meetforever/GRPO_CL huggingface dataset:
user: A **Beaver-number** is a positive 5-digit integer whose digit sum is divisible by 17. Call a pair of **Beaver-numbers** differing by exactly $1$ a **Beaver-pair**. The smaller number in a **Beaver-pair** is called an **MIT Beaver**, while the larger number is called a **CIT Beaver**. Find the positive difference between the largest and smallest **CIT Beavers** (over all **Beaver-pairs**).
Answer: 79200
user: Let $ABC$ be a triangle. Point $P$ lies in the interior of $\triangle ABC$ such that $\angle ABP = 20^\circ$ and $\angle ACP = 15^\circ$. Compute $\angle BPC - \angle BAC$.
Answer: 35
user: The centers of the three circles \(A\), \(B\), and \(C\) are collinear with the center of circle \(B\) lying between the centers of circles \(A\) and \(C\). Circles \(A\) and \(C\) are both externally tangent to circle \(B\), and the three circles share a common tangent line. Given that circle \(A\) has radius \(12\) and circle \(B\) has radius \(42\), find the radius of circle \(C\).
Answer: 147
user: Find the smallest possible value of $x + y$ where $x, y \ge 1$ and $x$ and $y$ are integers that satisfy $x^2 - 29y^2 = 1$.
Answer: 11621
user: How many integers between $2$ and $100$ inclusive cannot be written as $m \cdot n$, where $m$ and $n$ have no common factors and neither $m$ nor $n$ is equal to $1$?
Answer: 35
user: A group of $25$ friends were discussing a large positive integer. The first friend said, "It can be divided by $1$." The second friend said, "It can be divided by $2$." The third friend said, "And by $3$." The fourth friend added, "And by $4$." This pattern continued until everyone had made such a comment. If exactly $2$ friends were incorrect, and those two friends said consecutive numbers, what was the least possible integer they were discussing?
Answer: 787386600
user: An triangle with coordinates $(x_1,y_1)$, $(x_2, y_2)$, $(x_3,y_3)$ has centroid at $(1,1)$. The ratio between the lengths of the sides of the triangle is $3:4:5$. Given that \[x_1^3+x_2^3+x_3^3=3x_1x_2x_3+20\] and \[y_1^3+y_2^3+y_3^3=3y_1y_2y_3+21,\] the area of the triangle can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. What is $m+n$?
Answer: 107
user: Define the infinite products:  
\[ A = \prod_{i=2}^{\infty} \left(1-\frac{1}{n^3}\right) \text{ and }  B = \prod_{i=1}^{\infty}\left(1+\frac{1}{n(n+1)}\right). \]  
If \(\frac{A}{B} = \frac{m}{n}\) where \(m,n\) are relatively prime positive integers, determine \(100m+n\).
Answer: 103
user: Let $a, b, c$ be real numbers satisfying:
\[ 3ab + 2 = 6b, \quad 3bc + 2 = 5c, \quad 3ca + 2 = 4a. \]
Let $\mathbb{Q}$ denote the set of all rational numbers. Given that the product $abc$ can take two values $\frac{r}{s} \in \mathbb{Q}$ and $\frac{t}{u} \in \mathbb{Q}$, both in their lowest terms, find $r + s + t + u$.
Answer: 18
user: Since this is the $6$th Greek Math Olympiad and the year is $1989$, can you find the last two digits of $6^{1989}$?
Answer: 96
user: Show that the number \( 4\sin\frac{\pi}{34}\left(\sin\frac{3\pi}{34}+\sin\frac{7\pi}{34}+\sin\frac{11\pi}{34}+\sin\frac{15\pi}{34}\right) \) is an integer and determine it.
Answer: 1
user: Suppose $a$ and $b$ are positive integers for which $8a^a b^b = 27a^b b^a$. Find $a^2 + b^2$.
Answer: 117
user: Determine the value of $q(q(q(2000^{2000})))$, where $q(n)$ represents the sum of the digits of a natural number $n$.
Answer: 4
user: There are $1000$ rooms in a row along a long corridor. Initially, the first room contains $1000$ people, and the remaining rooms are empty. Each minute, the following happens: for each room containing more than one person, someone in that room decides it is too crowded and moves to the next room. All these movements are simultaneous (so nobody moves more than once within a minute). After one hour, how many different rooms will have people in them?
Answer: 61
user: Given that nonzero real numbers $a, b, c, d$ satisfy $a^b = c^d$ and $\frac{a}{2c} = \frac{b}{d} = 2$, compute $\frac{1}{c}$.
Answer: 16
user: Let $N = 30^{2015}$. Find the number of ordered 4-tuples of integers $(A, B, C, D) \in \{1, 2, \ldots, N\}^4$ (not necessarily distinct) such that for every integer $n$, $An^3 + Bn^2 + 2Cn + D$ is divisible by $N$.
Answer: 2
user: The diagram shows a circle with radius $24$ that contains two smaller circles, each with a radius of $12$. These two circles are tangent to each other and to the larger circle. There is also a smallest circle that is tangent to the three other circles. Determine the radius of this smallest circle.
Answer: 8
user: **Problem 1.** The least common multiple of the numbers $a$ and $b$ is $88$. If $a^2 + b^2 = 2000$, find the value of $(a + b)$.
