meetforever/GRPO_CL huggingface dataset:
user: Rectangle $ABCD$ has side lengths $AB = 6\sqrt{3}$ and $BC = 8\sqrt{3}$. The probability that a randomly chosen point inside the rectangle is closer to the diagonal $\overline{AC}$ than to the outside of the rectangle is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
Answer: 17
user: What are the last $2$ digits of the number $2018^{2018}$ when written in base $7$?
Answer: 44
user: In trapezoid $ABCD$, the diagonals intersect at $E$. The area of $\triangle ABE$ is 72 and the area of $\triangle CDE$ is 50. What is the area of trapezoid $ABCD$?
Answer: 242
user: In how many ways can you write $12$ as an ordered sum of integers where the smallest of those integers is equal to $2$? For example, $2 + 10$, $10 + 2$, and $3 + 2 + 2 + 5$ are three such ways.
Answer: 70
user: Points $A$, $B$, and $C$ lie on a circle $\Omega$ such that $A$ and $C$ are diametrically opposite each other. A line $\ell$ tangent to the incircle of $\triangle ABC$ at $T$ intersects $\Omega$ at points $X$ and $Y$. Suppose that $AB=30$, $BC=40$, and $XY=48$. Compute $TX \cdot TY$.
Answer: 460
user: Compute the remainder when $29^{30} + 31^{28} + 28! \cdot 30!$ is divided by $29 \cdot 31$.
Answer: 668
user: Find the remainder when $2+4+\cdots+2014$ is divided by $1+3+\cdots+2013$. Justify your answer.
Answer: 1007
user: Kayla draws three triangles on a sheet of paper. What is the maximum possible number of regions, including the exterior region, that the paper can be divided into by the sides of the triangles?
Answer: 20
user: Compute the largest prime factor of $357! + 358! + 359! + 360!$.
Answer: 379
user: Let $f(x) = (x-a)^3$. If the sum of all $x$ satisfying $f(x) = f(x-a)$ is $42$, find $a$.
Answer: 14
user: The lengths of the altitudes of $\triangle ABC$ are the roots of the polynomial $x^3 - 34x^2 + 360x - 1200.$ Find the area of $\triangle ABC.$
Answer: 100
user: Given that $a_1, a_2, a_3, \ldots, a_{99}$ is a permutation of $1, 2, 3, \ldots, 99$, find the maximum possible value of
\[
|a_1 - 1| + |a_2 - 2| + |a_3 - 3| + \dots + |a_{99} - 99|.
\]
Answer: 4900
user: Find all rational terms of the sequence defined by the formula: $$ a_n = \sqrt{\frac{9n-2}{n+1}}, \quad n \in \mathbb{N} $$
Answer: 3
user: Five persons wearing badges with numbers $1, 2, 3, 4, 5$ are seated on $5$ chairs around a circular table. In how many ways can they be seated so that no two persons whose badges have consecutive numbers are seated next to each other? (Two arrangements obtained by rotation around the table are considered different)
Answer: 10
user: Ditty can bench $80$ pounds today. Every week, the amount he benches increases by the largest prime factor of the weight he benched in the previous week. For example, since he started benching $80$ pounds, next week he would bench $85$ pounds. What is the minimum number of weeks from today it takes for Ditty to bench at least $2021$ pounds?
Answer: 69
user: Find the smallest positive integer $j$ such that for every polynomial $p(x)$ with integer coefficients and for every integer $k$, the integer 
\[p^{(j)}(k)=\left. \frac{d^j}{dx^j}p(x) \right|_{x=k}\]
(the $j$-th derivative of $p(x)$ at $k$) is divisible by $2016$.
Answer: 8
user: Suppose that $x$, $y$, and $z$ are complex numbers of equal magnitude that satisfy:
\[
x + y + z = -\frac{\sqrt{3}}{2} - i\sqrt{5}
\]
and
\[
xyz = \sqrt{3} + i\sqrt{5}.
