GULF 2013 Problem 3

There are $n$ people standing on a circular track. We want to perform a number of moves so that we end up with a situation where the distance between every two neighbours is the same. The move that is allowed consists in selecting two people and asking one of them to walk a distance $d$ on the circular track clockwise, and asking the other to walk the same distance on the track anticlockwise. The two people selected and the quantity $d$ can vary from move to move.

Prove that it is possible to reach the desired situation (where the distance between every two neighbours is the same) after at most $n-1$ moves.