Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Point $P$ is located inside a square $ABCD$ of side length $10$. Let $O_1$, $O_2$, $O_3$, $O_4$ be the circumcenters of $P AB$, $P BC$, $P CD$, and $P DA$, respectively. Given that $P A+P B +P C +P D = 23\sqrt2$ and the area of $O_1O_2O_3O_4$ is $50$, the second largest of the lengths $O_1O_2$, $O_2O_3$, $O_3O_4$, $O_4O_1$ can be written as $\sqrt{\frac{a}{b}}$, where $a$ and $b$ are relatively prime positive integers. Compute $100a + b$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Michel starts with the string HMMT. An operation consists of either replacing an occurrence of H with HM, replacing an occurrence of MM with MOM, or replacing an occurrence of T with MT. For example, the two strings that can be reached after one operation are HMMMT and HMOMT. Compute the number of distinct strings Michel can obtain after exactly $10$ operations.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Suppose $P(x)$ is a monic polynomial of degree $2023$ such that $P(k) = k^{2023}P(1-\frac{1}{k})$ for every positive integer $1 \leq k \leq 2023$. Then $P(-1) = \frac{a}{b}$ where $a$ and $b$ are relatively prime integers. Compute the unique integer $0 \leq n < 2027$ such that $bn-a$ is divisible by the prime $2027$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Richard starts with the string $HHMMMMTT$. A move consists of replacing an instance of $HM$ with $MH$, replacing an instance of $MT$ with $TM$, or replacing an instance of $TH$ with $HT$. Compute the number of possible strings he can end up with after performing zero or more moves.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Suppose $E$, $I$, $L$, $V$ are (not necessarily distinct) nonzero digits in base ten for which
[list]
[*] the four-digit number $\underline{E}\ \underline{V}\ \underline{I}\ \underline{L}$ is divisible by $73$, and

[*] the four-digit number $\underline{V}\ \underline{I}\ \underline{L}\ \underline{E}$ is divisible by $74$.
[/list]
Compute the four-digit number $\underline{L}\ \underline{I}\ \underline{V}\ \underline{E}$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Philena and Nathan are playing a game. First, Nathan secretly chooses an ordered pair $(x, y)$ of positive integers such that $x \leq 20$ and $y \leq 23$. (Philena knows that Nathan’s pair must satisfy $x \leq 20$ and $y \leq 23$.) The game then proceeds in rounds; in every round, Philena chooses an ordered pair $(a, b)$ of positive integers and tells it to Nathan; Nathan says YES if $x \leq a$ and $y \leq b$, and NO otherwise. Find, with proof, the smallest positive integer $N$ for which Philena has a strategy that guarantees she can be certain of Nathan’s pair after at most $N$ rounds.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

For any positive integers $a$ and $b$ with $b > 1$, let $s_b(a)$ be the sum of the digits of $a$ when it is written in base $b$. Suppose $n$ is a positive integer such that $$\sum^{\lfloor \log_{23} n\rfloor}_{i=1}
s_{20} \left( \left\lfloor \frac{n}{23^i} \right\rfloor \right)= 103 \,\,\, \text{and} \,\,\,
\sum^{\lfloor \log_{20} n\rfloor}_{i=1} s_{23} \left( \left\lfloor \frac{n}{20^i} \right\rfloor \right)= 115$$
Compute $s_{20}(n) - s_{23}(n)$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Let $S$ be the set of ordered pairs $(a, b)$ of positive integers such that $\gcd(a, b) = 1$. Compute \[ \sum_{(a, b) \in S} \left\lfloor \frac{300}{2a+3b} \right\rfloor. \]

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Elbert and Yaiza each draw $10$ cards from a $20$-card deck with cards numbered $1,2,3,\dots,20$. Then, starting with the player with the card numbered $1$, the players take turns placing down the lowest-numbered card from their hand that is greater than every card previously placed. When a player cannot place a card, they lose and the game ends.

Given that Yaiza lost and $5$ cards were placed in total, compute the number of ways the cards could have been initially distributed. (The order of cards in a player’s hand does not matter.)

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Each cell of a $3 \times 3$ grid is labeled with a digit in the set $\{1, 2, 3, 4, 5\}$ Then, the maximum entry in each row and each column is recorded. Compute the number of labelings for which every digit from $1$ to $5$ is recorded at least once.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Let $a_1, a_2, \ldots, a_{100}$ be positive integers, satisfying $$\frac{a_1^2+a_2^2+\ldots+a_{100}^2} {a_1+a_2+\ldots+a_{100}}=100.$$ What is the maximal value of $a_1$?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Sally the snail sits on the $3 \times  24$ lattice of points $(i, j)$ for all $1 \le  i  \le  3$ and $1 \le  j \le 24$. She wants to visit every point in the lattice exactly once. In a move, Sally can move to a point in the lattice exactly one unit away. Given that Sally starts at $(2, 1)$, compute the number of possible paths Sally can take.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Let $P(n)=(n-1^3)(n-2^3)\ldots (n-40^3)$ for positive integers $n$. Let $d$ be the largest positive integer such that $d \mid P(n)$ for any $n>2023$. If $d$ is product of $m$ not necessarily distinct primes, find $m$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

A [i]lame king[/i] is a chess piece that can move from a cell to any cell that shares at least one vertex with it, except for the cells in the same column as the current cell.

A lame king is placed in the top-left cell of a $7\times 7$ grid. Compute the maximum number of cells it can visit without visiting the same cell twice (including its starting cell).

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Compute the number of triples $(f,g,h)$ of permutations on $\{1,2,3,4,5\}$ such that \begin{align*}
 & f(g(h(x))) = h(g(f(x))) = g(x)  \\
 & g(h(f(x))) = f(h(g(x))) = h(x), \text{ and }  \\
 & h(f(g(x))) = g(f(h(x))) = f(x),  \\
\end{align*}  for all $x\in \{1,2,3,4,5\}$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

A polynomial $f(x)$ with integer coefficients is called $\textit{splitty}$ if and only if for every prime $p$, there exists polynomials $g_p, h_p$ with integer coefficients with degrees strictly smaller than the the degree of $f$, such that all coefficients of $f-g_ph_p$ are divisible by $p$. Compute the sum of all positive integers $n \leq 100$ such that $x^4+16x^2+n$ is $\textit{splitty}$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Nine distinct positive integers summing to $74$ are put into a $3 \times 3$ grid. Simultaneously, the number in each cell is replaced with the sum of the numbers in its adjacent cells. (Two cells are adjacent if they share an edge.) After this, exactly four of the numbers in the grid are $23$. Determine, with proof, all possible numbers that could have been originally in the center of the grid.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Compute the sum of all positive integers $50 \leq n \leq 100$ such that $2n+3 \nmid 2^{n!}-1$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Let $\Omega$ and $\omega$ be circles with radii $123$ and $61$, respectively, such that the center of $\Omega$ lies on $\omega$. A chord of $\Omega$ is cut by $\omega$ into three segments, whose lengths are in the ratio $1 : 2 : 3$ in that order. Given that this chord is not a diameter of $\Omega$, compute the length of this chord.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Let $ABC$ be a triangle with $\angle BAC = 90^o$. Let $D$, $E$, and $F$ be the feet of altitude, angle bisector, and median from $A$ to $BC$, respectively. If $DE = 3$ and $EF = 5$, compute the length of $BC$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

What is the largest integer which cannot be expressed as $2008x + 2009y + 2010z$ for some positive integers $x, y$, and $z$?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

In four-dimensional space, the $24$-cell of sidelength $\sqrt{2}$ is the convex hull of (smallest convex set containing) the $24$ points $(\pm 1, \pm 1, 0, 0)$ and its permutations. Find the four-dimensional volume of this region.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

If $f(x) = x^{x^{x^x}}$ , find the last two digits of $f(17) + f(18) + f(19) + f(20)$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Find the sum of the values of $x$ for which $\binom{x}{0}-\binom{x}{1}+\binom{x}{2}-...+\binom{x}{2008}=0$

