ÖMO 2015 Problem 6

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)