IMO Shortlist 2010 problem C5


Kvaliteta:
  Avg: 0.0
Težina:
  Avg: 8.0
Dodao/la: arhiva
June 23, 2013
LaTeX PDF
n \geq 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 w_i and l_i be respectively the number of wins and losses of the i-th player. Prove that

\sum^n_{i=1} \left(w_i - l_i\right)^3 \geq 0.

Proposed by Sung Yun Kim, South Korea
Source: Međunarodna matematička olimpijada, shortlist 2010