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Find the number of ordered pairs $(a, b)$ of positive integers that are solutions of the following equation: \[a^2 + b^2 = ab(a+b).\]

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Lines $l$ and $m$ are perpendicular. Line $l$ partitions a convex polygon into two parts of equal area, and partitions the projection of the polygon onto $m$ into two line segments of length $a$ and $b$ respectively. Determine the maximum value of $\left\lfloor \frac{1000a}{b} \right\rfloor$. (The floor notation $\lfloor x \rfloor$ denotes largest integer not exceeding $x$)

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Consider $\triangle ABC$ and a point $M$ in its interior so that $\angle MAB = 10^\circ$, $\angle MBA = 20^\circ$, $\angle MCA = 30^\circ$ and $\angle MAC = 40^\circ$. What is $\angle MBC$?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

The real numbers $x$, $y$, $z$, and $t$ satisfy the following equation:
\[2x^2 + 4xy + 3y^2 - 2xz -2 yz + z^2 + 1 = t + \sqrt{y + z - t} \]
Find 100 times the maximum possible value for $t$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

In a rectangular plot of land, a man walks in a very peculiar fashion. Labeling the corners $ABCD$, he starts at $A$ and walks to $C$. Then, he walks to the midpoint of side $AD$, say $A_1$. Then, he walks to the midpoint of side $CD$ say $C_1$, and then the midpoint of $A_1D$ which is $A_2$. He continues in this fashion, indefinitely. The total length of his path if $AB=5$ and $BC=12$ is of the form $a + b\sqrt{c}$. Find $\displaystyle\frac{abc}{4}$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

In regular hexagon $ABCDEF$, $AC$, $CE$ are two diagonals. Points $M$, $N$ are on $AC$, $CE$ respectively and satisfy $AC: AM = CE: CN = r$. Suppose $B, M, N$ are collinear, find $100r^2$.
[asy]
size(120); defaultpen(linewidth(0.7)+fontsize(10));
pair D2(pair P) {
dot(P,linewidth(3)); return P;
}
pair A=dir(0), B=dir(60), C=dir(120), D=dir(180), E=dir(240), F=dir(300), N=(4*E+C)/5,M=intersectionpoints(A--C,B--N)[0];
draw(A--B--C--D--E--F--cycle); draw(A--C--E); draw(B--N);
label("$A$",D2(A),plain.E);
label("$B$",D2(B),NE);
label("$C$",D2(C),NW);
label("$D$",D2(D),W);
label("$E$",D2(E),SW);
label("$F$",D2(F),SE);
label("$M$",D2(M),(0,-1.5));
label("$N$",D2(N),SE);
[/asy]

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

All the diagonals of a regular decagon are drawn.  A regular decagon satisfies the property that if three diagonals concur, then one of the three diagonals is a diameter of the circumcircle of the decagon.  How many distinct intersection points of diagonals are in the interior of the decagon?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

The expression $\sin2^\circ\sin4^\circ\sin6^\circ\cdots\sin90^\circ$ is equal to $p\sqrt{5}/2^{50}$, where $p$ is an integer. Find $p$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Triangle $ABC$ has $AB = 4$, $AC = 5$, and $BC = 6$.  An angle bisector is drawn from angle $A$, and meets $BC$ at $M$.  What is the nearest integer to $100 \frac{AM}{CM}$?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

On rectangular coordinates, point $A = (1,2)$, $B = (3,4)$. $P = (a, 0)$ is on $x$-axis. Given that $P$ is chosen such that $AP + PB$ is minimized, compute $60a$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Let $f(x)=3x^3-5x^2+2x-6$. If the roots of $f$ are given by $\alpha$, $\beta$, and $\gamma$, find
\[
\left(\frac{1}{\alpha-2}\right)^2+\left(\frac{1}{\beta-2}\right)^2+\left(\frac{1}{\gamma-2}\right)^2.
\]

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Let $\sigma$ be a random permutation of $\{0, 1, \ldots, 6\}$. Let $L(\sigma)$ be the length of the longest initial monotonic consecutive subsequence of $\sigma$ not containing $0$; for example, \[L(\underline{2,3,4},6,5,1,0) = 3,\ L(\underline{3,2},4,5,6,1,0) = 2,\ L(0,1,2,3,4,5,6) = 0.\] If the expected value of $L(\sigma)$ can be written as $\frac mn$, where $m$ and $n$ are relatively prime positive integers, then find $m + n$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Let $d(n)$ denote the number of divisors of $n$ (including itself). You are given that \[\sum_{n=1}^{\infty} \frac{1}{n^2} = \frac{\pi^2}{6}.\]  Find $p(6)$, where $p(x)$ is the unique polynomial with rational coefficients satisfying \[p(\pi) = \sum_{n=1}^{\infty} \frac{d(n)}{n^2}.\]

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Let $\ell_1$ and $\ell_2$ be two parallel lines, a distance of 15 apart. Points $A$ and $B$ lie on $\ell_1$ while points $C$  and $D$ lie on $\ell_2$ such that $\angle BAC = 30^\circ$ and $\angle ABD = 60^\circ$. The minimum value of $AD + BC$ is $a\sqrt b$, where $a$ and $b$ are integers and $b$ is squarefree. Find $a + b$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Let $f(x) = x^3-7x^2+16x-10$. As $x$ ranges over all integers, find the sum of distinct prime values taken on by $f(x)$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Let $a$ and $b$ be positive integers such that $a + bz = x^3 + y^4$ has no solutions for any integers $x, y, z$, with $b$ as small as possible, and $a$ as small as possible for the minimum $b$. Find $ab$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

A polynomial $p$ can be written as
\begin{align*}
p(x) = x^6+3x^5-3x^4+ax^3+bx^2+cx+d.
\end{align*}
Given that all roots of $p(x)$ are equal to either $m$ or $n$ where $m$ and $n$ are integers, compute $p(2)$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

At the start of the PUMaC opening ceremony in McCosh auditorium, the speaker counts $90$ people in the audience. Every minute afterwards, either one person enters the auditorium (due to waking up late) or leaves (in order to take a dreadful math contest). The speaker observes that in this time, exactly $100$ people enter the auditorium, $100$ leave, and $100$ was the largest audience size he saw. Find the largest integer $m$ such that $2^m$ divides the number of different possible sequences of entries and exits given the above information.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

For every integer $n$ from $0$ to $6$, we have $3$ identical weights with weight $2^n$. How many ways are there to form a total weight of 263 grams using only these given weights?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

What is the sum of all primes $p$ such that $7^p - 6^p + 2$ is divisible by 43?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Four circles are situated in the plane so that each is tangent to the other three. If three of the radii are $5$, $5$, and $8$, the largest possible radius of the fourth circle is $a/b$, where $a$ and $b$ are positive integers and gcd$(a, b) = 1$. Find $a + b$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

For how many ordered triplets of three positive integers is it true that their product is four more than twice their sum?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Three circles, with radii of $1, 1$, and $2$, are externally tangent to each other. The minimum possible area of a quadrilateral that contains and is tangent to all three circles can be written as $a + b\sqrt{c}$ where $c$ is not divisible by any perfect square larger than $1$. Find $a + b + c$

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Considering all numbers of the form $n = \lfloor \frac{k^3}{2012} \rfloor$, where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$, and $k$ ranges from $1$ to $2012$, how many of these $n$’s are distinct?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Two circles centered at $O$ and $P$ have radii of length $5$ and $6$ respectively. Circle $O$ passes through point $P$. Let the intersection points of circles $O$ and $P$ be $M$ and $N$. The area of triangle $\vartriangle MNP$ can be written in simplest form as $a/b$. Find $a + b$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