Answer: 52
user: Given a strictly increasing sequence $a_1 < a_2 < \dots$ of positive integers such that each term $a_k$ is divisible either by 1005 or 1006, but neither term is divisible by 97. Find the least possible value of the maximal difference of consecutive terms $a_{i+1} - a_i$.
Answer: 2010
user: Compute the number of positive integer divisors of $2121$ with a units digit of $1$.
Answer: 4
user: Let $\eta(m)$ be the product of all positive integers that divide $m$, including $1$ and $m$. If $\eta(\eta(\eta(10))) = 10^n$, compute $n$.
Answer: 450
user: Two math students play a game with $k$ sticks. Alternating turns, each player chooses a number from the set \{1, 3, 4\} and removes exactly that number of sticks from the pile. If the pile has fewer sticks than the chosen number, the player must choose a number that can be removed. The winner is the player who takes the last stick. For $1 \leq k \leq 100$, determine the number of cases in which the first player can guarantee a win.
Answer: 71
user: In an isosceles triangle $ABC$ with an angle $\angle A = 20^\circ$ and base $BC = 12$, a point $E$ on the side $AC$ is chosen such that $\angle ABE = 30^\circ$. Additionally, a point $F$ is on the side $AB$ such that $EF = FC$. Find the length of $FC$.
Answer: 12
user: Marisa has two identical cubical dice labeled with the numbers $\{1, 2, 3, 4, 5, 6\}$. However, the two dice are not fair, meaning that they can land on each face with different probabilities. Marisa rolls the two dice and calculates their sum. Given that the sum is $2$ with probability $0.04$, and $12$ with probability $0.01$, the maximum possible probability of the sum being $7$ is $p$. Compute $\lfloor 100p \rfloor$.
Answer: 28
user: A cuboctahedron is a solid with 6 square faces and 8 equilateral triangle faces, with each edge adjacent to both a square and a triangle. Suppose the ratio of the volume of an octahedron to a cuboctahedron with the same side length is $r$. Find $100r^2$.
Answer: 4
user: Let $A$ be the area of the largest semicircle that can be inscribed in a quarter-circle of radius $1$. Compute $\frac{120A}{\pi}$.
Answer: 15
user: Cyclic quadrilateral $ABCD$ has $AC \perp BD$, $AB + CD = 12$, and $BC + AD = 13$. Find the greatest possible area of $ABCD$.
Answer: 36
user: Laurie loves multiplying numbers in her head. One day, she decides to multiply two 2-digit numbers $x$ and $y$ such that $x \leq y$ and the two numbers collectively have at least three distinct digits. Unfortunately, she accidentally remembers the digits of each number in the opposite order (for example, instead of remembering $51$ she remembers $15$). Surprisingly, the product of the two numbers after flipping the digits is the same as the product of the two original numbers. How many possible pairs of numbers could Laurie have tried to multiply?
Answer: 28
user: Triangle $ABC$ has side lengths $AC=3$, $BC=4$, and $AB=5$. Let $R$ be a point on the incircle $\omega$ of $\triangle ABC$. The altitude from $C$ to $\overline{AB}$ intersects $\omega$ at points $P$ and $Q$. Then, the greatest possible area of $\triangle PQR$ is $\frac{m\sqrt{n}}{p}$, where $m$ and $p$ are relatively prime positive integers, and $n$ is a positive integer not divisible by the square of any prime. Find $m+n+p$.
Answer: 43
user: The real numbers $a_0, a_1, \dots, a_{2013}$ and $b_0, b_1, \dots, b_{2013}$ satisfy the recurrence relations:
\[
a_{n} = \frac{1}{63} \sqrt{2n+2} + a_{n-1} \quad \text{and} \quad b_{n} = \frac{1}{96} \sqrt{2n+2} - b_{n-1}
\]
for every integer $n = 1, 2, \dots, 2013$. Given the initial conditions $a_0 = b_{2013}$ and $b_0 = a_{2013}$, compute the following sum:
\[
\sum_{k=1}^{2013} \left( a_kb_{k-1} - a_{k-1}b_k \right).
\]
Answer: 671
user: In a triangle $ABC$, $\angle C = 2\angle A$. A point $D$ is marked on the side $AC$ such that $\angle ABD = \angle DBC$. Knowing that $AB = 10$ and $CD = 3$, what is the length of the side $BC$?
Answer: 7
user: In the convex and cyclic quadrilateral $ABCD$, we have $\angle B = 110^{\circ}$. The intersection of $AD$ and $BC$ is $E$, and the intersection of $AB$ and $CD$ is $F$. If the perpendicular from $E$ to $AB$ intersects the perpendicular from $F$ to $BC$ on the circumcircle of the quadrilateral at point $P$, what is $\angle PDF$ in degrees?
Answer: 70
user: Find the least positive integer that cannot be represented as $\frac{2^a-2^b}{2^c-2^d}$ for some positive integers $a, b, c, d$.
Answer: 11
user: Let $n \geq 3$ be an integer. Mario draws $20$ lines in the plane, such that no two lines are parallel.