\]
If $x = x_1 + ix_2$, $y = y_1 + iy_2$, and $z = z_1 + iz_2$ for real $x_1, x_2, y_1, y_2, z_1$, and $z_2$, then:
\[
(x_1x_2 + y_1y_2 + z_1z_2)^2
\]
can be written as $\frac{a}{b}$ for relatively prime positive integers $a$ and $b$. Compute $100a + b$.
Answer: 1516
user: A pulley system with two blocks is released from rest. The block on the table, with mass $m = 1.0 \, \text{kg}$, slides and hits the pulley, coming to a stop. The initial distance between the block and the pulley is $1.0 \, \text{m}$. During the slide, the rope makes a constant angle of $10.0^\circ$ with the horizontal. The hanging block has a mass of $M = 2.0 \, \text{kg}$. The table has a coefficient of friction of $0.50$ with the block. The pulley is frictionless, and the block remains on the ground throughout. Let $t$ be the time in seconds for the $1.0\text{-m}$ slide. Find $100t$, rounded to two significant figures.
Answer: 62
user: What is the smallest positive integer $n$ such that $2^n - 1$ is a multiple of $2015$?
Answer: 60
user: Determine the least real number $k$ such that the inequality
\[
\left(\frac{2a}{a-b}\right)^2 + \left(\frac{2b}{b-c}\right)^2 + \left(\frac{2c}{c-a}\right)^2 + k \geq 4\left(\frac{2a}{a-b} + \frac{2b}{b-c} + \frac{2c}{c-a}\right)
\]
holds for all real numbers $a, b, c$.
Answer: 8
user: Consider the sequence \( \{1, 3, 13, 31, \ldots\} \) that is obtained by following diagonally the array of numbers in a spiral. Find the number in the \(100\)th position of that sequence.
Answer: 39007
user: Consider all possible integers $n \ge 0$ such that $5 \cdot 3^m + 4 = n^2$ holds for some corresponding integer $m \ge 0$. Find the sum of all such $n$.
Answer: 10
user: Let $ABCD$ be a convex quadrilateral with $AC = 20$, $BC = 12$, and $BD = 17$. If $\angle CAB = 80^{\circ}$ and $\angle DBA = 70^{\circ}$, then find the area of $ABCD$.
Answer: 85
user: Let $a, b, c, d, e$ be nonnegative integers such that $625a + 250b + 100c + 40d + 16e = 15^3$. What is the maximum possible value of $a + b + c + d + e$?
Answer: 153
user: In a plane, 2014 lines are distributed into 3 groups. In each group, all the lines are parallel to each other. What is the maximum number of triangles that can be formed, such that each side of a triangle lies on one of these lines?
Answer: 302561952
user: Let $p(x) = 3x^2 + 1$. Compute the largest prime divisor of $p(100) - p(3)$.
Answer: 103
user: Let $ABC$ be an isosceles triangle with $AB=AC$ and incentre $I$. If $AI = 3$ and the distance from $I$ to $BC$ is $2$, what is the square of the length of $BC$?
