ÖMO 2015

Let I be the incenter of triangle $ABC$ and let $k$ be a circle through the points $A$ and $B$ . The circle intersects

• the line $AI$ in points $A$ and $P$ * the line $BI$ in points $B$ and $Q$ * the line $AC$ in points $A$ and $R$ * the line $BC$ in points $B$ and $S$ with none of the points $A,B,P,Q,R$ and $S$ coinciding and such that $R$ and $S$ are interior points of the line segments $AC$ and $BC$ , respectively.

Prove that the lines $PS$ , $QR$ , and $CI$ meet in a single point.

(Stephan Wagner)

Max has $2015$ jars labeled with the numbers $1$ to $2015$ and an unlimited supply of coins.

Consider the following starting configurations:

(a) All jars are empty. (b) Jar $1$ contains $1$ coin, jar $2$ contains $2$ coins, and so on, up to jar $2015$ which contains $2015$ coins. (c) Jar $1$ contains $2015$ coins, jar $2$ contains $2014$ coins, and so on, up to jar $2015$ which contains $1$ coin.

Now Max selects in each step a number $n$ from $1$ to $2015$ and adds $n$ to each jar except to the jar $n$.

Determine for each starting configuration in (a), (b), (c), if Max can use a finite, strictly positive number of steps to obtain an equal number of coins in each jar.

(Birgit Vera Schmidt)