Consider a tournament with n players where each player plays against every other player. Suppose each player wins at least once. Show that at least 2 of the players have the same number of wins.
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The number of wins for a player is at least and at the most
These numbers correspond to the pigeonholes to accomodate playesr (pigeons).
Thus, by the Pigeonhole Principle there must be at least two players occupying one pigeonhole (number of wins).
Hence, it is proved.