Answer: 80
user: The figure below shows a $9 \times 7$ arrangement of $2 \times 2$ squares. Alternate squares of the grid are split into two triangles, with one of the triangles shaded. Find the area of the shaded region.  \[ \text{[asy]} \] \[ \text{size}(5\text{cm}); \] \[ \text{defaultpen}(\text{linewidth}(.6)); \] \[ \text{fill}((0,1)--(1,1)--(1,0)--\text{cycle}^^(0,3)--(1,3)--(1,2)--\text{cycle}^^(1,2)--(2,2)--(2,1)--\text{cycle}^^(2,1)--(3,1)--(3,0)--\text{cycle},\text{rgb}(.76,.76,.76)); \] \[ \text{fill}((0,5)--(1,5)--(1,4)--\text{cycle}^^(1,4)--(2,4)--(2,3)--\text{cycle}^^(2,3)--(3,3)--(3,2)--\text{cycle}^^(3,2)--(4,2)--(4,1)--\text{cycle}^^(4,1)--(5,1)--(5,0)--\text{cycle},\text{rgb}(.76,.76,.76)); \] \[ \text{fill}((0,7)--(1,7)--(1,6)--\text{cycle}^^(1,6)--(2,6)--(2,5)--\text{cycle}^^(2,5)--(3,5)--(3,4)--\text{cycle}^^(3,4)--(4,4)--(4,3)--\text{cycle}^^(4,3)--(5,3)--(5,2)--\text{cycle}^^(5,2)--(6,2)--(6,1)--\text{cycle}^^(6,1)--(7,1)--(7,0)--\text{cycle},\text{rgb}(.76,.76,.76)); \] \[ \text{fill}((2,7)--(3,7)--(3,6)--\text{cycle}^^(3,6)--(4,6)--(4,5)--\text{cycle}^^(4,5)--(5,5)--(5,4)--\text{cycle}^^(5,4)--(6,4)--(6,3)--\text{cycle}^^(6,3)--(7,3)--(7,2)--\text{cycle}^^(7,2)--(8,2)--(8,1)--\text{cycle}^^(8,1)--(9,1)--(9,0)--\text{cycle},\text{rgb}(.76,.76,.76)); \] \[ \text{fill}((4,7)--(5,7)--(5,6)--\text{cycle}^^(5,6)--(6,6)--(6,5)--\text{cycle}^^(6,5)--(7,5)--(7,4)--\text{cycle}^^(7,4)--(8,4)--(8,3)--\text{cycle}^^(8,3)--(9,3)--(9,2)--\text{cycle},\text{rgb}(.76,.76,.76)); \] \[ \text{fill}((6,7)--(7,7)--(7,6)--\text{cycle}^^(7,6)--(8,6)--(8,5)--\text{cycle}^^(8,5)--(9,5)--(9,4)--\text{cycle},\text{rgb}(.76,.76,.76)); \] \[ \text{fill}((8,7)--(9,7)--(9,6)--\text{cycle},\text{rgb}(.76,.76,.76)); \] \[ \text{draw}((0,0)--(0,7)^^(1,0)--(1,7)^^(2,0)--(2,7)^^(3,0)--(3,7)^^(4,0)--(4,7)^^(5,0)--(5,7)^^(6,0)--(6,7)^^(7,0)--(7,7)^^(8,0)--(8,7)^^(9,0)--(9,7)); \] \[ \text{draw}((0,0)--(9,0)^^(0,1)--(9,1)^^(0,2)--(9,2)^^(0,3)--(9,3)^^(0,4)--(9,4)^^(0,5)--(9,5)^^(0,6)--(9,6)^^(0,7)--(9,7)); \] \[ \text{draw}((0,1)--(1,0)^^(0,3)--(3,0)^^(0,5)--(5,0)^^(0,7)--(7,0)^^(2,7)--(9,0)^^(4,7)--(9,2)^^(6,7)--(9,4)^^(8,7)--(9,6)); \]
Answer: 64
user: Let $\lfloor x \rfloor$ denote the greatest integer less than or equal to $x$. Find the value of the sum:
\[
\left\lfloor 2+\frac{1}{2^{2021}} \right\rfloor + \left\lfloor 2+\frac{1}{2^{2020}} \right\rfloor + \cdots + \left\lfloor 2+\frac{1}{2^1} \right\rfloor + \left\lfloor 2+\frac{1}{2^0} \right\rfloor.
\]
Answer: 4045
user: Let $A$ be a subset of $\{1, 2, 3, \ldots, 50\}$ with the property that for every $x, y \in A$ with $x \neq y$, it holds that \[\left| \frac{1}{x} - \frac{1}{y} \right| > \frac{1}{1000}.\] Determine the largest possible number of elements that the set $A$ can have.
Answer: 40
user: Let $a = -\sqrt{3} + \sqrt{5} + \sqrt{7}$, $b = \sqrt{3} - \sqrt{5} + \sqrt{7}$, $c = \sqrt{3} + \sqrt{5} - \sqrt{7}$. Evaluate the expression:
\[
\frac{a^4}{(a-b)(a-c)} + \frac{b^4}{(b-c)(b-a)} + \frac{c^4}{(c-a)(c-b)}.