When some number $a^2$ is written in base $b$, the result is $144_b$. $a$ and $b$ also happen to be integer side lengths of a right triangle. If $a$ and $b$ are both less than $20$, find the sum of all possible values of $a$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Two white pentagonal pyramids, with side lengths all the same, are glued to each other at their regular pentagon bases. Some of the resulting $10$ faces are colored black. How many rotationally distinguishable colorings may result?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Compute the smallest positive integer $a$ for which $$\sqrt{a +\sqrt{a +...}} - \frac{1}{a +\frac{1}{a+...}}> 7$$

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Let f be a polynomial of degree $3$ with integer coefficients such that $f(0) = 3$ and $f(1) = 11$.
If f has exactly $2$ integer roots, how many such polynomials $f$ exist?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Call a positive integer $x$ a leader if there exists a positive integer $n$ such that the decimal representation of $x^n$ starts ([u]not ends[/u])  with $2012$. For example, $586$ is a leader since $586^3 =201230056$. How many leaders are there in the set $\{1, 2, 3, ..., 2012\}$?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

If the probability that the sum of three distinct integers between $16$ and $30$ (inclusive) is even can be written as $\frac{m}{n}$ , where $m$ and $n$ are relatively prime positive integers, find $m + n$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

 How many (possibly empty) sets of lattice points $\{P_1, P_2, ... , P_M\}$, where each point $P_i =(x_i, y_i)$ for $x_i
, y_i \in \{0, 1, 2, 3, 4, 5, 6\}$, satisfy that the slope of the line $P_iP_j$ is positive for each $1 \le i < j \le M$ ? An infinite slope, e.g. $P_i$ is vertically above $P_j$ , does not count as positive.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Let $M$ be the smallest positive multiple of $2012$ that has $2012$ divisors.
Suppose $M$ can be written as  $\Pi_{k=1}^{n}p_k^{a_k}$ where the $p_k$’s are distinct primes and the $a_k$’s are positive integers.
Find $\Sigma_{k=1}^{n}(p_k + a_k)$

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Let $\vartriangle ABC$ be a triangle with $\angle BAC = 45^o, \angle BCA = 30^o$, and $AB = 1$. Point $D$ lies on segment $AC$ such that $AB = BD$. Find the square of the length of the common external tangent to the circumcircles of triangles $\vartriangle BDC$ and $\vartriangle ABC$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Compute $\Sigma_{n=1}^{\infty}\frac{n + 1}{n^2(n + 2)^2}$ .
Your answer in simplest form can be written as $a/b$, where $a, b$ are relatively-prime positive integers. Find $a + b$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Let A be a regular $12$-sided polygon. A new $12$-gon B is constructed by connecting the midpoints of the sides of A. The ratio of the area of B to the area of A can be written in simplest form as $(a +\sqrt{b})/c$, where $a, b, c$ are integers. Find $a + b + c$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Let $f_n(x) = n + x^2$. Evaluate the product $gcd\{f_{2001}(2002), f_{2001}(2003)\} \times gcd\{f_{2011}(2012), f_{2011}(2013)\}  \times gcd\{f_{2021}(2022), f_{2021}(2023)\}$, where $gcd\{x, y\}$ is the greatest common divisor of $x$ and $y$

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

What is the smallest natural number $n$ greater than $2012$ such that the polynomial $f(x) =(x^6 + x^4)^n - x^{4n} - x^6$ is divisible by $g(x) = x^4 + x^2 + 1$?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Let an be a sequence such that $a_0 = 0$ and:
$a_{3n+1} = a_{3n} + 1 = a_n + 1$
$a_{3n+2} = a_{3n} + 2 = a_n + 2$
for all natural numbers $n$. How many $n$ less than $2012$ have the property that $a_n = 7$?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

During chemistry labs, we oftentimes fold a disk-shaped filter paper twice, and then open up a flap of the quartercircle to form a cone shape, as in the diagram. What is the angle $\theta$, in degrees, of the bottom of the cone when we look at it from the side?
[img]https://cdn.artofproblemsolving.com/attachments/d/2/f8e3a7afb606dfd6fad277f547b116566a4a91.png[/img]

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Suppose $a,b,c>0$ are integers such that \[abc-bc-ac-ab+a+b+c=2013.\]  Find the number of possibilities for the ordered triple $(a,b,c)$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

(Following question 1) Now instead consider an infinite strip of squares, labeled with the integers $0, 1, 2, \ldots$ in that order. You start at the square labeled $0$. You want to end up at the square labeled $3$. In how many ways can this be done in exactly $15$ moves?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Let $d$ be the greatest common divisor of $2^{30^{10}}-2$ and $2^{30^{45}}-2$.  Find the remainder when $d$ is divided by $2013$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

What is the largest positive integer that cannot be expressed as a sum of non-negative integer multiple of $13$, $17$, and $23$?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Suppose the function $\psi$ satisfies $\psi(1)=\sqrt{2+\sqrt{2+\sqrt2}}$ and $\psi(3x)+3\psi(x)=\psi(x)^3$ for all real $x$.  Determine the greatest integer less than $\textstyle\prod_{n=1}^{100}\psi(3^n)$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Circle $w$ with center $O$ meets circle $\Gamma$ at $X,Y,$ and $O$ is on $\Gamma$.  Point $Z\in\Gamma$ lies outside $w$ such that $XZ=11$, $OZ=15$, and $YZ=13$.  If the radius of circle $w$ is $r$, find $r^2$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Mereduth has many red boxes and many blue boxes. Coloon has placed five green boxes in a row on the ground, and Mereduth wants to arrange some number of her boxes on top of his row. Assume that each box must be placed so that it straddles two lower boxes. Including the one with no boxes, how many arrangements can Mereduth make?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

You are eating at a fancy restaurant with a person you wish to impress. For some reason, you think that eating at least one spicy course and one meat-filled course will impress the person. The meal is five courses, with four options for each course. Each course has one option that is spicy and meat-filled, one option that is just spicy, one that is just meat-filled, and one that is neither spicy nor meat-filled. How many possible meals can you have?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

An integer sequence $a_1,a_2,\ldots,a_n$ has $a_1=0$, $a_n\leq 10$ and $a_{i+1}-a_i\geq 2$ for $1\leq i<n$.  How many possibilities are there for this sequence?  The sequence may be of any length.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

If $x,y$ are real, then the $\textit{absolute value}$ of the complex number $z=x+yi$ is \[|z|=\sqrt{x^2+y^2}.\]  Find the number of polynomials $f(t)=A_0+A_1t+A_2t^2+A_3t^3+t^4$ such that $A_0,\ldots,A_3$ are integers and all roots of $f$ in the complex plane have absolute value $\leq 1$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Three chords of a sphere, each having length $5,6,7$, intersect at a single point inside the sphere and are pairwise perpendicular.  For $R$ the maximum possible radius of this sphere, find $R^2$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Consider a random permutation of the set $\{1, 2, . . . , 2015\}$. In other words, for each $1 \le i \le 2015$, $i$ is sent to the element $a_i$ where $a_i \in \{1, 2, . . . , 2015\}$ and if $i \neq j$, then $a_i \neq a_j$. What is the expected number of ordered pairs $(a_i, a_j )$ with $i - j > 155$ and $a_i - a_j > 266$?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Jonathan has a magical coin machine which takes coins in amounts of $7, 8$, and $9$. If he puts in $7$ coins, he gets $3$ coins back; if he puts in $8$, he gets $11$ back; and if he puts in $9$, he gets $4$ back. The coin machine does not allow two entries of the same amount to happen consecutively. Starting with $15$ coins, what is the minimum number of entries he can make to end up with $4$ coins?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Define the sequence $a_i$ as follows: $a_1 = 1, a_2 = 2015$, and $a_n = \frac{na_{n-1}^2}{a_{n-1}+na_{n-2}}$ for $n > 2$. What is the least $k$ such that $a_k < a_{k-1}$?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Find the largest $r$ such that $4$ balls each of radius $r$ can be packed into a regular tetrahedron with side length $1$. In a packing, each ball lies outside every other ball, and every ball lies inside the boundaries of the tetrahedron. If $r$ can be expressed in the form $\frac{\sqrt{a}+b}{c}$ where $a, b, c$ are integers such that $\gcd(b, c) = 1$, what is $a + b + c$?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

