sungyub/dapo-math-17k-verl huggingface dataset:
user: Determine the remainder when
$$\sum_{i=0}^{2015} \left\lfloor \frac{2^i}{25} \right\rfloor$$
is divided by 100, where $\lfloor x \rfloor$ denotes the largest integer not greater than $x$.

Answer: 14
user: Call a positive integer $N \ge 2$ ``special'' if for every $k$ such that $2 \leq k \leq N$, $N$ can be expressed as a sum of $k$ positive integers that are relatively prime to $N$ (although not necessarily relatively prime to each other). How many special integers are there less than $100$?

Answer: 50
user: On the Cartesian plane $\mathbb{R}^2$, a circle is said to be \textit{nice}
if its center is at the origin $(0,0)$ and it passes through at least one lattice point
(i.e. a point with integer coordinates).
Define the points $A = (20,15)$ and $B = (20,16)$. How many nice circles intersect the open segment $AB$?

For reference, the numbers $601$, $607$, $613$, $617$, $619$, $631$, $641$, $643$, $647$, $653$, $659$, $661$, $673$, $677$, $683$, $691$ are the only prime numbers between $600$ and $700$.

Answer: 10
user: $\textbf{(Lucas Numbers)}$ The Lucas numbers are defined by $L_0 = 2$, $L_1 = 1$, and $L_{n+2} = L_{n+1} + L_n$ for every $n \ge 0$. There are $N$ integers $1 \le n \le 2016$ such that $L_n$ contains the digit $1$. Estimate $N$.

An estimate of $E$ earns $\left\lfloor 20 - 2|N-E| \right\rfloor$ or $0$ points, whichever is greater.

Answer: 1984
user: For positive integers $a,b$, $a\uparrow\uparrow b$ is defined as follows: $a\uparrow\uparrow 1=a$, and $a\uparrow\uparrow b=a^{a\uparrow\uparrow (b-1)}$ if $b>1$.

Find the smallest positive integer $n$ for which there exists a positive integer $a$ such that $a\uparrow\uparrow 6\not \equiv a\uparrow\uparrow 7$ mod $n$.

Answer: 329
user: Define the sequence $a_1, a_2 \dots$ as follows: $a_1=1$ and for every $n\ge 2$,
\[
a_n = \begin{cases}
n-2 & \text{if } a_{n-1} =0 \\
a_{n-1} -1 & \text{if } a_{n-1} \neq 0
\end{cases}
\]
A non-negative integer $d$ is said to be {\em jet-lagged} if there are non-negative integers $r,s$ and a positive integer $n$ such that $d=r+s$ and that $a_{n+r} = a_n +s$. How many integers in $\{1,2,\dots, 2016\}$ are jet-lagged?

Answer: 51
user: Kelvin the Frog was bored in math class one day, so he wrote all ordered triples $(a, b, c)$ of positive integers such that $abc=2310$ on a sheet of paper. Find the sum of all the integers he wrote down. In other words, compute $$\sum_{\stackrel{abc=2310}{a,b,c\in \mathbb{N}}} (a+b+c),$$ where $\mathbb{N}$ denotes the positive integers.

Answer: 49140
user: Find the number of ordered triples of positive integers $(a, b, c)$ such that
\[6a + 10b + 15c = 3000.\]

Answer: 4851
user: Let $P$ and $A$ denote the perimeter and area respectively of a right triangle with relatively prime integer side-lengths. Find the largest possible integral value of $\frac{P^2}{A}$

The official statement does not have the final period.[/color]

Answer: 45
user: Sam spends his days walking around the following $2\times 2$ grid of squares.
\begin{tabular}[t]{|c|c|}\hline
1&2\\ \hline
4&3 \\ \hline
\end{tabular}
Say that two squares are adjacent if they share a side. He starts at the square labeled $1$ and every second walks to an adjacent square. How many paths can Sam take so that the sum of the numbers on every square he visits in his path is equal to $20$ (not counting the square he started on)?

