# Task 8.4

Petya and Vasya made 5 shots each in a shooting range. With their first three shots, they scored the same number of points, and with their last three shots, Petya scored three times as many points as Vasya. The holes left on the target were for $10,9,9,8,8,5,4,4,3,2$ points. Where did each of them hit with their third shot? Provide all possible answers and prove that there are no others.

## Number of points 7
Answer:

Petya scored 10 points with the third shot, and Vasya scored 2 points.

Solution. In the last three shots, Vasya could not score more than 9 points (otherwise, Petya would have scored at least 30 points with his last three shots). Vasya also could not score less than 9 points, as the smallest sum for three shots is $2+3+4=9$. Therefore, Vasya scored 2, 3, and 4 points, while Petya scored 10, 9, and 8 points (there are no other ways to score 27 points with three shots). This means that with the first two shots, the boys scored 9, 8, 5, and 4 points. Given that Petya scored at least 8 points with the third shot, and Vasya scored no more than 4 points, and the total points after the first three shots were equal, Petya must have scored at least four points less than Vasya with the first two shots.

The only possibility is that Vasya scored 9 and 8 points, while Petya scored 5 and 4 points. Therefore, with the third shot, Vasya scored 2 points, and Petya scored 10 points.