What is the largest positive integer $n$ less than $10, 000$ such that in base $4$, $n$ and $3n$ have the same number of digits; in base $8$, $n$ and $7n$ have the same number of digits; and in base $16$, $n$ and $15n$ have the same number of digits? Express your answer in base $10$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

We define the function $f(x,y)=x^3+(y-4)x^2+(y^2-4y+4)x+(y^3-4y^2+4y)$. Then choose any distinct $a, b, c \in \mathbb{R}$ such that the following holds: $f(a,b)=f(b,c)=f(c,a)$. Over all such choices of $a, b, c$, what is the maximum value achieved by
\[\min(a^4 - 4a^3 + 4a^2, b^4 - 4b^3 + 4b^2, c^4 - 4c^3 + 4c^2)?\]

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Triangle $ABC$ has $\overline{AB} = 5, \overline{BC} = 4, \overline{CA} = 6$. Points $D$ and $E$ are on sides $AB$ and $AC$, respectively, such that $\overline{AD} = \overline{AE} = \overline{BC}$. Let $CD$ and $BE$ intersect at $F$ and let $AF$ and $DE$ intersect at $G$. The length of $FG$ can be expressed in the form $\tfrac{a\sqrt{b}}{c}$ in simplified form. What is $a + b + c$?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

We define the ridiculous numbers recursively as follows:
[list=a]
[*]1 is a ridiculous number.
[*]If $a$ is a ridiculous number, then $\sqrt{a}$ and $1+\sqrt{a}$ are also ridiculous numbers.
[/list]
A closed interval $I$ is ``boring'' if
[list]
[*]$I$ contains no ridiculous numbers, and
[*]There exists an interval $[b,c]$ containing $I$ for which $b$ and $c$ are both ridiculous numbers.
[/list]
The smallest non-negative $l$ such that there does not exist a boring interval with length $l$ can be represented in the form $\dfrac{a + b\sqrt{c}}{d}$ where $a, b, c, d$ are integers, $\gcd(a, b, d) = 1$, and no integer square greater than 1 divides $c$. What is $a + b + c + d$?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Princeton’s Math Club recently bought a stock for $\$2$ and sold it for $\$9$ thirteen days later. Given that the stock either increases or decreases by $\$1$ every day and never reached $\$0$, in how many possible ways could the stock have changed during those thirteen days?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Given a rational number $r$ that, when expressed in base-$10$, is a repeating, non-terminating decimal, we define $f(r)$ to be the number of digits in the decimal representation of $r$ that are after the decimal point but before the repeating part of $r$. For example, $f(1.2\overline{7}) = 0$ and $f(0.35\overline{2}) = 2$. What is the smallest positive integer $n$ such that $\tfrac{1}{n}, \tfrac{2}{n}$, and $\tfrac{4}{n}$ are non-terminating decimals, where $f\left( \tfrac{1}{n} \right) = 3, f\left( \tfrac{2}{n} \right) = 3$, and $f\left( \tfrac{4}{n} \right) = 2$?.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Ryan is messing with Brice’s coin. He weights the coin such that it comes up on one side twice as frequently as the other, and he chooses whether to weight heads or tails more with equal probability. Brice flips his modified coin twice and it lands up heads both times. The probability that the coin lands up heads on the next flip can be expressed in the form $\tfrac{p}{q}$ for positive integers $p, q$ satisfying $\gcd(p, q) = 1$, what is $p + q$?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Let $I$ be the incenter of a triangle $ABC$ with $AB = 20$, $BC = 15$, and $BI = 12$. Let $CI$ intersect the circumcircle $\omega_1$ of $ABC$ at $D \neq C $. Alice draws a line $l$ through $D$ that intersects $\omega_1$ on the minor arc $AC$ at $X$ and the circumcircle $\omega_2$ of $AIC$ at $Y$ outside $\omega_1$. She notices that she can construct a right triangle with side lengths $ID$, $DX$, and $XY$. Determine, with proof, the length of $IY$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Let $f$ be a function which takes in $0, 1, 2$ and returns $0, 1, $ or $2$. The values need not be distinct: for instance we could have $f(0) = 1, f(1) = 1, f(2) = 2$. How many such functions are there which satisfy \[f(2) + f(f(0)) + f(f(f(1))) = 5?\]

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Let $n = 2^{2015} - 1$. For any integer $1 \le x < n$, let \[f_n(x) = \sum\limits_p s_p(n-x) + s_p(x) - s_p(n),\] where $s_q(k)$ denotes the sum of the digits of $k$ when written in base $q$ and the summation is over all primes $p$. Let $N$ be the number of values of $x$ such that $4 | f_n(x)$. What is the remainder when $N$ is divided by $1000?$

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Let $P, A, B, C$ be points on circle $O$ such that $C$ does not lie on arc $\widehat{BAP}$, $\overline{P A} = 21, \overline{P B} = 56, \overline{P C} = 35$ and $m \angle BP C = 60^\circ$. Now choose point $D$ on the circle such that $C$ does not lie on arc $\widehat{BDP}$ and $\overline{BD} = 39$. What is $AD$?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Let $S$ be the set of ordered integer pairs $(x, y)$ such that $0 < x < y < 42$ and there exists some integer $n$ such that $x^6-y^6 \mid n^2+2015^2$. What is the sum $\sum_{(x_i, y_i) \in S}x_iy_i$?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Marie is painting a $4 \times 4$ grid of identical square windows. Initially, they are all orange but she wants to paint $4$ of them black. How many ways can she do this up to rotation and reflection?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

How many ordered pairs $(a, b)$ of positive integers with $1 \le a, b \le 10$ are there such that in the geometric sequence whose first term is $a$ and whose second term is $b$, the third term is an integer?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Every day, Heesu talks to Sally with some probability $p$. One day, after not talking to Sally the previous day, Heesu resolves to ask Sally out on a date. From now on, each day, if Heesu has talked to Sally each of the past four days, then Heesu will ask Sally out on a date. Heesu’s friend remarked that at this rate, it would take Heesu an expected $2800$ days to finally ask Sally out. Suppose $p=\tfrac{m}{n}$, where $\gcd(m, n) = 1$ and $m, n > 0$. What is $m + n$?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Given a positive integer $k$, let $f(k)$ be the sum of the $k$-th powers of the primitive roots of $73$. For how many positive integers $k < 2015$ is $f(k)$ divisible by $73?$

[i]Note: A primitive root of $r$ of a prime $p$ is an integer $1 \le r < p$ such that the smallest positive integer $k$ such that $r^k \equiv 1 \pmod{p}$ is $k = p-1$.[/i]

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Let $a\%b$ denote the remainder when $a$ is divided by $b$. Find $\Sigma_{i=1}^{100}(100\%i)$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Suppose $z^{3}=2+2i$, where $i=\sqrt{-1}$. The product of all possible values of the real part of $z$ can be written in the form $\frac{p}{q}$ where $p$ and $q$ are relatively prime positive integers. Find $p+q$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

The sequence of positive integers $a_1, a_2, \dots$ has the property that $\gcd(a_m, a_n) > 1$ if and only if $|m - n| = 1$. Find the sum of the four smallest possible values of $a_2$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

The area of parallelogram $ABCD$ is $51\sqrt{55}$ and $\angle{DAC}$ is a right angle. If the side lengths of the parallelogram are integers, what is the perimeter of the parallelogram?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Tristan is eating his favorite cereal, Tiger Crunch, which has marshmallows of two colors, black and orange. He eats the marshmallows by randomly choosing from those remaining one at a time, and he starts out with $17$ orange and $5$ black marshmallows. If $\frac{p}{q}$ is the expected number of marshmallows remaining the instant that there is only one color left, and $p$ and $q$ are relatively prime positive integers, find $p + q$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

