding $14$, and the sum of these numbers is $S$. Determine the greatest possible value of $S$ such that, irrespective of the specific integers on the balls, it is possible to divide the balls into two piles where the sum of the numbers on the balls in each pile does not exceed $129$.
Answer: 258
Problem: The diagram below shows two parallel rows with seven points in the upper row and nine points in the lower row. The points in each row are spaced one unit apart, and the two rows are two units apart. How many trapezoids, which are not parallelograms, have vertices in this set of $16$ points and have an area of at least six square units?

\begin{asy}
import graph; size(7cm); 
pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps);
pen dotstyle = black;
dot((-2,4),linewidth(6pt) + dotstyle); 
dot((-1,4),linewidth(6pt) + dotstyle); 
dot((0,4),linewidth(6pt) + dotstyle); 
dot((1,4),linewidth(6pt) + dotstyle); 
dot((2,4),linewidth(6pt) + dotstyle); 
dot((3,4),linewidth(6pt) + dotstyle); 
dot((4,4),linewidth(6pt) + dotstyle); 
dot((-3,2),linewidth(6pt) + dotstyle); 
dot((-2,2),linewidth(6pt) + dotstyle); 
dot((-1,2),linewidth(6pt) + dotstyle); 
dot((0,2),linewidth(6pt) + dotstyle); 
dot((1,2),linewidth(6pt) + dotstyle); 
dot((2,2),linewidth(6pt) + dotstyle); 
dot((3,2),linewidth(6pt) + dotstyle); 
dot((4,2),linewidth(6pt) + dotstyle); 
dot((5,2),linewidth(6pt) + dotstyle); 
\end{asy}
Answer: 361
Problem: Compute the number of real numbers $t$ such that \[ t = 50 \sin(t - \lfloor t \rfloor). \] Here, $\lfloor \cdot \rfloor$ denotes the greatest integer function.
Answer: 42
Problem: If distinct digits $D, E, L, M, Q$ (between $0$ and $9$ inclusive) satisfy the following equation:
\[
\begin{array}{c@{\,}c@{\,}c@{\,}c}
  &   & E & L \\
+ & M & E & M \\\hline
  & Q & E & D \\
\end{array}
\]
What is the maximum possible value of the three-digit integer $QED$?
Answer: 893
Problem: What is the largest factor of $130000$ that does not contain the digit $0$ or $5$?
Answer: 26
Problem: Suppose Alan, Michael, Kevin, Igor, and Big Rahul are in a running race. It is given that exactly one pair of people tie (for example, two people both get second place), so that no other pair of people end in the same position. Each competitor has equal skill; this means that each outcome of the race, given that exactly two people tie, is equally likely. The probability that Big Rahul gets first place (either by himself or he ties for first) can be expressed in the form $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $m+n$.
Answer: 5
Problem: If $a$ and $b$ are nonzero real numbers such that $|a| \neq |b|$, compute the value of the expression
\[
  \left( \frac{b^2}{a^2} + \frac{a^2}{b^2} - 2 \right) \times
  \left( \frac{a + b}{b - a} + \frac{b - a}{a + b} \right) \times
  \left( 
    \frac{\frac{1}{a^2} + \frac{1}{b^2}}{\frac{1}{b^2} - \frac{1}{a^2}}
      - \frac{\frac{1}{b^2} - \frac{1}{a^2}}{\frac{1}{a^2} + \frac{1}{b^2}}
  \right).
\]
Answer: -8
Problem: The two brothers, without waiting for the bus, decided to walk to the next stop. After passing $\frac{1}{3}$ of the way, they looked back and saw a bus approaching the stop. One of the brothers ran backwards, and the other ran forward at the same speed. It turned out that everyone ran to their stop exactly at the moment when the bus approached it. Find the speed of the brothers, if the bus speed is $30$ km/h, neglecting the bus stop time.
Answer: 10
Problem: Two circles have a radius of $9$, and one circle has a radius of $7$. Each circle is externally tangent to the other two circles, and each circle is internally tangent to two sides of an isosceles triangle, as shown in the figure. The sine of the base angle of the triangle is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
Answer: 7
Problem: Seongcheol has $3$ red shirts and $2$ green shirts, such that he cannot tell the difference between his three red shirts and he similarly cannot tell the difference between his two green shirts. In how many ways can he hang them in a row in his closet, given that he does not want the two green shirts next to each other?
Answer: 6
Problem: In how many ways can one choose distinct numbers $a$ and $b$ from the set \( \{1, 2, 3, \ldots, 2005\} \) such that \( a + b \) is a multiple of 5?
Answer: 401802
Problem: A $8 \times 8$ board is given, with sides directed north-south and east-west. It is divided into $1 \times 1$ cells in the usual manner. In each cell, there is at most one house. A house occupies only one cell.

A house is considered "in the shade" if there is a house in each of the cells directly to the south, east, and west of its cell. Importantly, no house placed on the south, east, or west edge of the board can be in the shade.

