## Task 4 - 130714

At a Circle Olympiad for Young Mathematicians, Anneliese, Bertram, Christiane, Detlev, Erich, and Franziska participated in the Olympiad class 7. Exactly two of them received prizes. In response to the question of which two participants they were, the following five answers were given:

(1) Anneliese and Christiane

(2) Bertram and Franziska

(3) Anneliese and Franziska

(4) Bertram and Erich

(5) Anneliese and Detlev.

It later turned out that in exactly one answer, both names were incorrectly given, while in each of the other four answers, exactly one name was correctly given.

What are the names of the two prize winners?
Assuming Anneliese did not receive a prize. Then, since according to the problem, one name in each of the answers (1), (3), (5) would be correct, two of the participants Christiane, Franziska, Detlev would be prize winners. However, this would mean that both statements in (4) and additionally in one of the statements (1), (3), (5) would be false, contradicting the problem.

Therefore, Anneliese is a prize winner. Then, according to the problem, Christiane, Franziska, and Detlev are not prize winners.

Since of the statements (2), (4) exactly one must contain two false names and Bertram appears in both statements, Erich is the other prize winner.

Thus, Anneliese and Erich are the two prize winners.

Solutions of the I. Round 1973 adopted from [5]

### 4.15.2 II. Round 1973, Class 7