Let triangle $ABC$ satisfy $2BC = AB+AC$ and have incenter $I$ and circumcircle $\omega$. Let $D$ be the intersection of $AI$ and $\omega$ (with $A, D$ distinct). Prove that $I$ is the midpoint of $AD$.

The vertices of a regular nonagon are colored such that $1)$ adjacent vertices are different colors and $2)$ if $3$ vertices form an equilateral triangle, they are all different colors. Let $m$ be the minimum number of colors needed for a valid coloring, and n be the total number of colorings using $m$ colors. Determine $mn$. (Assume each vertex is distinguishable.)

How many polynomials $P$ with integer coefficients and degree at most $5$ satisfy $0 \le P(x) < 120$ for all $x \in \{0,1,2,3,4,5\}$?

Tim and Allen are playing a match of [i]tenus[/i]. In a match of [i]tenus[/i], the two players play a series of games, each of which is won by one of the two players. The match ends when one player has won exactly two more games than the other player, at which point the player who has won more games wins the match. In odd-numbered games, Tim wins with probability $3/4$, and in the even-numbered games, Allen wins with probability $3/4$. What is the expected number of games in a match?

For an even positive integer $n$ Kevin has a tape of length $4n$ with marks at $-2n,-2n+1,\ldots,2n-1,2n$. He then randomly picks $n$ points in the set $-n,-n+1,-n+2,\ldots,n-1,n$ and places a stone on each of these points. We call a stone 'stuck' if it is on $2n$ or $-2n$, or either all the points to the right, or all the points to the left, all contain stones. Then, every minute, Kevin shifts the unstruck stones in the following manner: [list] [*]He picks an unstuck stone uniformly at random and then flips a fair coin. [*]If the coin came up heads, he then moves that stone and every stone in the largest contiguous set containing that stone one point to the left. If the coin came up tails, he moves every stone in that set one point right instead. [*]He repeats until all the stones are stuck.[/list] Let $p_n$ be the probability that at the end of the process there are exactly $k$ stones in the right half. Evaluate \[\dfrac{p_{n-1}-p_{n-2}+p_{n-3}+\ldots+p_3-p_2+p_1}{p_{n-1}+p_{n-2}+p_{n-3}+\ldots+p_3+p_2+p_1}\] in terms of $n$.

Let $ABCDEF$ be a convex equilateral hexagon such that lines $BC$, $AD$, and $EF$ are parallel. Let $H$ be the orthocenter of triangle $ABD$. If the smallest interior angle of the hexagon is $4$ degrees, determine the smallest angle of the triangle $HAD$ in degrees.

Let $z_0+z_1+z_2+\cdots$ be an infinite complex geometric series such that $z_0=1$ and $z_{2013}=\dfrac 1{2013^{2013}}$. Find the sum of all possible sums of this series.

Find all positive integers $n$ for which there do not exist $n$ consecutive composite positive integers less than $n!$.

Compute the prime factorization of $1007021035035021007001$. (You should write your answer in the form $p_1^{e_1}p_2^{e_2}\ldots p_k^{e_k}$ where $p_1,\ldots,p_k$ are distinct prime numbers and $e_1,\ldots,e_k$ are positive integers.)

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 $x\in \{1,2,3,4,5\}$.

Steph Curry is playing the following game and he wins if he has exactly $5$ points at some time. Flip a fair coin. If heads, shoot a $3$-point shot which is worth $3$ points. If tails, shoot a free throw which is worth $1$ point. He makes $\frac12$ of his $3$-point shots and all of his free throws. Find the probability he will win the game. (Note he keeps flipping the coin until he has exactly $5$ or goes over $5$ points)

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

An isosceles trapezoid $ABCD$ with bases $AB$ and $CD$ has $AB=13$, $CD=17$, and height $3$. Let $E$ be the intersection of $AC$ and $BD$. Circles $\Omega$ and $\omega$ are circumscribed about triangles $ABE$ and $CDE$. Compute the sum of the radii of $\Omega$ and $\omega$.

