\section*{Problem 4 - 041034}

Of six students from a school who participated in the second stage of the Mathematical Olympiad, two achieved full marks. The students are abbreviated as \(A, B, C, D, E\), and \(F\).

In response to the question of which two students achieved full marks, the following five different answers were given:

}
(1) A and C,
(2) B and F,
(3) F and A,
(4) B and E,
(5) D and A.

We now know that in exactly one answer, both statements are false, while in the other four answers, exactly one statement is true.

Which two students achieved full marks?
}

A good start would be to consider candidate A, as this one appears in 3 statements. If A does not have full marks, then one of these 3 statements must contain both false statements. If A and C are both false, then F and D must have received full marks. This would mean that B and E both came up empty, and thus in statement (4) we would again have 2 incorrect. This does not work.

The same reasoning applies to the case where A and F are both false, and where A and D are both false. A must therefore have received full marks, and C, F, and D did not.

Thus, there is still the statement missing where both statements are incorrect.

According to statement (2), B and F received full marks, and according to statement (4), B and E. B, therefore, cannot have received full marks, as in this case, one statement in both would be correct.

Thus, B did not receive full marks. Then E received full marks.

Therefore, A and E received full marks.