IMO 2010 Shortlist C5
n ©4 players participated in a tennis tournament. Any two players have played exactly one game, and there was no tie gam...
Category: Combinatorics
Problem
n ©4 players participated in a tennis tournament. Any two players have played exactly one game, and there was no tie game. We call a company of four players bad if one player was defeated by the other three players, and each of these three players won a game and lost another game among themselves. Suppose that there is no bad company in this tournament. Let wi and ℓi be respectively the number of wins and losses of the ith player. Prove that n ¸ i1 pwi ℓiq3 ©0. (1) (South Korea)