2. The participants of the Olympiad left 9 pens in the office. Among any four pens, at least two belong to the same owner. And among any five pens, no more than three belong to the same owner. How many students forgot their pens, and how many pens does each student have?
Answer. There are three students, each owning three pens.

Solution. No student owned more than three pens, as otherwise the condition "among any five pens, no more than three belonged to one owner" would not be met. There are a total of 9 pens, so there are no fewer than 3 students. On the other hand, among any four pens, there are two pens belonging to one student, so there are fewer than 4 students. Therefore, there are three students, and each forgot no more than three pens, with a total of 9 pens. This means each student forgot 3 pens.

Comment. A complete and justified solution - 7 points. Only proved that no student owned more than three pens - 1 point. Only proved that there are no fewer than 3 students - 2 points. Only proved that there are fewer than 4 students - 2 points. Proved that there are three students - 5 points. Provided the answer without justification or with incorrect justifications - 0 points.