Let $\Gamma$ be the circumcircle of triangle $ABC$, and let $D$ lie on side $BC$. The tangent to $\Gamma$ at $A$ meets the line through $D$ parallel to $BA$ at $E$. The segment $CE$ meets $\Gamma$ again at $F$. Suppose $B,D,F,E$ are concyclic. Prove that $AC,BF,DE$ are concurrent.
Show that $r=2$ is the largest real number $r$ with the following property: if a sequence $a_1,a_2,\ldots$ of positive integers satisfies
$$a_n\le a_{n+2}\le\sqrt{a_n^2+r a_{n+1}}$$
for every positive integer $n$, then there is a positive integer $M$ such that $a_{n+2}=a_n$ for every $n\ge M$.
Determine all positive integers $k$ for which there are a positive integer $m$ and a set $S$ of positive integers such that every integer $n\gt m$ can be written as a sum of distinct elements of $S$ in exactly $k$ ways.
Let $\mathbb Z$ be the integers. Find all polynomials $P(x)$ with integer coefficients such that, for every infinite sequence $a_1,a_2,\ldots$ containing each integer exactly once, there are indices $i\lt j$ and an integer $k$ satisfying
$$a_i+a_{i+1}+\cdots+a_j=P(k).$$
Fix an integer $n\ge3$. Initially the number $1$ is written $n$ times on a blackboard, and two buckets are empty. A move erases two numbers $a,b$, replaces them by $1,a+b$, adds one stone to the first bucket, and adds $\gcd(a,b)$ stones to the second. After finitely many moves the buckets contain positive integers $s,t$ stones. Find all possible values of $t/s$.
