2. At the Olympiad, for each solved problem, one could receive 3, 8, or 10 points. Vasya scored 45 points. What is the smallest number of problems he could have solved? (It is necessary to explain why he could not have solved fewer problems.)
Answer: 6 problems. Solution. Since 8 and 10 are even numbers, and 45 is an odd number, there must be a problem worth 3 points. It is impossible to score 42 points with 4 problems, even if each problem is worth 10 points. Therefore, a minimum of 5 problems are needed to sum up to 42 points, making the total number of problems 6. An example with 6 problems: $-3+8+8+8+8+10$.

Criteria. Only the answer - 0 points.

An example of when there can be 6 problems - 2 points.

Proven that the number of problems is no less than $6-5$ points.