For a sequence of $10$ coin flips, each pair of consecutive flips and count the number of “Heads-Heads”, “Heads-Tails”, “Tails-Heads”, and “Tails-Tails” sequences is recorded. These four numbers are then multiplied to get the [i]Tiger number[/i] of the sequence of flips. How many such sequences have a [i]Tiger number [/i] of $24$?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

A right regular hexagonal prism has bases $ABCDEF$, $A'B'C'D'E'F'$ and edges $AA'$, $BB'$, $CC'$, $DD'$, $EE'$, $FF'$, each of which is perpendicular to both hexagons. The height of the prism is $5$ and the side length of the hexagons is $6$. The plane $P$ passes through points $A$, $C'$, and $E$. The area of the portion of $P$ contained in the prism can be expressed as $m\sqrt{n}$, where $n$ is not divisible by the square of any prime. Find $m+n$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Define the sequences $a_n$ and $b_n$ as follows: $a_1 = 2017$ and $b_1 = 1$.
For $n > 1$, if there is a greatest integer $k > 1$ such that $a_n$ is a perfect $k$th power, then $a_{n+1} =\sqrt[k]{a_n}$, otherwise  $a_{n+1} = a_n + b_n$. If $a_{n+1} \ge a_n$ then $b_{n+1} = b_n$, otherwise $b_{n+1} = b_n + 1$. Find $a_{2017}$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Together, Kenneth and Ellen pick a real number $a$. Kenneth subtracts $a$ from every thousandth root of unity (that is, the thousand complex numbers $\omega$ for which $\omega^{1000}=1$) then inverts each, then sums the results. Ellen inverts every thousandth root of unity, then subtracts $a$ from each, and then sums the results. They are surprised to find that they actually got the same answer! How many possible values of $a$ are there?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Zack keeps cutting the interval $[0, 1]$ of the number line, each time cutting at a uniformly random point in the interval, until the interval is cut into pieces, none of which have length greater than $\frac35$ . The expected number of cuts that Zack makes can be written as $\frac{p}{q}$ for $p$ and $q$ relatively prime positive integers. Find $p + q$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Let
\[ \frac{p}{q} = \frac{2017}{2-\frac{1}{3-\frac{2}{2-\frac{1}{2-\frac{1}{3-\frac{2}{2-\frac{1}{2-\frac{1}{2-\frac{1}{3-\frac{2}{2-\frac{1}{2-\cdots}}}}}}}}}}}\]
where $p$ and $q$ are relatively prime positive integers. Find $p+q$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Find the minimum value attained by $\sum_{m=1}^{100} \gcd(M - m, 400)$ for $M$ an integer in the range $[1746, 2017]$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

There is a box containing $100$ balls, each of which is either orange or black. The box is equally likely to contain any number of black balls between $0$ and $100$, inclusive. A random black ball rolls out of the box. The probability that the next ball to roll out of the box is also black can be written in the form $\tfrac{p}{q}$ where $p$ and $q$ are relatively prime positive integers. Find $p + q$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Let $X =\{1, 2, ... , 2017\}$. Let $k$ be a positive integer. Given any $r$ such that $1\le  r \le k$, there exist $k$ subsets of $X$ such that the union of any $ r$ of them is equal to $X$ , but the union of any fewer than $r$ of them is not equal to $X$ . Find, with proof, the greatest possible value for $k$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Call an ordered triple $(a, b, c)$ of integers feral if $b -a, c - a$ and $c - b$ are all prime.
Find the number of feral triples where $1  \le a < b < c \le 20$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

 Triangle $ABC$ is so that $AB = 15$, $BC = 22$, and $AC = 20$. Let $D, E, F$ lie on $BC$, $AC$, and $AB$, respectively, so $AD$, $BE$, $CF$ all contain a point $K$. Let $L$ be the second intersection of the circumcircles of $BFK$ and $CEK$. Suppose that $\frac{AK}{KD} = \frac{11}{7}$ , and $BD = 6$. If $KL^2 =\frac{a}{b}$, where $a, b$ are relatively prime integers, find $a + b$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Billy the baker makes a bunch of loaves of bread every day, and sells them in bundles of size $1, 2$, or $3$. On one particular day, there are $375$ orders, $125$ for each bundle type. As such, Billy goes ahead and makes just enough loaves of bread to meet all the orders. Whenever Billy makes loaves, some get burned, and are not sellable. For nonnegative i less than or equal to the total number of loaves, the probability that exactly i loaves are sellable to customers is inversely proportional to $2^i$ (otherwise, it’s $0$). Once he makes the loaves, he distributes out all of the sellable loaves of bread to some subset of these customers (each of whom will only accept their desired bundle of bread), without worrying about the order in which he gives them out. If the expected number of ways Billy can distribute the bread is of the form $\frac{a^b}{2^c-1}$, find $a + b + c$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Suppose that f is a function $f : R_{\ge 0} \to R$ so that for all $x, y \in R_{\ge 0}$ (nonnegative reals) we have that $$f(x)+f(y) = f(x+y+xy)+f(x)f(y).$$ Given that $f\left(\frac{3}{5} \right) = \frac12$ and$ f(1) = 3$, determine
$$\lfloor \log_2 (-f(10^{2021} - 1)) \rfloor.$$

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Let $\gamma_1$ and $\gamma_2$ be circles centered at $O$ and $ P$ respectively, and externally tangent to each other at point $Q$. Draw point $D$ on $\gamma_1$ and point $E$ on $\gamma_2$ such that line $DE$ is tangent to both circles. If the length $OQ = 1$ and the area of the quadrilateral $ODEP$ is $520$, then what is the value of length $PQ$?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Cary has six distinct coins in a jar. Occasionally, he takes out three of the coins and adds a dot to each of them. Determine the number of orders in which Cary can choose the coins so that, eventually, for each number $i \in \{0, 1, . . . , 5\}$, some coin has exactly $i$ dots on it.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Find the number of ordered pairs of integers $(x, y)$ such that $2167$ divides $3x^2 + 27y^2 + 2021$ with $0 \le x, y \le 2166$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Let $C$ be a circle centered at point $O$, and let $P$ be a point in the interior of $C$. Let $Q$ be a point on the circumference of $C$ such that $PQ \perp OP$, and let $D$ be the circle with diameter $PQ$. Consider a circle tangent to $C$ whose circumference passes through point $P$. Let the curve $\Gamma$ be the locus of the centers of all such circles. If the area enclosed by $\Gamma$ is $1/100$ the area of $C$, then what is the ratio of the area of $C$ to the area of $D$?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Let $f$ be defined as below for integers $n \ge 0$ and $a_0, a_1, ...$ such that $\sum_{i\ge 0}a_i$ is finite:
$$f(n; a_0, a_1, ...) = \begin{cases} a_{2020}, & \text{ $n = 0$} \\
\sum_{i\ge 0} a_i f(n-1;a_0,...,a_{i-1},a_i-1,a_{i+1}+1,a_{i+2},...)/ \sum_{i\ge 0}a_i & \text{$n > 0$} \end{cases}$$.
Find the nearest integer to $f(2020^2; 2020, 0, 0, ...)$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Let $P$ be a $10$-degree monic polynomial with roots $r_1, r_2, . . . , r_{10} \ne $ and let $Q$ be a $45$-degree monic polynomial with roots $\frac{1}{r_i}+\frac{1}{r_j}-\frac{1}{r_ir_j}$ where $i < j$ and $i, j \in \{1, ... , 10\}$. If $P(0) = Q(1) = 2$, then $\log_2 (|P(1)|)$ can be written as $a/b$ for relatively prime integers $a, b$. Find $a + b$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Let $ABCD$ be a cyclic quadrilateral with circumcenter $O$ and radius $10$. Let sides $AB$, $BC$, $CD$, and $DA$ have midpoints $M, N, P$, and $Q$, respectively. If $MP = NQ$ and $OM + OP = 16$, then what is the area of triangle $\vartriangle OAB$?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Let there be a tiger, William, at the origin. William leaps $ 1$ unit in a random direction, then leaps $2$ units in a random direction, and so forth until he leaps $15$ units in a random direction to celebrate PUMaC’s 15th year.
There exists a circle centered at the origin such that the probability that William is contained in the circle (assume William is a point) is exactly $1/2$ after the $15$ leaps. The area of that circle can be written as $A\pi$. What is $A$?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Let $ABC$ be a triangle with sides $AB = 34$, $BC = 15$, $AC = 35$ and let $\Gamma$ be the circle of smallest possible radius passing through $A$ tangent to $BC$. Let the second intersections of $\Gamma$ and sides $AB$, $AC$ be the points $X, Y$ . Let the ray $XY$ intersect the circumcircle of the triangle $ABC$ at $Z$. If $AZ =\frac{p}{q}$ for relatively prime integers $p$ and $q$, find $p + q$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Let $\{x\} = x- \lfloor x \rfloor$ . Consider a function f from the set $\{1, 2, . . . , 2020\}$ to the half-open interval $[0, 1)$. Suppose that for all $x, y,$ there exists a $z$ so that $\{f(x) + f(y)\} = f(z)$. We say that a pair of integers $m, n$ is valid if $1 \le m, n \le 2020$ and there exists a function $f$ satisfying the above so $f(1) = \frac{m}{n}$. Determine the sum over all valid pairs $m, n$ of ${m}{n}$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Three (not necessarily distinct) points in the plane which have integer coordinates between $ 1$ and $2020$, inclusive, are chosen uniformly at random. The probability that the area of the triangle with these three vertices is an integer is $a/b$ in lowest terms. If the three points are collinear, the area of the degenerate triangle is $0$. Find $a + b$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Let $X, Y$ , and $Z$ be concentric circles with radii $1$, $13$, and $22$, respectively. Draw points $A, B$, and $C$ on $X$, $Y$ , and $Z$, respectively, such that the area of triangle $ABC$ is as large as possible. If the area of the triangle is $\Delta$, find $\Delta^2$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