You are given an unlimited supply of red, blue, and yellow cards to form a hand. Each card has a point value and your score is the sum of the point values of those cards. The point values are as follows: the value of each red card is 1, the value of each blue card is equal to twice the number of red cards, and the value of each yellow card is equal to three times the number of blue cards. What is the maximum score you can get with fifteen cards?

Given $w$ and $z$ are complex numbers such that $|w+z|=1$ and $|w^2+z^2|=14$, find the smallest possible value of $|w^3+z^3|$. Here $| \cdot |$ denotes the absolute value of a complex number, given by $|a+bi|=\sqrt{a^2+b^2}$ whenever $a$ and $b$ are real numbers.

Michael is trying to drive a bus from his home, $(0,0)$, to school, located at $(6,6)$. There are horizontal and vertical roads at every line $x=0,1,\ldots,6$ and $y=0,1,\ldots,6$. The city has placed $6$ roadblocks on lattice point intersections $(x,y)$ with $0\leq x,y \leq 6$. Michael notices that the only path he can take that only goes up and to the right is directly up from $(0,0)$ to $(0,6)$, and then right to $(6,6)$. How many sets of $6$ locations could the city have blocked?

Let $P_1P_2 \ldots P_8$ be a convex octagon. An integer $i$ is chosen uniformly at random from $1$ to $7$, inclusive. For each vertex of the octagon, the line between that vertex and the vertex $i$ vertices to the right is painted red. What is the expected number times two red lines intersect at a point that is not one of the vertices, given that no three diagonals are concurrent?

Find the number of integers $n$ such that \[1+\left\lfloor\dfrac{100n}{101}\right\rfloor=\left\lceil\dfrac{99n}{100}\right\rceil.\]

Can the set of lattice points $\{(x, y) \mid x, y \in \mathbb{Z}, 1 \le x, y \le 252, x \neq y\}$ be colored using 10 distinct colors such that for all $a \neq b$, $b \neq c$, the colors of $(a, b)$ and $(b, c)$ are distinct?

16. Create a cube $C_1$ with edge length $1$. Take the centers of the faces and connect them to form an octahedron $O_1$. Take the centers of the octahedron’s faces and connect them to form a new cube $C_2$. Continue this process infinitely. Find the sum of all the surface areas of the cubes and octahedrons. 17. Let $p(x) = x^2 - x + 1$. Let $\alpha$ be a root of $p(p(p(p(x)))$. Find the value of $$(p(\alpha) - 1)p(\alpha)p(p(\alpha))p(p(p(\alpha))$$ 18. An $8$ by $8$ grid of numbers obeys the following pattern: 1) The first row and first column consist of all $1$s. 2) The entry in the $i$th row and $j$th column equals the sum of the numbers in the $(i - 1)$ by $(j - 1)$ sub-grid with row less than i and column less than $j$. What is the number in the 8th row and 8th column?

Let $S$ be the set of integers of the form $2^x+2^y+2^z$, where $x,y,z$ are pairwise distinct non-negative integers. Determine the $100$th smallest element of $S$.

If $a$ and $b$ satisfy the equations $a +\frac1b=4$ and $\frac1a+b=\frac{16}{15}$, determine the product of all possible values of $ab$.

A rectangular pool table has vertices at $(0, 0) (12, 0) (0, 10),$ and $(12, 10)$. There are pockets only in the four corners. A ball is hit from $(0, 0)$ along the line $y = x$ and bounces off several walls before eventually entering a pocket. Find the number of walls that the ball bounces off of before entering a pocket.

In a game, there are three indistinguishable boxes; one box contains two red balls, one contains two blue balls, and the last contains one ball of each color. To play, Raj first predicts whether he will draw two balls of the same color or two of different colors. Then, he picks a box, draws a ball at random, looks at the color, and replaces the ball in the same box. Finally, he repeats this; however, the boxes are not shuffled between draws, so he can determine whether he wants to draw again from the same box. Raj wins if he predicts correctly; if he plays optimally, what is the probability that he will win?

What is the largest factor of $130000$ that does not contain the digit $0$ or $5$?