Answer: 167
user: Consider the graph in $3$-space of
\[0 = xyz(x + y)(y + z)(z + x)(x - y)(y - z)(z - x).\]
This graph divides $3$-space into $N$ connected regions. What is $N$?

Answer: 48
user: Kelvin and $15$ other frogs are in a meeting, for a total of $16$ frogs. During the meeting, each pair of distinct frogs becomes friends with probability $\frac{1}{2}$. Kelvin thinks the situation after the meeting is [I]cool[/I] if for each of the $16$ frogs, the number of friends they made during the meeting is a multiple of $4$. Say that the probability of the situation being cool can be expressed in the form $\frac{a}{b}$, where $a$ and $b$ are relatively prime. Find $a$.

Answer: 1167
user: Sean is a biologist, and is looking at a strng of length $66$ composed of the letters $A$, $T$, $C$, $G$. A [i]substring[/i] of a string is a contiguous sequence of letters in the string. For example, the string $AGTC$ has $10$ substrings: $A$, $G$, $T$, $C$, $AG$, $GT$, $TC$, $AGT$, $GTC$, $AGTC$. What is the maximum number of distinct substrings of the string Sean is looking at?

Answer: 2100
user: Kelvin the Frog and $10$ of his relatives are at a party. Every pair of frogs is either [i]friendly[/i] or [i]unfriendly[/i]. When $3$ pairwise friendly frogs meet up, they will gossip about one another and end up in a [i]fight[/i] (but stay [i]friendly[/i] anyway). When $3$ pairwise unfriendly frogs meet up, they will also end up in a [i]fight[/i]. In all other cases, common ground is found and there is no fight. If all $\binom{11}{3}$ triples of frogs meet up exactly once, what is the minimum possible number of fights?

Answer: 28
user: Kelvin the Frog likes numbers whose digits strictly decrease, but numbers that violate this condition in at most one place are good enough. In other words, if $d_i$ denotes the $i$th digit, then $d_i\le d_{i+1}$ for at most one value of $i$. For example, Kelvin likes the numbers $43210$, $132$, and $3$, but not the numbers $1337$ and $123$. How many $5$-digit numbers does Kelvin like?

Answer: 14034
user: Let $ABCD$ be a convex quadrilateral with $AB=5$, $BC=6$, $CD=7$, and $DA=8$. Let $M$, $P$, $N$, $Q$ be the midpoints of sides $AB$, $BC$, $CD$, $DA$ respectively. Compute $MN^2-PQ^2$.

Answer: 13
user: Start by writing the integers $1, 2, 4, 6$ on the blackboard. At each step, write the smallest positive integer $n$ that satisfies both of the following properties on the board.
[*] $n$ is larger than any integer on the board currently.
[*] $n$ cannot be written as the sum of $2$ distinct integers on the board.
[/list]
Find the $100$-th integer that you write on the board. Recall that at the beginning, there are already $4$ integers on the board.

Answer: 388
user: 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
user: Let $\mathbb{N}$ denote the natural numbers. Compute the number of functions $f:\mathbb{N}\rightarrow \{0, 1, \dots, 16\}$ such that $$f(x+17)=f(x)\qquad \text{and} \qquad f(x^2)\equiv f(x)^2+15 \pmod {17}$$ for all integers $x\ge 1$.

Answer: 12066
user: There are $2017$ jars in a row on a table, initially empty. Each day, a nice man picks ten consecutive jars and deposits one coin in each of the ten jars. Later, Kelvin the Frog comes back to see that $N$ of the jars all contain the same positive integer number of coins (i.e. there is an integer $d>0$ such that $N$ of the jars have exactly $d$ coins). What is the maximum possible value of $N$?

Answer: 2014
user: Let $ABCD$ be a quadrilateral with an inscribed circle $\omega$ and let $P$ be the intersection of its diagonals $AC$ and $BD$. Let $R_1$, $R_2$, $R_3$, $R_4$ be the circumradii of triangles $APB$, $BPC$, $CPD$, $DPA$ respectively. If $R_1=31$ and $R_2=24$ and $R_3=12$, find $R_4$.

