3. A certain mechanism consists of 30 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 20 parts - at least one large part. How many of each type of part does the mechanism contain?
Answer: 11 large parts and 19 small parts.

Solution. Since among any 12 parts there is a small one, there are no more than 11 large parts. Since among any 20 parts there is a large one, there are no more than 19 small parts. If there were fewer than 11 large parts or fewer than 19 small parts, there would be fewer than 30 parts in total, but according to the condition, there are 30. Therefore, there are 11 large parts and 19 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.