For each equilateral triangle formed by three of these lines, Mario receives three coins. For each isosceles and non-equilateral triangle formed by three of these lines, Mario receives one coin. What is the maximum number of coins that Mario can receive?
Answer: 380
user: Let a rad number be a palindrome such that the square root of the sum of its digits is irrational. Find the number of $4$-digit rad numbers.
Answer: 79
user: A number $N$ has 2009 positive factors. What is the maximum number of positive factors that $N^2$ could have?
Answer: 13689
user: Let $ABC$ be an equilateral triangle. Let $P$ and $S$ be points on $AB$ and $AC$, respectively, and let $Q$ and $R$ be points on $BC$ such that $PQRS$ is a rectangle. If $PQ = \sqrt{3} \cdot PS$ and the area of $PQRS$ is $28\sqrt{3}$, what is the length of $PC$?
Answer: 14
user: \( \triangle ABC \) has \( AB = 5 \), \( BC = 12 \), and \( AC = 13 \). A circle is inscribed in \( \triangle ABC \), and \( MN \) tangent to the circle is drawn such that \( M \) is on \( \overline{AC} \), \( N \) is on \( \overline{BC} \), and \( \overline{MN} \parallel \overline{AB} \). The area of \( \triangle MNC \) is \( \frac{m}{n} \), where \( m \) and \( n \) are relatively prime positive integers. Find \( m + n \).
Answer: 43
user: Find the number of ordered tuples \((C,A,M,B)\) of non-negative integers such that \[ C! + C! + A! + M! = B! \]
Answer: 7
user: Let $a, b, c > 1$ be positive real numbers such that $a^{\log_b c} = 27$, $b^{\log_c a} = 81$, and $c^{\log_a b} = 243$. Then the value of $\log_3{abc}$ can be written as $\sqrt{x} + \sqrt{y} + \sqrt{z}$ for positive integers $x, y,$ and $z$. Find $x + y + z$.
Answer: 47
user: Let $O$ be a circle with diameter $AB = 2$. Circles $O_1$ and $O_2$ have centers on $\overline{AB}$ such that $O$ is tangent to $O_1$ at $A$ and to $O_2$ at $B$, and $O_1$ and $O_2$ are externally tangent to each other. The minimum possible value of the sum of the areas of $O_1$ and $O_2$ can be written in the form $\frac{m\pi}{n}$ where $m$ and $n$ are relatively prime positive integers. Compute $m + n$.
Answer: 3
user: Given that $a$, $b$, and $c$ are distinct positive integers such that $a \cdot b \cdot c = 2020$, determine the minimum possible positive value of $\frac{1}{a} - \frac{1}{b} - \frac{1}{c}$. This value can be expressed as $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
Answer: 2101
user: Find all positive integers $n$ for which $1^3 + 2^3 + \cdots + 16^3 + 17^n$ is a perfect square.
Answer: 3
user: Sohom constructs a square $BERK$ with side length $10$. Darlnim adds points $T$, $O$, $W$, and $N$, which are the midpoints of $\overline{BE}$, $\overline{ER}$, $\overline{RK}$, and $\overline{KB}$, respectively. Lastly, Sylvia constructs square $CALI$ whose edges contain the vertices of $BERK$, such that $\overline{CA}$ is parallel to $\overline{BO}$. Compute the area of $CALI$.
Answer: 180
user: For each positive integer $n$, let $s(n)$ be the sum of the digits of $n$. Find the smallest positive integer $k$ such that \[ s(k) = s(2k) = s(3k) = \cdots = s(2013k) = s(2014k). \]
Answer: 9999
user: Two circles $S_1$ and $S_2$, with radii $6$ units and $3$ units respectively, are tangent to each other externally. Let $AC$ and $BD$ be their direct common tangents, with $A$ and $B$ on $S_1$, and $C$ and $D$ on $S_2$. Find the area of quadrilateral $ABDC$ to the nearest integer.
Answer: 68
user: A particle is at the origin of the Cartesian plane, $(0, 0)$. In each step, the particle can move $1$ unit in any of the four directions: left, right, up, or down. Determine the number of ways the particle can move from $(0, 0)$ to $(0, 2)$ in exactly $6$ steps. 