Answer: 19
user: A polynomial $P$ of degree $2015$ satisfies the equation $P(n)=\frac{1}{n^2}$ for $n=1, 2, \dots, 2016$. Find $\lfloor 2017P(2017)\rfloor$.

Answer: -9
user: For positive integers $a$ and $N$, let $r(a, N) \in \{0, 1, \dots, N - 1\}$ denote the remainder of $a$ when divided by $N$. Determine the number of positive integers $n \le 1000000$ for which
\[r(n, 1000) > r(n, 1001).\]

Answer: 499500
user: Jeffrey writes the numbers $1$ and $100000000 = 10^8$ on the blackboard. Every minute, if $x, y$ are on the board, Jeffery replaces them with
\[\frac{x + y}{2} \quad \text{and} \quad 2 \left(\frac{1}{x} + \frac{1}{y}\right)^{-1}.\]
After $2017$ minutes the two numbers are $a$ and $b$. Find $\min(a, b)$ to the nearest integer.

Answer: 10000
user: Find the number of ordered triples of nonnegative integers $(a, b, c)$ that satisfy
\[(ab + 1)(bc + 1)(ca + 1) = 84.\]

Answer: 12
user: Let $ABCD$ be a convex quadrilateral with $AC = 7$ and $BD = 17$. Let $M$, $P$, $N$, $Q$ be the midpoints of sides $AB$, $BC$, $CD$, $DA$ respectively. Compute $MN^2 + PQ^2$

The official problem statement does not have the final period.[/color]

Answer: 169
user: At a recent math contest, Evan was asked to find $2^{2016} \pmod{p}$ for a given prime number $p$ with $100 < p < 500$. Evan has forgotten what the prime $p$ was, but still remembers how he solved it:
[*] Evan first tried taking $2016$ modulo $p - 1$, but got a value $e$ larger than $100$.
[*] However, Evan noted that $e - \frac{1}{2}(p - 1) = 21$, and then realized the answer was $-2^{21} \pmod{p}$.
[/list]
What was the prime $p$?

Answer: 211
user: Let $m$ be a positive integer, and let $T$ denote the set of all subsets of $\{1, 2, \dots, m\}$. Call a subset $S$ of $T$ $\delta$-[I]good[/I] if for all $s_1, s_2\in S$, $s_1\neq s_2$, $|\Delta (s_1, s_2)|\ge \delta m$, where $\Delta$ denotes the symmetric difference (the symmetric difference of two sets is the set of elements that is in exactly one of the two sets). Find the largest possible integer $s$ such that there exists an integer $m$ and $ \frac{1024}{2047}$-good set of size $s$.

Answer: 2048
user: In convex quadrilateral $ABCD$ we have $AB=15$, $BC=16$, $CD=12$, $DA=25$, and $BD=20$. Let $M$ and $\gamma$ denote the circumcenter and circumcircle of $\triangle ABD$. Line $CB$ meets $\gamma$ again at $F$, line $AF$ meets $MC$ at $G$, and line $GD$ meets $\gamma$ again at $E$. Determine the area of pentagon $ABCDE$.

Answer: 396
user: There are two prime numbers $p$ so that $5p$ can be expressed in the form $\left\lfloor \dfrac{n^2}{5}\right\rfloor$ for some positive integer $n.$ What is the sum of these two prime numbers?

Answer: 52
user: For some real number $c,$ the graphs of the equation $y=|x-20|+|x+18|$ and the line $y=x+c$ intersect at exactly one point. What is $c$?

Answer: 18
user: Lily has a $300\times 300$ grid of squares. She now removes $100 \times 100$ squares from each of the four corners and colors each of the remaining $50000$ squares black and white. Given that no $2\times 2$ square is colored in a checkerboard pattern, find the maximum possible number of (unordered) pairs of squares such that one is black, one is white and the squares share an edge.