\]
Answer: 30
user: Dan the dog spots Cate the cat 50 meters away. At that instant, Cate begins running away from Dan at 6 m/s, and Dan begins running toward Cate at 8 m/s. Both of them accelerate instantaneously and run in straight lines. Compute the number of seconds it takes for Dan to reach Cate.
Answer: 25
user: Two ants sit at the vertex of the parabola $y = x^2$. One starts walking northeast (i.e., upward along the line $y = x$) and the other starts walking northwest (i.e., upward along the line $y = -x$). Each time they reach the parabola again, they swap directions and continue walking. Both ants walk at the same speed. When the ants meet for the eleventh time (including the time at the origin), their paths will enclose 10 squares. What is the total area of these squares?
Answer: 770
user: Let $\tau(x)$ be the number of positive divisors of $x$ (including $1$ and $x$). Find $\tau\left( \tau\left( \dots \tau\left(2024^{2024^{2024}}\right) \right)\right)$, where there are $4202^{4202^{4202}}$ $\tau$'s.
Answer: 2
user: A positive integer $n$ is defined as a \textit{stepstool number} if $n$ has one less positive divisor than $n + 1$. For example, $3$ is a stepstool number, as $3$ has $2$ divisors and $4$ has $2 + 1 = 3$ divisors. Find the sum of all stepstool numbers less than $300$.
Answer: 687
user: The figure below shows a large square divided into $9$ congruent smaller squares. A shaded square, bounded by some of the diagonals of those smaller squares, has an area of $14$. Find the area of the large square.
Answer: 63
user: Find the last three digits of the number $2003^{2002^{2001}}$.
Answer: 241
user: For natural numbers $x$ and $y$, let $(x,y)$ denote the greatest common divisor of $x$ and $y$. How many pairs of natural numbers $x$ and $y$ with $x \le y$ satisfy the equation $xy = x + y + (x, y)$?
Answer: 3
user: Sara wrote on the board an integer with less than thirty digits and ending in $2$. Celia erases the $2$ from the end and writes it at the beginning. The number that remains written is equal to twice the number that Sara had written. What number did Sara write?
Answer: 105263157894736848
user: In order to save money on gas and use up less fuel, Hannah has a special battery installed in the family van. Before the installation, the van averaged $18$ miles per gallon of gas. After the conversion, the van got $24$ miles per gallon of gas.

Michael notes, "The amount of money we will save on gas over any time period is equal to the amount we would save if we were able to convert the van to go from $24$ miles per gallon to $m$ miles per gallon. It is also the same that we would save if we were able to convert the van to go from $m$ miles per gallon to $n$ miles per gallon."

Assuming Michael is correct, compute $m+n$. In this problem, assume that gas mileage is constant over all speeds and terrain and that the van gets used the same amount regardless of its present state of conversion.
Answer: 108
user: Evaluate the sum of the number of decimal digits in $2^{1997}$ and $5^{1997}$. Let $m$ be the number of decimal digits in $2^{1997}$, and $n$ be the number of decimal digits in $5^{1997}$. Find $m + n$.
Answer: 1998
user: Let us consider a set of $6050$ points in the plane, where no three points are collinear. Determine the maximum number $k$ of non-overlapping triangles that can be formed such that no two triangles share a vertex.
Answer: 2016
user: Evaluate the sum: 
\[ \sqrt{1 - \frac{1}{2} \cdot \sqrt{1 \cdot 3}} + \sqrt{2 - \frac{1}{2} \cdot \sqrt{3 \cdot 5}} + \sqrt{3 - \frac{1}{2} \cdot \sqrt{5 \cdot 7}} + \dots + \sqrt{40 - \frac{1}{2} \cdot \sqrt{79 \cdot 81}} \]
Answer: 4
user: For a real number $x$, let $\lfloor x \rfloor$ denote the greatest integer less than or equal to $x$, and let $\{x\} = x - \lfloor x \rfloor$ denote the fractional part of $x$. The sum of all real numbers $\alpha$ that satisfy the equation $$\alpha^2 + \{\alpha\} = 21$$ can be expressed in the form $$\frac{\sqrt{a} - \sqrt{b}}{c} - d$$ where $a$, $b$, $c$, and $d$ are positive integers, and $a$ and $b$ are not divisible by the square of any prime. Compute $a + b + c + d$.
