Question 71
Twenty players took part in a chess tournament. Each player faced every other player exactly once, and each game ended either with a win for one of the players or in a draw. In this tournament it was noticed that, for each game that ended in a draw, each of the other 18 players beat at least one of the two players involved in it. We also know that at least two games ended in a draw. Show that it is possible to name the players $P_{1}, P_{2}, \ldots, P_{20}$ so that player $P_{k}$ beat player $P_{k+1}$ for each $k \in\{1,2,3, \ldots, 19\}$.