In chess, the winner receives 1 point and the loser 0 points. In the event of a draw (remis), each player receives $1 / 2$ point.
Fourteen chess players, no two of whom were the same age, participated in a tournament where each played against every other. After the tournament, a ranking list was created. In the case of two players with the same number of points, the younger one received a better ranking.
After the tournament, Jan noticed that the top three players had a total number of points exactly equal to the total number of points of the last nine players. Jörg added that the number of games that ended in a draw was maximized. Determine the number of drawn games.
The total number of points of the last nine players is at least (9*8):2 = 36 points (because if each of the nine played only against another of the nine, then there would already be 36 points in total, as one point is awarded in each match). The total number of points of the top three, however, is at most $13+12+11=36$ points, if they win all their games against the remaining 13 players.
It follows that the last nine players do not win any of their games against the others, and the top three win all their games, although they can play to a draw against each other.
The number of drawn matches (which, according to Jörg, is maximized!) must therefore be $3+1+36=40$.