Answer: 169
user: Let $n$ be a positive integer. E. Chen and E. Chen play a game on the $n^2$ points of an $n \times n$ lattice grid. They alternately mark points on the grid such that no player marks a point that is on or inside a non-degenerate triangle formed by three marked points. Each point can be marked only once. The game ends when no player can make a move, and the last player to make a move wins. Determine the number of values of $n$ between $1$ and $2013$ (inclusive) for which the first player can guarantee a win, regardless of the moves that the second player makes.
Answer: 1007
user: Consider the number obtained by writing the numbers $1, 2, \ldots, 1990$ one after another. In this number, every digit at an even position is omitted. In the resulting number, every digit at an odd position is omitted. Then, in the new number, every digit at an even position is omitted, and so on. What will be the last remaining digit?
Answer: 9
user: A triangle with side lengths $16$, $18$, and $21$ has a circle with radius $6$ centered at each vertex. Find $n$ so that the total area inside the three circles but outside of the triangle is $n\pi$.
Answer: 90
user: Compute the number of positive integers less than or equal to $2015$ that are divisible by $5$ or $13$, but not both.
Answer: 496
user: In the USA, standard letter-size paper is 8.5 inches wide and 11 inches long. What is the largest integer that cannot be written as a sum of a whole number (possibly zero) of 8.5's and a whole number (possibly zero) of 11's?
Answer: 159
user: Peter wants to create a unique die where each face displays a different positive integer. For any two adjacent faces, the numbers must differ by at least two. Determine the minimum possible sum of the six numbers on the die.
Answer: 36
user: Find the smallest positive $\alpha$ (in degrees) for which all the numbers $\cos{\alpha},\cos{2\alpha},\ldots,\cos{2^n\alpha},\ldots$ are negative.
Answer: 120
user: How many permutations of the word "ACADEMY" have two vowels separated by an odd distance? 

For example, in the sequence XAY, X and Y are separated by an even distance, while in XABY, X and Y are separated by an odd distance. Note: The vowels are A, E, I, O, and U. Y is NOT considered a vowel.
Answer: 2160
user: Starting at $12:00:00$ AM on January $1, 2022$, after $13!$ seconds it will be $y$ years (including leap years) and $d$ days later, where $d < 365$. Find $y + d$.
Answer: 317
user: $P(x)$ is a polynomial of degree at most $6$ such that $P(1)$, $P(2)$, $P(3)$, $P(4)$, $P(5)$, $P(6)$, and $P(7)$ are $1$, $2$, $3$, $4$, $5$, $6$, and $7$ in some order. What is the maximum possible value of $P(8)$?
Answer: 312
user: The diagram below shows a rectangle with side lengths $36$ and $48$. Each of the sides is trisected, and edges are added between the trisection points as shown. Then the shaded corner regions are removed, leaving the octagon which is not shaded in the diagram. Find the perimeter of this octagon.
Answer: 136
user: For each integer $n \geq 1$, let $S_n$ be the set of integers $k > n$ such that $k$ divides $30n - 1$. How many elements of the set $$\mathcal{S} = \bigcup_{i \geq 1} S_i = S_1 \cup S_2 \cup S_3 \cup \ldots$$ are less than $2016$?
Answer: 536
user: A lumberjack is building a non-degenerate triangle out of logs. Two sides of the triangle have lengths $\log 101$ and $\log 2018$. The last side of his triangle has side length $\log n$, where $n$ is an integer. How many possible values are there for $n$?
Answer: 203798
user: The function $f(x)$ is defined on the reals such that \( f\left(\frac{1-4x}{4-x}\right) = 4-xf(x) \) for all \( x \neq 4 \). There exist two distinct real numbers \( a, b \neq 4 \) such that \( f(a) = f(b) = \frac{5}{2} \). The sum \( a+b \) can be represented as \( \frac{p}{q} \) where \( p, q \) are relatively prime positive integers. Find \( 10p + q \).