Note: Two paths are considered different if the order of traversal of each point differs, even if the same set of points is traversed.
Answer: 225
user: A positive integer $N$ is divided into $n$ parts inversely proportional to the numbers $2, 6, 12, 20, \ldots$. The smallest part is equal to $\frac{1}{400} N$. Find the value of $n$.
Answer: 20
user: Every one of the six trucks of a construction company drove for $8$ hours, and they all together spent $720$ litres of oil. How many litres should $9$ trucks spend if every one of them drives for $6$ hours?
Answer: 810
user: Let $ABCD$ be a convex quadrilateral for which $DA = AB$ and $CA = CB$. Set $I_0 = C$ and $J_0 = D$, and for each nonnegative integer $n$, let $I_{n+1}$ and $J_{n+1}$ denote the incenters of $\triangle I_nAB$ and $\triangle J_nAB$, respectively.

Suppose that \[ \angle DAC = 15^{\circ}, \quad \angle BAC = 65^{\circ} \quad \text{and} \quad \angle J_{2013}J_{2014}I_{2014} = \left( 90 + \frac{2k+1}{2^n} \right)^{\circ} \] for some nonnegative integers $n$ and $k$. Compute $n+k$.
Answer: 2021
user: \textbf{Problem 1.} There are less than $400$ marbles. If they are distributed among $3$ children, there is one left over. If they are distributed among $7$ children, there are 2 left over. Finally, if they are distributed among $5$ children, there are none left over. What is the largest number of marbles you have?
Answer: 310
user: Emily draws six dots on a piece of paper such that no three lie on a straight line. She then draws a line segment connecting each pair of dots and colors five of these segments red. Her coloring is said to be $\emph{red-triangle-free}$ if for every set of three points from her six drawn points, there exists an uncolored segment connecting two of the three points. In how many ways can Emily color her drawing such that it is red-triangle-free?
Answer: 1773
user: Given a Pacman of radius $1$, and a mouth opening angle of $90^\circ$, what is the largest circular pellet it can eat? The pellet must lie entirely outside the yellow portion and entirely inside the circumcircle of the Pacman. Let the radius be equal to $a\sqrt{b}+c$, where $b$ is square-free. Find $a+b+c$.
Answer: 2
user: Consider real numbers $A$, $B$, \dots, $Z$ such that:
\[\begin{align*}
  EVIL &= \frac{5}{31}, \\
  LOVE &= \frac{6}{29}, \\
  IMO  &= \frac{7}{3}.
\end{align*}\]
If $OMO = \frac{m}{n}$ for relatively prime positive integers $m$ and $n$, find the value of $m+n$.
Answer: 579
user: Find the smallest positive multiple of $77$ such that the last four digits of the number are $2020$.
Answer: 482020
user: The roots of the polynomial $P(x) = x^3 + 5x + 4$ are $r$, $s$, and $t$. Evaluate $(r+s)^4 (s+t)^4 (t+r)^4$.
Answer: 256
user: On a $5 \times 5$ grid, cover the area with several S-Tetrominos (also known as Z-Tetrominos) such that in every square, there are two or fewer tiles covering it (tiles can overlap). Determine the maximum possible number of squares that can be covered by at least one tile.
Answer: 24
user: $P(x)$ is a polynomial in $x$ with non-negative integer coefficients. If $P(1) = 5$ and $P(P(1)) = 177$, what is the sum of all possible values of $P(10)$?
Answer: 1202
user: The sequence of letters **TAGC** is written in succession 55 times on a strip, as shown below. The strip is to be cut into segments between letters, leaving strings of letters on each segment, which we call words. For example, a cut after the first G, after the second T, and after the second C would yield the words **TAG**, **CT**, and **AGC**. At most, how many distinct words could be found if the entire strip were cut? Justify your answer.

