## Task 1 - 160711

At the 3rd stage of the 15th Mathematics Olympiad, six Thälmann Pioneers, Anita, Bernd, Christine, Doris, Erich, and Fritz, each received a prize. Exactly two of them received full marks. In response to the question of which two Pioneers received full marks, the following five answers were given:

(1) Anita and Christine;

(2) Anita and Fritz;

(3) Bernd and Fritz;

(4) Anita and Doris;

(5) Bernd and Erich.

It was subsequently determined that in exactly one of these five answers, both statements are false, while in the other four, one statement is true and one is false.

Based on this finding, what are the names of the two prize winners who received full marks? Check whether this question can be answered uniquely from the given answers!
Assuming Anita did not achieve a full score, then according to the findings about answers (1), (2), (4), exactly two of the students Christine, Fritz, and Doris would have achieved a full score, meaning that in exactly one of the statements (1), (2), (4), both statements would be false.

However, this would contradict the problem, as both statements in (5) would also have to be false. Therefore, Anita must have achieved a full score, and consequently, Christine, Fritz, and Doris did not.

If Bernd had also achieved a full score, then in none of the answers (1) to (5) would both statements be false. Therefore, Bernd did not achieve a full score. This means that apart from Anita, only Erich could have achieved a full score.

Indeed, this answer is consistent with all the conditions of the problem: in (3), both statements are false, and in (1), (2), (4), (5), exactly one statement is false.

Thus, the prize winners with a full score can be uniquely determined from the given information; they are named Anita and Erich.