Answer: 180
user: Find the greatest integer $n$ such that $10^n$ divides \[ \frac{2^{10^5} \cdot 5^{2^{10}}}{10^{5^2}} \]
Answer: 999
user: If we divide the number $1^{1990} + 2^{1990} + 3^{1990} + \ldots + 1990^{1990}$ by $10$, what is the remainder?
Answer: 5
user: Bill bought 13 notebooks, 26 pens, and 19 markers for $25. Paula bought 27 notebooks, 18 pens, and 31 markers for $31. How many dollars would it cost Greg to buy 24 notebooks, 120 pens, and 52 markers?
Answer: 88
user: Starting on April 15, 2008, you can go one day backward and one day forward to get the dates 14 and 16. If you go 15 days backward and 15 days forward, you get the dates 31 (from March) and 30 (from April). Find the least positive integer $k$ so that if you go $k$ days backward and $k$ days forward you get two calendar dates that are the same.
Answer: 14
user: Let $a, b, c, d$ be four real numbers such that $a + b + c + d = 20$ and $ab + bc + cd + da = 16$. Find the maximum possible value of $abc + bcd + cda + dab$.
Answer: 80
user: An equilateral triangle $ABC$ has a side length of $7$. Point $P$ is located inside triangle $ABC$ such that $PB = 3$ and $PC = 5$. The distance between the circumcenters of triangles $ABC$ and $PBC$ can be expressed in the form $\frac{m\sqrt{n}}{p}$, where $n$ is not divisible by the square of any prime number, and $m$ and $p$ are relatively prime positive integers. Determine the value of $m + n + p$.
Answer: 13
user: In the Game of Life, each square in an infinite grid of squares is either shaded or blank. Every day, if a square shares an edge with exactly zero or four shaded squares, it becomes blank the next day. If a square shares an edge with exactly two or three shaded squares, it becomes shaded the next day. Otherwise, it does not change. On day $1$, each square is randomly shaded or blank with equal probability. If the probability that a given square is shaded on day 2 is $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers, find $a + b$.
Answer: 7
user: Let $f$ be a function satisfying $f(x+y+z) = f(x) + f(y) + f(z)$ for all integers $x$, $y$, $z$. Suppose $f(1) = 1$, $f(2) = 2$. Then $\lim \limits_{n \to \infty} \frac{1}{n^3} \sum \limits_{r=1}^n 4rf(3r)$ equals:

1. 4
2. 6
3. 12
4. 24
Answer: 4
user: Let $A=\{1,2,3,\cdots ,17\}$. A mapping $f:A\rightarrow A$ is defined as follows: $f^{[1]}(x)=f(x)$, $f^{[k+1]}(x)=f(f^{[k]}(x))$ for $k\in\mathbb{N}$. Suppose that $f$ is bijective and that there exists a natural number $M$ such that:

1. When $m<M$ and $1\le i\le 16$, we have $f^{[m]}(i+1)- f^{[m]}(i) \not=\pm 1\pmod{17}$ and  $f^{[m]}(1)- f^{[m]}(17) \not=\pm 1\pmod{17}$;
2. When $1\le i\le 16$, we have $f^{[M]}(i+1)- f^{[M]}(i)=\pm 1 \pmod{17}$ and $f^{[M]}(1)- f^{[M]}(17)=\pm 1\pmod{17}$.

Find the maximal value of $M$.
Answer: 8
user: A container contains five red balls. On each turn, one of the balls is selected at random, painted blue, and returned to the container. The expected number of turns it will take before all five balls are colored blue is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
Answer: 149
user: Andrew and Blair are bored in class and decide to play a game. They pick a pair \((a, b)\) with \(1 \le a, b \le 100\). Andrew says the next number in the geometric series that begins with \(a, b\) and Blair says the next number in the arithmetic series that begins with \(a, b\). For how many pairs \((a, b)\) is Andrew's number minus Blair's number a positive perfect square?