What is the smallest integer $a_0$ such that, for every positive integer $n$, there exists a sequence of positive integers $a_0, a_1, ..., a_{n-1}, a_n$ such that the first $n-1$ are all distinct, $a_0 = a_n$, and for $0 \le i \le n -1$, $a_i^{a_{i+1}}$ ends in the digits $\overline{0a_i}$ when expressed without leading zeros in base $10$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Given two positive integers $a \ne b$, let $f(a, b)$ be the smallest integer that divides exactly one of $a, b$, but not both. Determine the number of pairs of positive integers $(x, y)$, where $x \ne y$, $1\le x, y, \le 100$ and $\gcd(f(x, y), \gcd(x, y)) = 2$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

 You are walking along a road of constant width with sidewalks on each side. You can only walk on the sidewalks or cross the road perpendicular to the sidewalk. Coming up on a turn, you realize that you are on the “outside” of the turn; i.e., you are taking the longer way around the turn. The turn is a circular arc. Assuming that your destination is on the same side of the road as you are currently, let $\theta$ be the smallest turn angle, in radians, that would justify crossing the road and then crossing back after the turn to take the shorter total path to your destination. What is $\lfloor 100 \cdot \theta \rfloor$ ?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

How many ordered triples of nonzero integers $(a, b, c)$ satisfy $2abc = a + b + c + 4$?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Will and Lucas are playing a game. Will claims that he has a polynomial $f$ with integer coefficients in mind, but Lucas doesn’t believe him. To see if Will is lying, Lucas asks him on minute $i$ for the value of $f(i)$, starting from minute $ 1$. If Will is telling the truth, he will report $f(i)$. Otherwise, he will randomly and uniformly pick a positive integer from the range $[1,(i+1)!]$. Now, Lucas is able to tell whether or not the values that Will has given are possible immediately, and will call out Will if this occurs. If Will is lying, say the probability that Will makes it to round $20$ is $a/b$. If the prime factorization of $b$ is $p_1^{e_1}... p_k^{e_k}$ , determine the sum $\sum_{i=1}^{k} e_i$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Find the number of points $P \in Z^2$ that satisfy the following two conditions:
1) If $Q$ is a point on the circle of radius $\sqrt{2020}$ centered at the origin such that the line $PQ$ is tangent to the circle at $Q$, then $PQ$ has integral length.
2) The x-coordinate of $P$ is $38$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Jacob has a piece of bread shaped like a figure $8$, marked into sections and all initially connected as one piece of bread. The central part of the “$8$” is a single section, and each of the two loops of “$8$” is divided into an additional $1010$ pieces. For each section, there is a $50$ percent chance that Jacob will decide to cut it out and give it to a friend, and this is done independently for each section. The remaining sections of bread form some number of connected pieces. If $E$ is the expected number of these pieces, and $k$ is the smallest positive integer so that $2^k(E - \lfloor E \rfloor ) \ge 1$, find $\lfloor E \rfloor +k$. (Here, we say that if Jacob donates all pieces, there are $0$ pieces left).

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Consider a regular $2020$-gon circumscribed into a circle of radius $ 1$. Given three vertices of this polygon such that they form an isosceles triangle, let $X$ be the expected area of the isosceles triangle they create. $X$ can be written as $\frac{1}{m \tan((2\pi)/n)}$ where $m$ and $n$ are integers. Compute $m + n$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Runey is speaking his made-up language, Runese, that consists only of the “letters” zap, zep, zip, zop, and zup. Words in Runese consist of anywhere between $1$ and $5$ letters, inclusive. As well, Runey can choose to add emphasis on any letter(s) that he chooses in a given word, hence making it a totally distinct word! What is the maximum number of possible words in Runese?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

There are n lilypads in a row labeled $1, 2, \dots, n$ from left to right. Fareniss the Frog picks a lilypad at random to start on, and every second she jumps to an adjacent lilypad; if there are two such lilypads, she is twice as likely to jump to the right as to the left. After some finite number of seconds, there exists two lilypads $A$ and $B$ such that Fareniss is more than $1000$ times as likely to be on $A$ as she is to be on $B$. What is the minimal number of lilypads $n$ such that this situation must occur?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

We say that a polynomial $p$ is respectful if $\forall x, y \in Z$, $y - x$ divides $p(y) - p(x)$, and $\forall x \in Z$, $p(x)  \in Z$. We say that a respectful polynomial is disguising if it is nonzero, and all of its non-zero coefficients lie between $0$ and $ 1$, exclusive. Determine $\sum deg(f)\cdot  f(2)$, where the sum includes all disguising polynomials $f$ of degree at most $5$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Select two distinct diagonals at random from a regular octagon. What is the probability that the two diagonals intersect at a point strictly within the octagon? Express your answer as $a + b$, where the probability is $\tfrac{a}{b}$ and $a$ and $b$ are relatively prime positive integers.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Suppose that $f : Z\times Z \to R$, satisfies the equation $f(x, y) = f(3x+y, 2x+ 2y)$ for all $x, y \in Z$. Determine the maximal number of distinct values of $f(x, y)$ for $1 \le x, y \le  100$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Determine the number of pairs $(a, b)$, where $1 \le a \le b \le 100$ are positive integers, so that $\frac{a^3+b^3}{a^2+b^2}$ is an integer.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

