Task 3. During the break, one of the five students wrote an incorrect equation on the board. When asked by the class teacher who did it, the students gave the following answers.

Anton: "It was Borya or Vova."

Borya: "Neither Dima nor I did this."

Vova: "Anton and Borya are both lying."

Gosha: "One of Anton and Borya is lying, and the other is telling the truth."

Dima: "Gosha is lying."

The class teacher knows that three of them always tell the truth, while the other two always lie. Who wrote the incorrect equation? (20 points)
Answer: Vova.

Solution. Note that one of the two students - Gosha and Dima - is telling the truth, while the other is lying. If Vova is telling the truth, then already three children are lying - Anton, Borya, and one of Gosha and Dima. But according to the condition, there are only two liars, so Vova is lying.

In this case, there are already two liars - Vova and one of Gosha and Dima. Therefore, Anton and Borya must be telling the truth. Then Gosha is lying, and consequently, Dima is telling the truth. We have determined who is who.

It remains to understand who wrote the equation. Borya is telling the truth that neither he nor Dima did it. And since Anton is telling the truth, Vova must have done it.

Note. Generally speaking, from the fact that Vova is lying, it does not follow that Anton and Borya are both telling the truth. Therefore, such a conclusion cannot be made directly without counting the number of liars.

Criteria. Only the correct answer (without proof or with completely incorrect reasoning) - 2 points.

All children are correctly identified - 12 points.

Partial progress:

Noted that Gosha and Dima are a knight and a liar in some order - 3 points.

Proved that Vova is lying - another 3 points.

Proved that Anton and Borya are both telling the truth - another 3 points.

If not all children are correctly identified, but the correct answer to this question is given and then correctly determined who among the children wrote on the board - another 3 points to the criteria for partial progress. All points of the last four criteria are summed.