II Caucasus Mathematical Olympiad, 9.5
Source: II Caucasus Mathematical Olympiad
March 20, 2018
combinatorics
Problem Statement
In a football tournament teams participated, each pair of teams played exactly one game. For the win the team is awarded points, for the draw -- point, for the lose no points awarded. The total number of points of all teams in the tournament is . Prove that there exist teams each having at least one draw.