US-TST 2009 Problem 6

Let $N > M > 1$ be fixed integers. There are $N$ people playing in a chess tournament; each pair of players plays each other once, with no draws. It turns out that for each sequence of $M + 1$ distinct players $P_0, P_1, \ldots P_M$ such that $P_{i - 1}$ beat $P_i$ for each $i = 1, \ldots, M$ , player $P_0$ also beat $P_M$ . Prove that the players can be numbered $1,2, \ldots, N$ in such a way that, whenever $a \geq b + M - 1$ , player $a$ beat player $b$ .

Gabriel Carroll.