Determine the maximum number of houses that can be placed on the board such that no house is in the shade.
Answer: 50
Problem: In a game, Jimmy and Jacob each randomly choose to either roll a fair six-sided die or to automatically roll a $1$ on their die. If the product of the two numbers face up on their dice is even, Jimmy wins the game. Otherwise, Jacob wins. The probability Jimmy wins $3$ games before Jacob wins $3$ games can be written as $\frac{p}{2^q}$, where $p$ and $q$ are positive integers, and $p$ is odd. Find the remainder when $p+q$ is divided by $1000$.
Answer: 360
Problem: Let \[ S = \sum_{i = 1}^{2012} i!. \] The tens and units digits of \( S \) (in decimal notation) are \( a \) and \( b \), respectively. Compute \( 10a + b \).
Answer: 13
Problem: Let $P(x)$ be a polynomial of degree at most 6 that satisfies $P(k) = k!$ for $k = 0, 1, 2, 3, 4, 5, 6$. Compute the value of $P(7)$.
Answer: 3186
Problem: Let $f(x) = (x - 5)(x - 12)$ and $g(x) = (x - 6)(x - 10)$. Find the sum of all integers $n$ such that $\frac{f(g(n))}{f(n)^2}$ is defined and an integer.
Answer: 23
Problem: A $3 \times 3 \times 3$ cube composed of $27$ unit cubes rests on a horizontal plane. Determine the number of ways of selecting two distinct unit cubes from a $3 \times 3 \times 1$ block (the order is irrelevant) with the property that the line joining the centers of the two cubes makes a $45^\circ$ angle with the horizontal plane.
Answer: 60
Problem: Three parallel lines $L_1$, $L_2$, and $L_3$ are drawn in the plane such that the perpendicular distance between $L_1$ and $L_2$ is $3$, and the perpendicular distance between lines $L_2$ and $L_3$ is also $3$. A square $ABCD$ is constructed such that $A$ lies on $L_1$, $B$ lies on $L_3$, and $C$ lies on $L_2$. Find the area of the square.
Answer: 45
Problem: Square $ABCD$ is divided into four rectangles by lines $EF$ and $GH$. Line $EF$ is parallel to $AB$, and line $GH$ is parallel to $BC$. It is given that $\angle BAF = 18^\circ$. The lines $EF$ and $GH$ intersect at point $P$. The area of rectangle $PFCH$ is twice that of rectangle $AGPE$. Given that the value of $\angle FAH$ in degrees is $x$, find the nearest integer to $x$.
Answer: 45
Problem: The natural numbers from $1$ to $50$ are written on the blackboard. What is the minimum number of these numbers that should be deleted so that the sum of any two remaining numbers is not a prime?
Answer: 25
Problem: Let $w, x, y, z$ be integers from $0$ to $3$ inclusive. Find the number of ordered quadruples $(w, x, y, z)$ such that $5x^2 + 5y^2 + 5z^2 - 6wx - 6wy - 6wz$ is divisible by $4$.
Answer: 32
Problem: For a positive integer $n$, let $f(n)$ denote the number of ways to pair $n$ objects, considering only which objects are paired together. Find the sum of all $f(f(2k))$, where $k$ ranges from 1 to 2023.
Answer: 0
Problem: Seven boys and three girls are playing basketball. In how many different ways can they make two teams of five players each, ensuring that both teams have at least one girl?
Answer: 105
Problem: Find the number of pairs of positive integers $(m, n)$, with $m \leq n$, such that the least common multiple (LCM) of $m$ and $n$ equals $600$.
Answer: 53
Problem: Let $ABC$ be a triangle with area $5$ and $BC = 10$. Let $E$ and $F$ be the midpoints of sides $AC$ and $AB$ respectively, and let $BE$ and $CF$ intersect at $G$. Suppose that quadrilateral $AEGF$ can be inscribed in a circle. Determine the value of $AB^2 + AC^2$.
Answer: 200
Problem: For any real number $x$, we let $\lfloor x \rfloor$ be the unique integer $n$ such that $n \leq x < n+1$. For example, $\lfloor 31.415 \rfloor = 31$. Compute \[2020^{2021} - \left\lfloor \frac{2020^{2021}}{2021} \right\rfloor \times 2021.\]
Answer: 2020
Problem: Let $a, b$ be positive integers satisfying $a^3 - b^3 - ab = 25$. Find the largest possible value of $a^2 + b^3$.
Answer: 43
Problem: For each positive integer $n$ (written with no leading zeros), let $t(n)$ equal the number formed by reversing the digits of $n$. For example, $t(461) = 164$ and $t(560) = 65$. Determine the number of three-digit positive integers $m$ for which $m + t(t(m))$ is odd.
Answer: 50
Problem: The smallest possible volume of a cylinder that will fit nine spheres of radius 1 can be expressed as $x\pi$ for some value of $x$. Compute $x$.
Answer: 18
Problem: Compute the sum of all real solutions to $4^x - 2021 \cdot 2^x + 1024 = 0$.
Answer: 10
Problem: The total number of edges in two regular polygons is $2014$, and the total number of diagonals is $1,014,053$. How many edges does the polygon with the smaller number of edges have?
Answer: 952
Problem: How many $6$-digit positive integers have their digits in non-decreasing order from left to right? Note that $0$ cannot be a leading digit.
Answer: 3003
Problem: Find the number of pairs \((m,n)\) of integers with \(-2014 \le m,n \le 2014\) such that \(x^3+y^3 = m + 3nxy\) has infinitely many integer solutions \((x,y)\).
Answer: 25
Problem: Two players play a game on a pile of \( n \) beans. On each player's turn, they may take exactly \( 1 \), \( 4 \), or \( 7 \) beans from the pile. One player goes first, and then the players alternate until somebody wins. A player wins when they take the last bean from the pile. For how many \( n \) between \( 2014 \) and \( 2050 \) (inclusive) does the second player win?
Answer: 14
Problem: There are 300 points in space. Four planes $A$, $B$, $C$, and $D$ each have the property that they split the 300 points into two equal sets. (No plane contains one of the 300 points.) What is the maximum number of points that can be found inside the tetrahedron whose faces are on $A$, $B$, $C$, and $D$?
Answer: 100