OBM 2014

Let $ABCD$ be a convex quadrilateral. Diagonals $AC$ and $BD$ meet at point $P$ . The inradii of triangles $ABP$ , $BCP$ , $CDP$ and $DAP$ are equal. Prove that $ABCD$ is a rhombus.

Find all integers $n$ , $n>1$ , with the following property: for all $k$ , $0\le k < n$ , there exists a multiple of $n$ whose digits sum leaves a remainder of $k$ when divided by $n$ .

Let $N$ be an integer, $N>2$ . Arnold and Bernold play the following game: there are initially $N$ tokens on a pile. Arnold plays first and removes $k$ tokens from the pile, $1\le k < N$ . Then Bernold removes $m$ tokens from the pile, $1\le m\le 2k$ and so on, that is, each player, on its turn, removes a number of tokens from the pile that is between $1$ and twice the number of tokens his opponent took last. The player that removes the last token wins.

For each value of $N$ , find which player has a winning strategy and describe it.

The infinite sequence $P_0(x),P_1(x),P_2(x),\ldots,P_n(x),\ldots$ is defined as $$P_0(x)=x, P_n(x) = P_{n-1}(x-1)\cdot P_{n-1}(x+1), n\ge 1.$$ Find the largest $k$ such that $P_{2014}(x)$ is divisible by $x^k$ .

Let $ABC$ be a triangle with incenter $I$ and incircle $\omega$ . Circle $\omega_A$ is externally tangent to $\omega$ and tangent to sides $AB$ and $AC$ at $A_1$ and $A_2$ , respectively. Let $r_A$ be the line $A_1A_2$ . Define $r_B$ and $r_C$ in a similar fashion. Lines $r_A$ , $r_B$ and $r_C$ determine a triangle $XYZ$ . Prove that the incenter of $XYZ$ , the circumcenter of $XYZ$ and $I$ are collinear.