Answer: 49998
user: Consider a $2\times 3$ grid where each entry is either $0$, $1$, or $2$. For how many such grids is the sum of the numbers in every row and in every column a multiple of $3$? One valid grid is shown below:
$$\begin{bmatrix} 1 & 2 & 0 \\ 2 & 1 & 0 \end{bmatrix}$$

Answer: 9
user: Rachel has the number $1000$ in her hands. When she puts the number $x$ in her left pocket, the number changes to $x+1.$ When she puts the number $x$ in her right pocket, the number changes to $x^{-1}.$ Each minute, she flips a fair coin. If it lands heads, she puts the number into her left pocket, and if it lands tails, she puts it into her right pocket. She then takes the new number out of her pocket. If the expected value of the number in Rachel's hands after eight minutes is $E,$ compute $\left\lfloor\frac{E}{10}\right\rfloor.$

Answer: 13
user: Let $\{\omega_1,\omega_2,\cdots,\omega_{100}\}$ be the roots of $\frac{x^{101}-1}{x-1}$ (in some order). Consider the set $$S=\{\omega_1^1,\omega_2^2,\omega_3^3,\cdots,\omega_{100}^{100}\}.$$ Let $M$ be the maximum possible number of unique values in $S,$ and let $N$ be the minimum possible number of unique values in $S.$ Find $M-N.$

Answer: 98
user: A $4\times 4$ window is made out of $16$ square windowpanes. How many ways are there to stain each of the windowpanes, red, pink, or magenta, such that each windowpane is the same color as exactly two of its neighbors? Two different windowpanes are neighbors if they share a side.

Answer: 24
user: Distinct prime numbers $p,q,r$ satisfy the equation $$2pqr+50pq=7pqr+55pr=8pqr+12qr=A$$ for some positive integer $A.$ What is $A$?

Answer: 1980
user: Let $ABC$ be an equilateral triangle with side length $8.$ Let $X$ be on side $AB$ so that $AX=5$ and $Y$ be on side $AC$ so that $AY=3.$ Let $Z$ be on side $BC$ so that $AZ,BY,CX$ are concurrent. Let $ZX,ZY$ intersect the circumcircle of $AXY$ again at $P,Q$ respectively. Let $XQ$ and $YP$ intersect at $K.$ Compute $KX\cdot KQ.$

Answer: 304
user: How many ways can you fill a $3 \times 3$ square grid with nonnegative integers such that no [i]nonzero[/i] integer appears more than once in the same row or column and the sum of the numbers in every row and column equals 7?

Answer: 216
user: Convex hexagon $ABCDEF$ is drawn in the plane such that $ACDF$ and $ABDE$ are parallelograms with area 168. $AC$ and $BD$ intersect at $G$. Given that the area of $AGB$ is 10 more than the area of $CGB$, find the smallest possible area of hexagon $ABCDEF$.

Answer: 196
user: Let $N = 2^{\left(2^2\right)}$ and $x$ be a real number such that $N^{\left(N^N\right)} = 2^{(2^x)}$. Find $x$.

Answer: 66
user: Let $x$ and $y$ be positive real numbers. Define $a = 1 + \tfrac{x}{y}$ and $b = 1 + \tfrac{y}{x}$. If $a^2 + b^2 = 15$, compute $a^3 + b^3$.

Answer: 50
user: In triangle $ABC$, $AB = 13$, $BC = 14$, $CA = 15$. Squares $ABB_1A_2$, $BCC_1B_2$, $CAA_1B_2$ are constructed outside the triangle. Squares $A_1A_2A_3A_4$, $B_1B_2B_3B_4$ are constructed outside the hexagon $A_1A_2B_1B_2C_1C_2$. Squares $A_3B_4B_5A_6$, $B_3C_4C_5B_6$, $C_3A_4A_5C_6$ are constructed outside the hexagon $A_4A_3B_4B_3C_4C_3$. Find the area of the hexagon $A_5A_6B_5B_6C_5C_6$.

Answer: 19444
user: Tessa the hyper-ant has a 2019-dimensional hypercube. For a real number $k$, she calls a placement of nonzero real numbers on the $2^{2019}$ vertices of the hypercube [i]$k$-harmonic[/i] if for any vertex, the sum of all 2019 numbers that are edge-adjacent to this vertex is equal to $k$ times the number on this vertex. Let $S$ be the set of all possible values of $k$ such that there exists a $k$-harmonic placement. Find $\sum_{k \in S} |k|$.