\[ \boxed{\textbf{T A G C T A G C T A G}} \ldots \boxed{\textbf{C T A G C}} \]
Answer: 16
user: Let $F_0 = 0$, $F_1 = 1$, and for $n \ge 1$, $F_{n+1} = F_n + F_{n-1}$. Define $a_n = \left(\frac{1 + \sqrt{5}}{2}\right)^n \cdot F_n$. Then there are rational numbers $A$ and $B$ such that:

$$\frac{a_{30} + a_{29}}{a_{26} + a_{25}} = A + B \sqrt{5}.$$ 

Find $A + B$.
Answer: 34
user: Nathan has discovered a new way to construct chocolate bars, but it's expensive! He starts with a single $1 \times 1$ square of chocolate and then adds more rows and columns from there. If his current bar has dimensions $w \times h$ (where $w$ is the number of columns and $h$ is the number of rows), then it costs $w^2$ dollars to add another row and $h^2$ dollars to add another column. What is the minimum cost to get his chocolate bar to size $20 \times 20$?
Answer: 5339
user: The cross-section of a prism with an index of refraction $1.5$ is an equilateral triangle, as shown in the diagram. A ray of light enters horizontally from the air into the prism and exits at an angle $\theta$ with respect to the surface of the triangle. Find $\theta$ in degrees and round to the nearest whole number.
Answer: 13
user: For a complex number constant $c$, and real number constants $p$ and $q$, there exist three distinct complex values of $x$ that satisfy the equation:
\[
x^3 + cx + p(1 + qi) = 0
\]
Suppose $c$, $p$, and $q$ were chosen such that all three complex roots $x$ satisfy the inequality:
\[
\frac{5}{6} \leq \frac{\mathrm{Im}(x)}{\mathrm{Re}(x)} \leq \frac{6}{5}
\]
where $\mathrm{Im}(x)$ and $\mathrm{Re}(x)$ are the imaginary and real parts of $x$, respectively. The largest possible value of $|q|$ can be expressed as a common fraction \( \frac{m}{n} \), where $m$ and $n$ are relatively prime positive integers. Compute $m + n$.
Answer: 11
user: There are sixteen buildings all on the same side of a street. How many ways can we choose a nonempty subset of the buildings such that there is an odd number of buildings between each pair of buildings in the subset?
Answer: 510
user: Six boxes are numbered $1$, $2$, $3$, $4$, $5$, and $6$. Suppose that there are $N$ balls distributed among these six boxes. Find the least $N$ for which it is guaranteed that for at least one $k$, box number $k$ contains at least $k^2$ balls.
Answer: 86
user: Let $\Omega$ be a circle with radius $18$ and let $\mathcal{S}$ be the region inside $\Omega$ that the centroid of $\triangle XYZ$ sweeps through as $X$ varies along all possible points lying outside of $\Omega$, $Y$ varies along all possible points lying on $\Omega$ and $XZ$ is tangent to the circle. Compute the greatest integer less than or equal to the area of $\mathcal{S}$.
Answer: 904
user: Let $\varphi$ be the positive solution to the equation $x^2 = x + 1$. For $n \ge 0$, let $a_n$ be the unique integer such that $\varphi^n - a_n \varphi$ is also an integer. Compute $\sum_{n=0}^{10} a_n$.
Answer: 143
user: Compute the sum \[ S = 1 + 2 + 3 - 4 - 5 + 6 + 7 + 8 - 9 - 10 + \dots - 2010 \] where every three consecutive additions (\(+\)) are followed by two subtractions (\(-\)).
Answer: 401799
user: Triangle $ABC$ has side lengths $AB=75$, $BC=100$, and $CA=125$. Point $D$ is the foot of the altitude from $B$, and $E$ lies on segment $BC$ such that $DE \perp BC$. Find the area of triangle $BDE$.
Answer: 864
user: Determine the largest natural number such that all its decimal digits are different and it is divisible by each of its digits.
Answer: 9867312
user: In a group of 2017 persons, any pair of persons has exactly one common friend (other than the pair of persons). Determine the smallest possible value of the difference between the numbers of friends of the person with the most friends and the person with the least friends in such a group.
Answer: 2014
user: Suppose $x$ and $y$ are nonzero real numbers that simultaneously satisfy the equations:

\[ x + \frac{2018}{y} = 1000 \]

and

\[ \frac{9}{x} + y = 1 \]

