3. A certain mechanism consists of 25 parts, some of which are large, and some are small. It is known that among any 12 parts taken, there will be at least one small part, and among any 15 parts, there will be at least one large part. How many of each type of part does the mechanism contain?
Answer: 11 large parts and 14 small parts.

Solution. Since among any 12 parts there is a small one, there are no more than 11 large parts. Since among any 15 parts there is a large one, there are no more than 14 small parts. If there were fewer than 11 large parts or fewer than 14 small parts, the total number of parts would be less than 25, but according to the condition, there are 25. Therefore, there are 11 large parts and 14 small parts.

Grading. 13 points for a correct solution. 2 points for a correct answer (without justification). If the correctness of the answer is shown but its uniqueness is not justified, 6 points.