Answer: 2040200
user: Let $\mathbb{N}$ be the set of positive integers, and let $f: \mathbb{N} \to \mathbb{N}$ be a function satisfying
[*] $f(1) = 1$,
[*] for $n \in \mathbb{N}$, $f(2n) = 2f(n)$ and $f(2n+1) = 2f(n) - 1$.
[/list]
Determine the sum of all positive integer solutions to $f(x) = 19$ that do not exceed 2019.

Answer: 1899
user: What is the smallest positive integer that cannot be written as the sum of two nonnegative palindromic integers? (An integer is [i]palindromic[/i] if the sequence of decimal digits are the same when read backwards.)

Answer: 21
user: For a positive integer $N$, we color the positive divisors of $N$ (including 1 and $N$) with four colors. A coloring is called [i]multichromatic[/i] if whenever $a$, $b$ and $\gcd(a, b)$ are pairwise distinct divisors of $N$, then they have pairwise distinct colors. What is the maximum possible number of multichromatic colorings a positive integer can have if it is not the power of any prime?

Answer: 192
user: For positive reals $p$ and $q$, define the [i]remainder[/i] when $p$ and $q$ as the smallest nonnegative real $r$ such that $\tfrac{p-r}{q}$ is an integer. For an ordered pair $(a, b)$ of positive integers, let $r_1$ and $r_2$ be the remainder when $a\sqrt{2} + b\sqrt{3}$ is divided by $\sqrt{2}$ and $\sqrt{3}$ respectively. Find the number of pairs $(a, b)$ such that $a, b \le 20$ and $r_1 + r_2 = \sqrt{2}$.

Answer: 16
user: How many distinct permutations of the letters in the word REDDER are there that do not contain a palindromic substring of length at least two? (A [i]substring[/i] is a continuous block of letters that is part of the string. A string is [i]palindromic[/i] if it is the same when read backwards.)

Answer: 6
user: A polynomial $P(x)$ is a \emph{base-$n$ polynomial} if it is of the form $a_dx^d+a_{d-1}x^{d-1}+\cdots + a_1x+a_0$, where each $a_i$ is an integer between $0$ and $n-1$ inclusive and $a_d>0$. Find the largest positive integer $n$ such that for any real number $c$, there exists at most one base-$n$ polynomial $P(x)$ for which $P(\sqrt 2+\sqrt 3)=c$.

Proposed by James Lin.[/i]

Answer: 9
user: How many positive integers at most $420$ leave different remainders when divided by each of $5$, $6$, and $7$?

Proposed by Milan Haiman.[/i]

Answer: 250
user: For positive integers $n$ and $k$, let $\mho(n,k)$ be the number of distinct prime divisors of $n$ that are at least $k$. For example, $\mho(90, 3)=2$, since the only prime factors of $90$ that are at least $3$ are $3$ and $5$. Find the closest integer to
\[\sum_{n=1}^\infty \sum_{k=1}^\infty \frac{\mho(n,k)}{3^{n+k-7}}.\]

Proposed by Daniel Zhu.[/i]

Answer: 167
user: Circles $\omega_a, \omega_b, \omega_c$ have centers $A, B, C$, respectively and are pairwise externally tangent at points $D, E, F$ (with $D\in BC, E\in CA, F\in AB$). Lines $BE$ and $CF$ meet at $T$. Given that $\omega_a$ has radius $341$, there exists a line $\ell$ tangent to all three circles, and there exists a circle of radius $49$ tangent to all three circles, compute the distance from $T$ to $\ell$.

Proposed by Andrew Gu.[/i]

Answer: 294
user: Let $P(x)=x^3+x^2-r^2x-2020$ be a polynomial with roots $r,s,t$. What is $P(1)$?