Find the maximum possible value of $x + 1000y$.
Answer: 1991
user: An equilateral triangle with side length $20$ is divided by a series of parallel lines into $400$ equilateral triangles, each with side length $1$. What is the maximum number of these small triangles that can be crossed (internally) by one line?
Answer: 39
user: Pedro must choose two irreducible fractions, each with a positive numerator and denominator, such that:

1. The sum of the fractions is equal to $2$.
2. The sum of the numerators of the fractions is equal to $1000$.

In how many ways can Pedro do this?
Answer: 200
user: Determine the number of 3-digit numbers in base 10 having at least one 5 and at most one 3.
Answer: 249
user: If $a, b, c, d$ are real numbers such that $a^2 + b^2 + c^2 + d^2 \leq 1$, find the maximum of the expression:
\[(a + b)^4 + (a + c)^4 + (a + d)^4 + (b + c)^4 + (b + d)^4 + (c + d)^4.\]
Answer: 6
user: 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
user: An *access pattern* $\pi$ is a permutation of $\{1,2,\dots,50\}$ describing the order in which some 50 memory addresses are accessed. We define the *locality* of $\pi$ to be how much the program jumps around the memory, or numerically, 
\[ \sum_{i=2}^{50}\left\lvert\pi(i)-\pi(i-1)\right\rvert. \]
If $\pi$ is a uniformly randomly chosen access pattern, what is the expected value of its locality?
Answer: 833
user: Taotao wants to buy a bracelet. The bracelets have 7 different beads on them, arranged in a circle. Two bracelets are considered the same if one can be rotated or flipped to become the other. If she can choose the colors and placement of the beads, and the beads come in orange, white, and black, how many possible bracelets can she buy?
Answer: 198
user: For $n \in \mathbb{N}$, consider non-negative valued functions $f$ on $\{1,2, \cdots , n\}$ satisfying $f(i) \geq f(j)$ for $i > j$ and $\sum_{i=1}^{n} (i + f(i)) = 2023.$ Choose $n$ such that $\sum_{i=1}^{n} f(i)$ is minimized. How many such functions exist in that case?
Answer: 15
user: Find the largest number $n$ such that $(2004!)!$ is divisible by $((n!)!)!$.
Answer: 6
user: For the NEMO, Kevin needs to compute the product:
\[ 9 \times 99 \times 999 \times \cdots \times 999999999. \]
Kevin takes exactly $ab$ seconds to multiply an $a$-digit integer by a $b$-digit integer. Compute the minimum number of seconds necessary for Kevin to evaluate the expression together by performing eight such multiplications.
Answer: 870
user: In a tournament bracket, a player must be defeated three times to be eliminated. If there are 512 contestants entering the tournament, what is the greatest number of games that could be played?
Answer: 1535
user: Let $x_1, x_2, \ldots, x_5$ be real numbers. Find the least positive integer $n$ with the following property: if some $n$ distinct sums of the form $x_p + x_q + x_r$ (with $1 \leq p < q < r \leq 5$) are equal to $0$, then $x_1 = x_2 = \cdots = x_5 = 0$.
Answer: 7
user: How many ordered pairs of integers \((x, y)\) satisfy the equation:
\[
8(x^3 + x^2y + xy^2 + y^3) = 15(x^2 + y^2 + xy + 1)
\]
Answer: 2
user: Call a positive integer \textit{prime-simple} if it can be expressed as the sum of the squares of two distinct prime numbers. How many positive integers less than or equal to $100$ are prime-simple?
Answer: 6
user: How many ways are there to rearrange the letters of \( CCARAVEN \)?
Answer: 10080
user: Find the number of integers $a$ with $1 \le a \le 2012$ for which there exist nonnegative integers $x, y, z$ satisfying the equation:
\[ x^2(x^2 + 2z) - y^2(y^2 + 2z) = a. \]
Answer: 1256