CZ-SK-MO 2009 Problem 5

At every vertex $A_k(1\le k\le n)$ of a regular $n$ -gon, $k$ coins are placed. We can do the following operation: in each step, one can choose two arbitrarily coins and move them to their adjacent vertices respectively, one clockwise and one anticlockwise. Find all positive integers $n$ such that after a finite number of operations, we can reach the following configuration: there are $n+1-k$ coins at vertex $A_k$ for all $1\le k\le n$ .