What is the maximum integer $n$ such that $\frac{50!}{2^n}$ is an integer?

Let $ABCD$ be a square and let $X$ be any point on side $BC$ between $B$ and $C$. Let $Y$ be the point on line $CD$ such that $BX = YD$ and $D$ is between $C$ and $Y$ . Prove that the midpoint of $XY$ lies on diagonal $BD$.

You have an unlimited supply of square tiles with side length $ 1$ and equilateral triangle tiles with side length $ 1$. For which n can you use these tiles to create a convex $n$-sided polygon? The tiles must fit together without gaps and may not overlap.

Determine all prime numbers $p$ such that $p^2 - 6$ and $p^2 + 6$ are both prime numbers.

Find all functions $f:\mathbb R \to \mathbb R$ such that for all $x,y\in \mathbb R$ $f(x+f(y))=2x+2f(y+1)$

Let $\vartriangle ABC$ be an acute triangle with $AB > AC$. Let $P$ be the foot of the altitude from $C$ to $AB$ and let $Q$ be the foot of the altitude from $B$ to $AC$. Let $X$ be the intersection of $PQ$ and $BC$. Let the intersection of the circumcircles of triangle $\vartriangle AXC$ and triangle $\vartriangle PQC$ be distinct points: $C$ and $Y$ . Prove that $PY$ bisects $AX$.

Josie and Ross are playing a game on a $20 \times 20$ chessboard. Initially the chessboard is empty. The two players alternately take turns, with Josie going first. On Josie’s turn, she selects any two different empty cells, and places one white stone in each of them. On Ross’ turn, he chooses any one white stone currently on the board, and replaces it with a black stone. If at any time there are $ 8$ consecutive cells in a line (horizontally or vertically) all of which contain a white stone, Josie wins. Is it possible that Ross can stop Josie winning - regardless of how Josie plays?

For a positive integer $x$, define a sequence $a_0, a_1, a_2, . . .$ according to the following rules: $a_0 = 1$, $a_1 = x + 1$ and $$a_{n+2} = xa_{n+1} - a_n$$for all $n \ge 0$. Prove that there exist infinitely many positive integers x such that this sequence does not contain a prime number.

Let $P(x) = x^3 - 2x + 1$ and let $Q(x) = x^3 - 4x^2 + 4x - 1$. Show that if $P(r) = 0$ then $Q(r^2) = 0$.

Find the smallest positive integer $N$ satisfying the following three properties. $\bullet$ N leaves a remainder of $5$ when divided by $7$. $\bullet$ N leaves a remainder of $6$ when divided by $ 8$. $\bullet$ N leaves a remainder of $7$ when divided by $9$.

There are $13$ marked points on the circumference of a circle with radius $13$. Prove that we can choose three of the marked points which form a triangle with area less than $13$.

Let $\Gamma_1$ and $\Gamma_2$ be circles internally tangent at point $A$, with $\Gamma_1$ inside $\Gamma_2$. Let $BC$ be a chord of $\Gamma_2$ which is tangent to $\Gamma_1$ at point $D$. Prove that line $AD$ is the angle bisector of $\angle BAC$.

A sequence of $A$s and $B$s is called antipalindromic if writing it backwards, then turning all the $A$s into $B$s and vice versa, produces the original sequence. For example $ABBAAB$ is antipalindromic. For any sequence of $A$s and $B$s we define the cost of the sequence to be the product of the positions of the $A$s. For example, the string $ABBAAB$ has cost $1\cdot 4 \cdot 5 = 20$. Find the sum of the costs of all antipalindromic sequences of length $2020$.