3. While tidying up the children's room before the guests arrived, mom found 9 socks. Among any four socks, at least two belong to the same owner. And among any five socks, no more than three belong to the same owner. How many children scattered the socks, and how many socks does each child own?

Answer. There are three children, and each owns three socks.
Solution. No child owned more than three socks, as otherwise the condition "among any five socks, no more than three belonged to one owner" would not be met. There are a total of 9 socks, so there are no fewer than three children. On the other hand, among any four socks, there are two socks belonging to one child, so there are fewer than four children. Therefore, there are three children in the family, and each scattered no more than three socks, with a total of 9 socks. This means that each child owns 3 of the socks found by the mother.

Grading Criteria.

- Complete answer with correct explanation - 7 points.
- Justified that there are three children - 5 points.
- Correct reasoning, but the solution is not fully completed - 1-2 points.
- Answer without justification - 0 points.