## Task 1 - 130831

Anja, Brigitte, Cathrin, Daja, and Eva played several four-player games among themselves. In each game, there was one winner and three losers. Each of the girls played the same number of times. After all the games were completed, it was observed:

(1) Cathrin won exactly half, Daja exactly one third, and Eva exactly one quarter of the games they participated in.

(2) The number of wins of the girl who achieved the third-best result was a prime number.

(3) None of the girls lost all the games.

Determine the exact number of games that were played and state how many games each girl won in total!
Let $z$ be the total number of games. Since each girl participated in exactly 4 out of every 5 games, each girl played in $\frac{4}{5} z$ of all games. This number is divisible by 3 and 4, and thus by 12. Therefore, there exists a natural number $n$ such that $\frac{4}{5} z = 12 n$; from this, it follows that $z = 15 n$.

Out of the 15n games, Cathrin won exactly 6n, Daja exactly 4n, and Eva exactly 3n games.

Thus, Anja and Brigitte together won exactly $2 n$ of all games. This implies that Eva, due to $6 n > 4 n > 3 n > 2 n$, achieved the third-best result. Since the number $3 n$ of Eva's wins was a prime number according to (2), it follows that $n = 1$.

Therefore, exactly 15 games were played; Cathrin won exactly 6, Daja exactly 4, Eva exactly 3 of these games, and Anja and Brigitte each won exactly 1 game according to (3).