At a chess tournament the winner gets 1 point and the defeated one 0 points. A tie makes both obtaining 1/2 points. 14 players, none of them equally aged, participated in a competition where everybody played against all the other players. After the competition a ranking was carried out. Of the two players with the same number of points the younger received the better ranking. After the competition Jan realizes that the best three players together got as many points as the last 9 players obtained points together. And Joerg noted that the number of ties was maximal. Determine the number of ties.
There are 14*13/2 = 91 games, so 91 points.
The players fall into three groups: the top 3, the bottom 9, and the middle 2.
The bottom 9 play 9*8/2 = 36 games against each other, all ties, for 36 points. So they have at least 36 points.
The top 3 play 3 games against each other (all ties) for 3 points, and 3*11 = 33 games against others, so they have at most 36 points.
Since the maximum score for the top 3 and the minimum for the bottom 9 is 36, and since they are equal, we know that the top 3 won all their matches against other groups and the bottom 3 lost all their match against the other groups.
The middle 2 play 1 tie match in their group, lose 6 matches against the top 3, and win 18 matches against the bottom 9, for a total of 19 points.
The only ties are within the groups, so the maximal number of ties are 3 + 1 + 36 = 40 ties.
Final answer.
The top 3 players finish with 12 points each, the middle 2 have 9.5 points each, and the bottom 9 have 4 points each.
Checkmate (spoiler)