The roots of the polynomial $f(x) = x^8 +x^7 -x^5 -x^4 -x^3 +x+ 1 $ are all roots of unity. We say that a real number $r \in [0, 1)$ is nice if $e^{2i \pi r} = \cos 2\pi r + i \sin 2\pi r$ is a root of the polynomial $f$ and if $e^{2i \pi r}$  has positive imaginary part. Let $S$ be the sum of the values of nice real numbers $r$. If $S =\frac{p}{q}$  for relatively prime positive integers $p, q$, find $p + q$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Nelson is having his friend drop his unique bouncy ball from a $12$ foot building, and Nelson will only catch the ball at the peak of its trajectory between bounces. On any given bounce, there is an $80\%$ chance that the next peak occurs at $\frac13$ the height of the previous peak and a $20\%$ chance that the next peak occurs at $3$ times the height of the previous peak (where the first peak is at $12$ feet). If Nelson can only reach $4$ feet into the air and will catch the ball as soon as possible, the probability that Nelson catches the ball after exactly $13$ bounces is $2^a \times 3^b  \times 5^c  \times 7^d  \times 11^e$ for integers $a, b, c, d$, and $e$. Find $|a| + |b| + |c| + |d| + |e|$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Compute the remainder when $2^{3^5}+ 3^{5^2}+ 5^{2^3}$ is divided by $30$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Eighteen people are standing in a (socially-distanced) line to enter a grocery store. Five people are wearing a black mask, $6$ are wearing a gray mask, and $7$ are wearing a white mask. Suppose that these $18$ people got on line in a random order. The expected number of pairs of adjacent people wearing different-colored masks can be given by $\tfrac{a}{b}$, where $\gcd(a, b) = 1$. Compute $a+b$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Kris is asked to compute $\log_{10} (x^y)$, where $y$ is a positive integer and $x$ is a positive real number. However, they misread this as $(\log_{10} x)^y$ , and compute this value. Despite the reading error, Kris still got the right answer. Given that $x > 10^{1.5}$ , determine the largest possible value of $y$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Given an integer $a_0$, we define a sequence of real numbers $a_0, a_1, . . .$ using the relation $$a^2_i = 1 + ia^2_{i-1},$$
for $i \ge 1$. An index $j$ is called [i]good [/i] if $a_j$ can be an integer for some $a_0$. Determine the sum of the indices $j$ which lie in the interval $[0, 99]$ and which are not good.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

 Abby and Ben have a little brother Carl who wants candy. Abby has $7$ different pieces of candy and Ben has $15$ different pieces of candy. Abby and Ben then decide to give Carl some candy. As Ben wants to be a better sibling than Abby, so he decides to give two more pieces of candy to Carl than Abby does. Let $N$ be the number of ways Abby and Ben can give Carl candy. Compute the number of positive divisors of $N$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Let $\alpha, \beta, \gamma \in C$ be the roots of the polynomial $x^3 - 3x2 + 3x + 7$. For any complex number $z$, let $f(z)$ be defined as follows:
$$f(z) = |z -\alpha | + |z - \beta|+ |z-\gamma | - 2 \underbrace{\max}_{w \in \{\alpha, \beta, \gamma}\} |z - w|.$$
Let $A$ be the area of the region bounded by the locus of all $z \in C$ at which $f(z)$ attains its global minimum. Find $\lfloor A \rfloor$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Let $\vartriangle ABC$ be a triangle. Let $Q$ be a point in the interior of $\vartriangle ABC$, and let $X, Y,Z$ denote the feet of the altitudes from $Q$ to sides $BC$, $CA$, $AB$, respectively. Suppose that $BC = 15$, $\angle ABC = 60^o$, $BZ = 8$, $ZQ = 6$, and  $\angle QCA = 30^o$. Let line $QX$ intersect the circumcircle of  $\vartriangle XY Z$ at the point $W\ne  X$. If the ratio $\frac{ WY}{WZ}$ can be expressed as $\frac{p}{q}$ for relatively prime positive integers $p, q$, find $p + q$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Provided that $\{a_i\}^{28}_{i=1}$ are the $28$ distinct roots of $29x^{28} + 28x^{27} + ... + 2x + 1 = 0$, then the absolute value of $\sum^{28}_{i=1}\frac{1}{(1-a_i)^2}$ can be written as $\frac{p}{q}$ for relatively prime positive integers $p, q$. Find $p + q$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Let $\vartriangle ABC$ have sidelengths $BC = 7$, $CA = 8$, and, $AB = 9$, and let $\Omega$ denote the circumcircle of $\vartriangle ABC$. Let circles $\omega_A$, $\omega_B$, $\omega_C$ be internally tangent to the minor arcs $BC$, $CA$, $AB$ of $\Omega$, respectively, and tangent to the segments $BC$, $CA$, $AB$ at points $X$, $Y$, and, $Z$, respectively. Suppose that $\frac{BX}{XC} = \frac{CY}{Y A} = \frac{AZ}{ZB} = \frac12$ . Let $t_{AB}$ be the length of the common external tangent of $\omega_A$ and $\omega_B$, let $t_{BC}$ be the length of the common external tangent of $\omega_B$ and $\omega_C$, and let $t_{CA}$ be the length of the common external tangent of $\omega_C$ and $\omega_A$. If $t_{AB} + t_{BC} + t_{CA}$ can be expressed as $\frac{m}{n}$ for relatively prime positive integers $m, n$, find $m + n$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

A sequence of integers $a_1, a_2, \ldots, a_n$ is said to be [i]sub-Fibonacci[/i] if $a_1=a_2=1$ and $a_i \le a_{i-1}+a_{i-2}$ for all $3 \le i \le n.$ How many sub-Fibonacci sequences are there with $10$ terms such that the last two terms are both $20$?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

The set of real values $a$ such that the equation $x^4-3ax^3+(2a^2+4a)x^2-5a^2x+3a^2$ has exactly two nonreal solutions is the set of real numbers between $x$ and $y,$ where $x<y.$ If $x+y$ can be written as $\tfrac{m}{n}$ for relatively prime positive integers $m,n,$ find $m+n.$

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

6. Let $f(p)$ denote the number of ordered tuples $\left(x_{1}, x_{2}, \ldots, x_{p}\right)$ of nonnegative integers satisfying $\sum_{i=1}^{p} x_{i}=2022$, where $x_{i} \equiv i(\bmod p)$ for all $1 \leq i \leq p$. Find the remainder when $\sum_{p \in \mathcal{S}} f(p)$ is divided by 1000, where $\mathcal{S}$ denotes the set of all primes less than 2022 .

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Let $\vartriangle ABC$ be a triangle with $AB = 13$, $BC = 14$, and $CA = 15$. Let $D$, $E$, and $F$ be the midpoints of $AB$, $BC$, and $CA$ respectively. Imagine cutting $\vartriangle ABC$ out of paper and then folding $\vartriangle AFD$ up along $FD$, folding $\vartriangle BED$ up along $DE$, and folding $\vartriangle CEF$ up along $EF$ until $A$, $B$, and $C$ coincide at a point $G$. The volume of the tetrahedron formed by vertices $D$, $E$, $F$, and $G$ can be expressed as $\frac{p\sqrt{q}}{r}$ , where $p$, $q$, and $r$ are positive integers, $p$ and $r$ are relatively prime, and $q$ is square-free. Find $p + q + r$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Let $S$ be the set of degree $4$ polynomials $f$ with complex number coefficients satisfying $f(1)=f(2)^2=f(3)^3$ $=$ $f(4)^4=f(5)^5=1.$ Find the mean of the fifth powers of the constant terms of all the members of $S.$