Answer: 281
user: Sam has 1 Among Us task left. He and his task are located at two randomly chosen distinct vertices of a 2021-dimensional unit hypercube. Let $E$ denote the expected distance he has to walk to get to his task, given that he is only allowed to walk along edges of the hypercube. Compute $\lceil 10E \rceil$.
Answer: 10105
user: Let $N$ be the number of sequences of natural numbers $d_1, d_2, \dots, d_{10}$ such that the following conditions hold: $d_1 \mid d_2$, $d_2 \mid d_3$, $\dots$, $d_9 \mid d_{10}$ and $d_{10} \mid 6^{2018}$. Evaluate the remainder when $N$ is divided by $2017$.
Answer: 121
user: Triangle $ABC$ is inscribed in circle $\omega$ with $AB = 5$, $BC = 7$, and $AC = 3$. The bisector of angle $A$ meets side $BC$ at $D$ and circle $\omega$ at a second point $E$. Let $\gamma$ be the circle with diameter $DE$. Circles $\omega$ and $\gamma$ meet at $E$ and a second point $F$. Then $AF^2 = \frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m + n$.
Answer: 919
user: Let $A$ be the set $\{1,2,\ldots,n\}$, where $n \geq 2$. Find the least number $n$ for which there exist permutations $\alpha$, $\beta$, $\gamma$, $\delta$ of the set $A$ such that:
\[ \sum_{i=1}^n \alpha(i) \beta(i) = \frac{19}{10} \sum_{i=1}^n \gamma(i)\delta(i) . \]
Answer: 28
user: In a triangle $ABC$, let $E$ be the midpoint of $AC$ and $F$ be the midpoint of $AB$. The medians $BE$ and $CF$ intersect at $G$. Let $Y$ and $Z$ be the midpoints of $BE$ and $CF$, respectively. If the area of triangle $ABC$ is 480, find the area of triangle $GYZ$.
Answer: 10
user: Joshua rolls two dice and records the product of the numbers face up. The probability that this product is composite can be expressed as \( \frac{m}{n} \) for relatively prime positive integers \( m \) and \( n \). Compute \( m+n \).
Answer: 65
user: Elfs and trolls are seated at a round table, with a total of 60 creatures. Trolls always lie, and all elfs always speak the truth, except when they make a little mistake. Everyone claims to sit between an elf and a troll, but exactly two elfs made a mistake. How many trolls are there at this table?
Answer: 20
user: Let $f$ be a one-to-one function from the set of natural numbers to itself such that $f(mn) = f(m)f(n)$ for all natural numbers $m$ and $n$. What is the least possible value of $f(999)$?
Answer: 24
user: Two distinct squares of the $8 \times 8$ chessboard $C$ are said to be adjacent if they have a vertex or side in common. Also, $g$ is called a $C$-gap if for every numbering of the squares of $C$ with all the integers $1, 2, \ldots, 64$, there exist two adjacent squares whose numbers differ by at least $g$. Determine the largest $C$-gap $g$.
Answer: 9
user: Find the number of positive integers $n$ between $1$ and $1000$, inclusive, satisfying
\[ \lfloor \sqrt{n - 1} \rfloor + 1 = \left\lfloor \sqrt{n + \sqrt{n} + 1}\right\rfloor\]
where $\lfloor n \rfloor$ denotes the greatest integer not exceeding $n$.
Answer: 535
user: Let $n = 2188 = 3^7+1$ and let $A_0^{(0)}, A_1^{(0)}, \ldots, A_{n-1}^{(0)}$ be the vertices of a regular $n$-gon (in that order) with center $O$. For $i = 1, 2, \dots, 7$ and $j=0,1,\dots,n-1$, let $A_j^{(i)}$ denote the centroid of the triangle \( \triangle A_j^{(i-1)} A_{j+3^{7-i}}^{(i-1)} A_{j+2 \cdot 3^{7-i}}^{(i-1)} \). Here the subscripts are taken modulo $n$. If \( \frac{|OA_{2014}^{(7)}|}{|OA_{2014}^{(0)}|} = \frac{p}{q} \) for relatively prime positive integers $p$ and $q$, find $p+q$.
Answer: 2188