Here is another problem I have trying to solve, with no success.
Fifteen teams play in a series. Each team plays one match against each of the other team exactly once. If a team wins they get 2 points, 1 point for a draw and 0 for a loss. When the serie ends, all teams got different scores. The team that came last had 7 points. Show that the team that won played at least a draw.
there are a total of 105 games played in the season (this is 15 choose 2). every match will give 2 points: either 2 to the winner, and none to the loser, or 1 each to both in the case of a draw. this means there are 210 points up for grabs during the season.
since every team had a unique score, and the lowest-ranking team had 7 points for the season, the smallest possible way to distribute the points is:
7,8,9,10,11,12,13,14,15,16,17,18,19,20, and 21. the sum of these numbers is 210, which means these must have been the actual scores of the teams.
so the winning team scored 21 points during the season. this number is odd, which can only happen if they had at least 1 draw (0 = 2 mod 2).