## Task 2 - 110722

Andreas, Birgit, and Claudia played a small chess tournament among themselves. The following is known about it:

(1) Each played the same number of games against each other.

(2) No game ended in a draw (remis).

(3) Andreas won exactly $\frac{2}{3}$ of his games.

(4) Birgit won exactly $\frac{3}{4}$ of her games.

(5) Claudia won exactly one game.

Determine the total number of games played in the tournament!
(6) Since there are exactly 3 ways to select a pair of players to play against each other from the three players, the total number of games, according to (1), is three times the number of games each such pair played against each other.

(7) Twice this number of games and thus $\frac{2}{3}$ of all games in the tournament is the number of games in which each player participated, i.e., each of the 3 players participated in exactly $\frac{2}{3}$ of all games.

Due to (3) and (7), Andreas consequently won exactly $\frac{2}{3} \cdot \frac{2}{3}=\frac{4}{9}$ of all games, and due to (4) and (7), Birgit won exactly $\frac{3}{4} \cdot \frac{2}{3}=\frac{1}{2}$ of all games.

Thus, Andreas and Birgit together won exactly $\frac{17}{18}$ of all games. Therefore, and because according to (2) each game had to be won by exactly one player, Claudia won exactly $\frac{1}{18}$ of all games.

Since this, on the other hand, was exactly one game according to (5), a total of 18 games were played in this tournament.