1. Six students, let's call them $A, B, C, D, E, F$, solved a problem. The problem was solved by two of them. When asked: Who solved the problem?, they gave five answers, stating that the problem was solved by:
1) A and $C$,
2) B and $F$,
3) $F$ and $A$,
4) $B$ and $E$,
5) $D$ and $A$.

In four of these five answers, one part is correct and the other is incorrect, while in one answer both parts are incorrect. Which students solved the problem?
Solution. For the sake of simplified notation for the statement: Student $A$ solved the problem, we will write $A=1$, and the statement: Student $A$ did not solve the problem, we will write $A=0$. Similarly, we write for the statements about other students.

According to the condition of the problem, in one of the five answers, both statements are incorrect, i.e., both letters are equal to 0, while in the remaining four, one is equal to 0 and the other to 1.

1) Let's assume that both statements in the first answer are incorrect, i.e., $A=0$ and $C=0$. Then from the third and fifth answers, we have $F=1$ and $D=1$. Now, from the second answer, we have $B=0$, and from the fourth answer, we have $E=1$. Thus, we have three students who correctly solved the problem ($F, D, E$), which contradicts the condition that only two students solved the problem.
2) Let's assume that both statements in the second answer are incorrect, i.e., $B=0$ and $F=0$. Now, from the third and fourth answers, we get $A=1$ and $E=1$, and from the first answer, it follows that $C=0$, and from the fifth answer, $D=0$. Therefore, the problem was solved by students $A$ and $E$.

All other assumptions lead to a contradiction that only two students solved the problem.

Finally, the problem was correctly solved by students $A$ and $E$.