Proposed by James Lin.[/i]

Answer: -4038
user: Given an $8\times 8$ checkerboard with alternating white and black squares, how many ways are there to choose four black squares and four white squares so that no two of the eight chosen squares are in the same row or column?

Proposed by James Lin.[/i]

Answer: 20736
user: Find the sum of all positive integers $n$ for which 
\[\frac{15\cdot n!^2+1}{2n-3}\]
is an integer.

Proposed by Andrew Gu.[/i]

Answer: 90
user: Farmer James wishes to cover a circle with circumference $10\pi$ with six different types of colored arcs. Each type of arc has radius $5$, has length either $\pi$ or $2\pi$, and is colored either red, green, or blue. He has an unlimited number of each of the six arc types. He wishes to completely cover his circle without overlap, subject to the following conditions:

[*] Any two adjacent arcs are of different colors.
[*] Any three adjacent arcs where the middle arc has length $\pi$ are of three different colors. [/list]
Find the number of distinct ways Farmer James can cover his circle. Here, two coverings are equivalent if and only if they are rotations of one another. In particular, two colorings are considered distinct if they are reflections of one another, but not rotations of one another.

Proposed by James Lin.[/i]

Answer: 93
user: Let $ABCD$ and $AEF G$ be unit squares such that the area of their intersection is $\frac{20}{21}$ . Given that $\angle BAE < 45^o$, $\tan \angle BAE$ can be expressed as $\frac{a}{b}$ for relatively prime positive integers $a$ and $b$. Compute $100a + b$.

Answer: 4940
user: Random sequences $a_1, a_2, . . .$ and $b_1, b_2, . . .$ are chosen so that every element in each sequence is chosen independently and uniformly from the set $\{0, 1, 2, 3, . . . , 100\}$. Compute the expected value of the smallest nonnegative integer $s$ such that there exist positive integers $m$ and $n$ with $$s =\sum^m_{i=1} a_i =\sum^n_{j=1}b_j .$$

Answer: 2550
user: Parallel lines $\ell_1$, $\ell_2$, $\ell_3$, $\ell_4$ are evenly spaced in the plane, in that order. Square $ABCD$ has the property that $A$ lies on $\ell_1$ and $C$ lies on $\ell_4$. Let $P$ be a uniformly random point in the interior of $ABCD$ and let $Q$ be a uniformly random point on the perimeter of $ABCD$. Given that the probability that $P$ lies between $\ell_2$ and $\ell_3$ is $\frac{53}{100}$ , the probability that $Q$ lies between $\ell_2$ and $\ell_3$ can be expressed as $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. Compute $100a + b$.

Answer: 6100
user: Positive integers $a$, $b$, and $c$ are all powers of $k$ for some positive integer $k$. It is known that the equation $ax^2-bx+c=0$ has exactly one real solution $r$, and this value $r$ is less than $100$. Compute the maximum possible value of $r$.

Answer: 64
user: Let triangle $ABC$ be such that $AB = AC = 22$ and $BC = 11$. Point $D$ is chosen in the interior of the triangle such that $AD = 19$ and $\angle ABD + \angle ACD = 90^o$ . The value of $BD^2 + CD^2$ can be expressed as $\frac{a}{b}$ , where $a$ and $b$ are relatively prime positive integers. Compute $100a + b$.

Answer: 36104
user: Sets $A, B$, and $C$ satisfy $|A| = 92$, $|B| = 35$, $|C| = 63$, $|A\cap B| = 16$, $|A\cap C| = 51$, $|B\cap C| = 19$. Compute the number of possible values of$ |A \cap B \cap C|$.

Answer: 10
user: The numbers $1, 2, . . . , 10$ are randomly arranged in a circle. Let $p$ be the probability that for every positive integer $k < 10$, there exists an integer $k' > k$ such that there is at most one number between $k$ and $k'$ in the circle. If $p$ can be expressed as $\frac{a}{b}$ for relatively prime positive integers $a$ and $b$, compute $100a + b$.

Answer: 1390
user: Compute the number of ways to color $3$ cells in a $3\times 3$ grid so that no two colored cells share an edge.

