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15 teams in a soccer league

Source: OIFMAT I 2010 day 2 p7 - Chilean Math Forum FMAT Olympiad https://artofproblemsolving.com/community/c2484778_oifmat

September 25, 2021
combinatorics

Problem Statement

15 15 teams participate in a soccer league. Each team plays each of the remaining teams exactly once. If a team beats another team in a match they receive 3 3 points, while the loser receives 1 1 point. In the event of a tie, both teams receive 2 2 points. When all possible league matches are held, the following can be observed:
∙\bullet No two teams have finished with the same amount of points. ∙\bullet Each team finished the league with at least 21 21 points.
Let WW be the team that finished the league with the highest score. Determine how many points WW scored and show that there were at least four ties in the league.
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