What is the minimum possible perimeter of a triangle two of whose sides are along the x- and y-axes and such that the third contains the point $(1,2)$?

In triangle $ABC$ with $AB < AC$, let $H$ be the orthocenter and $O$ be the circumcenter. Given that the midpoint of $OH$ lies on $BC$, $BC = 1$, and the perimeter of $ABC$ is 6, find the area of $ABC$.

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

Call a polygon [i]normal[/i] if it can be inscribed in a unit circle. How many non-congruent normal polygons are there such that the square of each side length is a positive integer?

Let $a$ and $b$ be real numbers, and let $r$, $s$, and $t$ be the roots of $f(x)=x^3+ax^2+bx-1$. Also, $g(x)=x^3+mx^2+nx+p$ has roots $r^2$, $s^2$, and $t^2$. If $g(-1)=-5$, find the maximum possible value of $b$.

Prove that for all positive integers $n$, all complex roots $r$ of the polynomial \[P(x) = (2n)x^{2n} + (2n-1)x^{2n-1} + \dots + (n+1)x^{n+1} + nx^n + (n+1)x^{n-1} + \dots + (2n-1)x + 2n\] lie on the unit circle (i.e. $|r| = 1$).

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?

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 $n$.

A positive integer is written on each corner of a square such that numbers on opposite vertices are relatively prime while numbers on adjacent vertices are not relatively prime. What is the smallest possible value of the sum of these $4$ numbers?

The three points $A, B, C$ form a triangle. $AB=4, BC=5, AC=6$. Let the angle bisector of $\angle A$ intersect side $BC$ at $D$. Let the foot of the perpendicular from $B$ to the angle bisector of $\angle A$ be $E$. Let the line through $E$ parallel to $AC$ meet $BC$ at $F$. Compute $DF$.

Let $p > 2$ be a prime number. $\mathbb{F}_p[x]$ is defined as the set of polynomials in $x$ with coefficients in $\mathbb{F}_p$ (the integers modulo $p$ with usual addition and subtraction), so that two polynomials are equal if and only if the coefficients of $x^k$ are equal in $\mathbb{F}_p$ for each nonnegative integer $k$. For example, $(x+2)(2x+3) = 2x^2 + 2x + 1$ in $\mathbb{F}_5[x]$ because the corresponding coefficients are equal modulo 5. Let $f, g \in \mathbb{F}_p[x]$. The pair $(f, g)$ is called [i]compositional[/i] if \[f(g(x)) \equiv x^{p^2} - x\] in $\mathbb{F}_p[x]$. Find, with proof, the number of compositional pairs.

Let $a$ and $b$ be integers (not necessarily positive). Prove that $a^3+5b^3 \neq 2016$.

You have some white one-by-one tiles and some black and white two-bye-one tiles as shown below. There are four different color patterns that can be generated when using these tiles to cover a three-by-one rectangoe by laying these tiles side by side (WWW, BWW, WBW, WWB). How many different color patterns can be generated when using these tiles to cover a ten-by-one rectangle? [asy] import graph; size(5cm); real labelscalefactor = 0.5; pen dps = linewidth(0.7) + fontsize(10); defaultpen(dps); draw((12,0)--(12,1)--(11,1)--(11,0)--cycle); fill((13.49,0)--(13.49,1)--(12.49,1)--(12.49,0)--cycle, black); draw((13.49,0)--(13.49,1)--(14.49,1)--(14.49,0)--cycle); draw((15,0)--(15,1)--(16,1)--(16,0)--cycle); fill((17,0)--(17,1)--(16,1)--(16,0)--cycle, black); [/asy]

Let $L$ be the length of the altitude to the hypotenuse of a right triangle with legs $5$ and $12$. Find the least integer greater than $L$.

Let $A_1,A_2,\ldots,A_m$ be finite sets of size $2012$ and let $B_1,B_2,\ldots,B_m$ be finite sets of size $2013$ such that $A_i\cap B_j=\emptyset$ if and only if $i=j$. Find the maximum value of $m$.

Given points $a$ and $b$ in the plane, let $a\oplus b$ be the unique point $c$ such that $abc$ is an equilateral triangle with $a,b,c$ in the clockwise orientation. Solve $(x\oplus (0,0))\oplus(1,1)=(1,-1)$ for $x$.

Mark and William are playing a game with a stored value. On his turn, a player may either multiply the stored value by $2$ and add $1$ or he may multiply the stored value by $4$ and add $3$. The first player to make the stored value exceed $2^{100}$ wins. The stored value starts at $1$ and Mark goes first. Assuming both players play optimally, what is the maximum number of times that William can make a move? (By optimal play, we mean that on any turn the player selects the move which leads to the best possible outcome given that the opponent is also playing optimally. If both moves lead to the same outcome, the player selects one of them arbitrarily.)

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

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

1. If five fair coins are flipped simultaneously, what is the probability that at least three of them show heads? 2. How many perfect squares divide $10^{10}$? 3. Evaluate $\frac{2016!^2}{2015!2017!}$ . Here $n!$ denotes $1 \times 2 \times \ldots \times n$.

Contessa is taking a random lattice walk in the plane, starting at $(1,1)$. (In a random lattice walk, one moves up, down, left, or right $1$ unit with equal probability at each step.) If she lands on a point of the form $(6m,6n)$ for $m,n \in \mathbb{Z}$, she ascends to heaven, but if she lands on a point of the form $(6m+3,6n+3)$ for $m,n \in \mathbb{Z}$, she descends to hell. What is the probability she ascends to heaven?

The sequence $(z_n)$ of complex numbers satisfies the following properties: [list] [*]$z_1$ and $z_2$ are not real. [*]$z_{n+2}=z_{n+1}^2z_n$ for all integers $n\geq 1$. [*]$\dfrac{z_{n+3}}{z_n^2}$ is real for all integers $n\geq 1$. [*]$\left|\dfrac{z_3}{z_4}\right|=\left|\dfrac{z_4}{z_5}\right|=2$. [/list] Find the product of all possible values of $z_1$.

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

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)

Sherry and Val are playing a game. Sherry has a deck containing $2011$ red cards and $2012$ black cards, shuffled randomly. Sherry flips these cards over one at a time, and before she flips each card over, Val guesses whether it is red or black. If Val guesses correctly, she wins $1$ dollar; otherwise, she loses $1$ dollar. In addition, Val must guess red exactly $2011$ times. If Val plays optimally, what is her expected profit from this game?