Answer: 22
user: Let f be a function from $\{1, 2, . . . , 22\}$ to the positive integers such that $mn | f(m) + f(n)$ for all $m, n \in \{1, 2, . . . , 22\}$. If $d$ is the number of positive divisors of $f(20)$, compute the minimum possible value of $d$.

Answer: 2016
user: Given a positive integer $k$, let $||k||$ denote the absolute difference between $k$ and the nearest perfect square. For example, $||13||=3$ since the nearest perfect square to $13$ is $16$. Compute the smallest positive integer $n$ such that $\frac{||1|| + ||2|| + ...+ ||n||}{n}=100$.

Answer: 89800
user: Positive integers $a_1, a_2, ... , a_7, b_1, b_2, ... , b_7$ satisfy $2 \leq a_i \leq 166$ and $a_i^{b_i} \cong a_{i+1}^2$ (mod 167) for each $1 \leq i \leq 7$ (where $a_8=a_1$). Compute the minimum possible value of $b_1b_2 ... b_7(b_1 + b_2 + ...+ b_7)$.

Answer: 675
user: Let $(x_1, y_1)$, $(x_2, y_2)$, $(x_3, y_3)$, $(x_4, y_4)$, and $(x_5, y_5)$ be the vertices of a regular pentagon centered at $(0, 0)$. Compute the product of all positive integers k such that the equality $x_1^k+x_2^k+x_3^k+x_4^k+x_5^k=y_1^k+y_2^k+y_3^k+y_4^k+y_5^k$ must hold for all possible choices of the pentagon.

Answer: 1152
user: Five cards labeled $1, 3, 5, 7, 9$ are laid in a row in that order, forming the five-digit number $13579$ when read from left to right. A swap consists of picking two distinct cards, and then swapping them. After three swaps, the cards form a new five-digit number n when read from left to right. Compute the expected value of $n$.

Answer: 50208
user: Let $S = \{(x, y) \in Z^2 | 0 \le x \le 11, 0\le y \le 9\}$. Compute the number of sequences $(s_0, s_1, . . . , s_n)$ of elements in $S$ (for any positive integer $n \ge 2$) that satisfy the following conditions:
$\bullet$ $s_0 = (0, 0)$ and $s_1 = (1, 0)$,
$\bullet$ $s_0, s_1, . . . , s_n$ are distinct,
$\bullet$ for all integers $2 \le i \le n$, $s_i$ is obtained by rotating $s_{i-2}$ about $s_{i-1}$ by either $90^o$ or $180^o$ in the
clockwise direction.

Answer: 646634
user: Let $ABCD$ be a rectangle inscribed in circle $\Gamma$, and let $P$ be a point on minor arc $AB$ of $\Gamma$. Suppose that $P A \cdot P B = 2$, $P C \cdot P D = 18$, and $P B \cdot P C = 9$. The area of rectangle $ABCD$ can be expressed as $\frac{a\sqrt{b}}{c}$ , where $a$ and $c$ are relatively prime positive integers and $b$ is a squarefree positive integer. Compute $100a + 10b + c$.

Answer: 21055
user: Compute the number of positive integers that divide at least two of the integers in the set $\{1^1,2^2,3^3,4^4,5^5,6^6,7^7,8^8,9^9,10^{10}\}$.

Answer: 22
user: Let $(a_1, a_2, ..., a_8)$ be a permutation of $(1, 2, ... , 8)$. Find, with proof, the maximum possible number of elements of the set $$\{a_1, a_1 + a_2, ... , a_1 + a_2 + ... + a_8\}$$ that can be perfect squares.

Answer: 5
user: Let $E$ be an ellipse with foci $A$ and $B$. Suppose there exists a parabola $P$ such that
$\bullet$ $P$ passes through $A$ and $B$,
$\bullet$ the focus $F$ of $P$ lies on $E$,
$\bullet$ the orthocenter $H$ of $\vartriangle F AB$ lies on the directrix of $P$.
If the major and minor axes of $E$ have lengths $50$ and $14$, respectively, compute $AH^2 + BH^2$.

Answer: 2402