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

10. The sum $\sum_{k=1}^{2020} k \cos \left(\frac{4 k \pi}{4041}\right)$ can be written in the form

$$
\frac{a \cos \left(\frac{p \pi}{q}\right)-b}{c \sin ^{2}\left(\frac{p \pi}{q}\right)}
$$

where $a, b, c$ are relatively prime positive integers and $p, q$ are relatively prime positive integers where $p<q$. Determine $a+b+c+p+q$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Let $\vartriangle ABC$ have $AB = 15$, $AC = 20$, and $BC = 21$. Suppose $\omega$ is a circle passing through $A$ that is tangent to segment $BC$. Let point $D\ne A$ be the second intersection of AB with $\omega$, and let point $E \ne A$ be the second intersection of $AC$ with $\omega$. Suppose $DE$ is parallel to $BC$. If $DE = \frac{a}{b}$ , where $a$, $b$ are relatively prime positive integers, find $a + b$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Let $\triangle{ABC}$ be an isosceles triangle with $AB = AC =\sqrt{7}, BC=1$. Let $G$ be the centroid of $\triangle{ABC}$. Given $ j\in \{0,1,2\}$, let $T_{j}$ denote the triangle obtained by rotating $\triangle{ABC}$ about $G$ by $\frac{2\pi j}{3}$ radians. Let $\mathcal{P}$ denote the intersection of the interiors of triangles $T_0,T_1,T_2$. If $K$ denotes the area of $\mathcal{P}$, then $K^2=\frac{a}{b}$ for relatively prime positive integers $a, b$. Find $a + b$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Let $\vartriangle ABC$ have $AB = 14$, $BC = 30$, $AC = 40$ and $\vartriangle AB'C'$ with $AB' = 7\sqrt6$, $B'C' = 15\sqrt6$, $AC' = 20\sqrt6$ such that $\angle BAB' = \frac{5\pi}{12}$ . The lines $BB'$  and $CC'$  intersect at point $D$. Let $O$ be the circumcenter of $\vartriangle BCD$, and let $O' $ be the circumcenter of $\vartriangle B'C'D$. Then the length of segment $OO'$ can be expressed as $\frac{a+b \sqrt{c}}{ d}$ , where $a$, $b$, $c$, and $d$ are positive integers such that $a$ and $d$ are relatively prime, and $c$ is not divisible by the square of any prime. Find $a+b+c+d$

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

The integers from $1$ to $25,$ inclusive, are randomly placed into a $5$ by $5$ grid such that in each row, the numbers are increasing from left to right. If the columns from left to right are numbered $1,2,3,4,$ and $5,$ then the expected column number of the entry $23$ can be written as $\tfrac{a}{b}$ where $a$ and $b$ are relatively prime positive integers. Find $a+b.$

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

A quadratic polynomial $f(x)$ is called sparse if its degree is exactly 2 , if it has integer coefficients, and if there exists a nonzero polynomial $g(x)$ with integer coefficients such that $f(x) g(x)$ has degree at most 3 and $f(x) g(x)$ has at most two nonzero coefficients. Find the number of sparse quadratics whose coefficients lie between 0 and 10, inclusive.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

A polynomial $p(x) = \sum_{j=1}^{2n-1} a_j x^j$ with real coefficients is called [i]mountainous[/i] if $n \ge 2$ and there exists a real number such that the polynomial's coefficients satisfy $a_1=1, a_{j+1}-a_j=k$ for $1 \le j \le n-1,$ and $a_{j+1}-a_j=-k$ for $n \le j \le 2n-2;$ we call $k$ the [i]step size[/i] of $p(x).$ A real number $k$ is called [i]good[/i] if there exists a mountainous polynomial $p(x)$ with step size $k$ such that $p(-3)=0.$ Let $S$ be the sum of all good numbers $k$ satisfying $k \ge 5$ or $k \le 3.$ If $S=\tfrac{b}{c}$ for relatively prime positive integers $b,c,$ find $b+c.$

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Alice, Bob, and Carol each independently roll a fair six-sided die and obtain the numbers $a, b, c$, respectively. They then compute the polynomial $f(x)=x^{3}+p x^{2}+q x+r$ with roots $a, b, c$. If the expected value of the sum of the squares of the coefficients of $f(x)$ is $\frac{m}{n}$ for relatively prime positive integers $m, n$, find the remainder when $m+n$ is divided by 1000 .

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

If  \[\frac{1}{1+2} + \frac{1}{1+2+3} + \ldots + \frac{1}{1+2 + \ldots + 20} = \frac{m}{n}\]
where $m$ and $n$ are positive integers with no common divisor, find $m + n$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

A circle radius $320$ is tangent to the inside of a circle radius $1000$. The smaller circle is tangent to a diameter of the larger circle at a point $P$. How far is the point $P$ from the outside of the larger circle?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Let $r$ be a real number such that $\sqrt[3]{r} - \frac{1}{\sqrt[3]{r}}=2$. Find $r^3 - \frac{1}{r^3}$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Let $x_1$ and $x_2$ be the roots of the equation $x^2 + 3x + 1 = 0$. Compute \[\left(\frac{x_1}{x_2 + 1}\right)^2 + \left(\frac{x_2}{x_1 + 1}\right)^2\]

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Let $f$ be a real-valued function of real and positive argument such that $f(x) + 3xf(\tfrac1x) = 2(x + 1)$ for all real numbers $x > 0$. Find $f(2003)$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

How many real numbers are roots of the polynomial \[x^9 - 37x^8 - 2x^7 + 74x^6 + x^4 - 37x^3 - 2x^2 + 74x?\]

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

One rainy afternoon you write the number $1$ once, the number $2$ twice, the number $3$ three times, and so forth until you have written the number $99$ ninety-nine times. What is the $2005$ th digit that you write?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

How many non-overlapping $2$ by $2$ squares will fit into a circle with radius $8$?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

How many positive integers less that $200$ are relatively prime to either $15$ or $24$?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

In the addition problem
\[ \setlength{\tabcolsep}{1mm}\begin{tabular}{cccccc}& W & H & I & T & E\\ + & W & A & T & E & R \\\hline P & I & C & N & I & C\end{tabular} \] each distinct letter represents a diﬀerent digit. Find the number represented by the answer PICNIC.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Find $x$ so that $2^{2^{3^{2^{2}}}} = 4^{4^{x}}$.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Find $x$ such that \[\frac{\frac{5}{x-50}+ \frac{7}{x+25}}{\frac{2}{x-50}- \frac{3}{x+25}} = 17.\]

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Find the number of nonnegative integers $n$ for which $(n^2 - 3n + 1)^2 + 1$ is a prime number

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

We glue together $990$ one inch cubes into a $9$ by $10$ by $11$ inch rectangular solid. Then we paint the outside of the solid. How many of the original $990$ cubes have just one of their sides painted?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

The work team was working at a rate fast enough to process $1250$ items in ten hours. But after working for six hours, the team was given an additional $150$ items to process. By what percent does the team need to increase its rate so that it can still complete its work within the ten hours?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

The number $1$ is special. The number $2$ is special because it is relatively prime to $1$. The number $3$ is not special because it is not relatively prime to the sum of the special numbers less than it, $1 + 2$. The number $4$ is special because it is relatively prime to the sum of the special numbers less than it. So, a number bigger than $1$ is special only if it is relatively prime to the sum of the special numbers less than it. Find the twentieth special number.

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Jerry is mowing a rectangular lawn which is $77$ feet north to south by $83$ feet east to west. His lawn mower cuts a path $18$ inches wide. Jerry mows the grass by cutting a path from west to east across the north side of the lawn and then making a right turn cutting a path along the east side of the lawn. When he completes mowing each side of the lawn, he continues by making right turns to mow a path along the next side. How many right turns will he make?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

The cost of producing each item is inversely proportional to the square root of the number of items produced. The cost of producing ten items is $ \$2100$. If items sell for $ \$30$ each, how many items need to be sold so that the producers break even?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

A palindrome is a number that reads the same forwards and backwards such as $3773$ or $42924$. What is the smallest $9$ digit palindrome which is a multiple of $3$ and has at least two digits which are $5$'s and two digits which are $7$'s?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

Find the number of quadruples $(a,b,c,d)$ of integers which satisfy both \begin{align*}\frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{1}{d} &= \frac{1}{2}\qquad\text{and}\\\\2(a+b+c+d) &= ab + cd + (a+b)(c+d) + 1.\end{align*}

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

A tailor met a tortoise sitting under a tree. When the tortoise was the tailor’s age, the tailor was only a quarter of his current age. When the tree was the tortoise’s age, the tortoise was only a seventh of its current age. If the sum of their ages is now $264$, how old is the tortoise?

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Solve the following math problem step by step. The last line of your response should be of the form Answer: \boxed{$Answer} where $Answer is the answer to the problem.

The work team was working at a rate fast enough to process $1250$ items in ten hours. But after working for six hours, the team was given an additional $165$ items to process. By what percent does the team need to increase its rate so that it can still complete its work within the ten hours?

Remember to put your answer on its own line after "Answer:".