CGMO 2007 Problem 8

In a round robin chess tournament each player plays every other player exactly once. The winner of each game gets $1$ point and the loser gets $0$ points. If the game is tied, each player gets $0.5$ points. Given a positive integer $m$ , a tournament is said to have property $P(m)$ if the following holds: among every set $S$ of $m$ players, there is one player who won all her games against the other $m-1$ players in $S$ and one player who lost all her games against the other $m - 1$ players in $S$ . For a given integer $m \ge 4$ , determine the minimum value of $n$ (as a function of $m$ ) such that the following holds: in every $n$ -player round robin chess tournament with property $P(m)$ , the final scores of the $n$ players are all distinct.