ELMO SL 2013 C10

Let $N\ge2$ be a fixed positive integer. There are $2N$ people, numbered $1,2,...,2N$ , participating in a tennis tournament. For any two positive integers $i,j$ with $1\le i<j\le 2N$ , player $i$ has a higher skill level than player $j$ . Prior to the first round, the players are paired arbitrarily and each pair is assigned a unique court among $N$ courts, numbered $1,2,...,N$ .

During a round, each player plays against the other person assigned to his court (so that exactly one match takes place per court), and the player with higher skill wins the match (in other words, there are no upsets). Afterwards, for $i=2,3,...,N$ , the winner of court $i$ moves to court $i-1$ and the loser of court $i$ stays on court $i$ ; however, the winner of court 1 stays on court 1 and the loser of court 1 moves to court $N$ .

Find all positive integers $M$ such that, regardless of the initial pairing, the players $2, 3, \ldots, N+1$ all change courts immediately after the $M$ th round.

Proposed by Ray Li