10.1. In a contest with 5 participants, there were several questions. $N a$ for each question, one of the participants gave a wrong answer, while the others answered correctly. The number of correct answers given by Petya is 10 - this is less than that of any other participant. The number of correct answers given by Vasya is 13 - this is more than that of any other participant. How many questions were there in the contest? Justify your answer.
Solution: For each question, there were four correct answers, which means the total number of correct answers is a multiple of four. The maximum number of correct answers is $13+3 \cdot 12+10=59$, and the minimum is $-13+3 \cdot 11+10=56$. In the range from 56 to 59, there is only one number that is a multiple of 4: the number 56. Therefore, the total number of questions is $56 \div 4 = 14$. A situation with exactly 14 questions is possible. For example, Vasya answered only the first question incorrectly, Petya answered the second, third, fourth, and fifth questions incorrectly, and each of the three remaining boys answered three of their own questions incorrectly (for instance, one on questions №6-8, the second on №9-11, and the third on №12-14).

Answer: 14 questions.

Recommendations for checking:

| included in the work | points |
| :--- | :--- |
| Correct and justified answer | 7 points |
| Not justified that the situation described in the problem is possible | points not reduced |
| In an exhaustive solution, not all cases are considered, but the correct answer and example are obtained | 3 points |
| Established either the boundaries of the possible number of correct answers (numbers 56 and 59) or the divisibility of the number of correct answers by 4 | 2 points |
| Provided a correct example (and gave the correct answer), but the uniqueness of the answer is not justified | 1 point |
| Answer without justification | 0 points |