Problem 9.7. In a school chess tournament, 4 people participated: Andrey, Vanya, Dima, and Sasha. Each played twice against each of their opponents. In each game, 1 point was awarded for a win, 0.5 points for a draw, and 0 points for a loss.

It is known that at the end of the tournament:

- all the participants scored a different number of points;
- Andrey took first place, Dima took second, Vanya took third, and Sasha took fourth;
- Andrey won as many games as Sasha did.

How many points did each of the participants score?
Answer: Andrey scored 4 points, Vanya - 2.5, Dima - 3.5, Sasha - 2.

Solution. In each game, two participants together receive 1 point. It is clear that there were a total of $6 \cdot 2=12$ games, so the total points accumulated by all participants is exactly 12. It is also clear that the final number of points for each participant is either an integer or an integer increased by 0.5.

If Sasha had scored at least 2.5 points, then Vanya would have scored at least 3 points, Dima at least 3.5 points, and Andrey at least 4 points. Then together they would have scored at least $2.5+3+3.5+4=13$ points, which is a contradiction. Therefore, Sasha scored no more than 2 points. Consequently, he won no more than 2 games, and thus Andrey also won no more than 2 games.

Andrey played a total of 6 games, so he scored no more than $2 \cdot 1+4 \cdot 0.5=4$ points. If Andrey had scored no more than 3.5 points, then Dima would have scored no more than 3 points, Vanya no more than 2.5 points, and Sasha no more than 2 points. Then together they would have scored no more than $3.5+3+2.5+2=11$ points, which is a contradiction. Therefore, Andrey scored exactly 4 points, which is possible only if he won 2 times and drew 4 times. Then Sasha also won 2 times, and he scored no more than 2 points. Therefore, Sasha scored exactly 2 points.

Dima and Vanya together scored the remaining exactly $12-4-2=6$ points, with Vanya scoring at least 2.5 points and Dima no more than 3.5 points. Since they scored different numbers of points and Vanya scored fewer points than Dima, Vanya scored exactly 2.5 points, and Dima exactly 3.5 points.

Let's provide an example of how the described tournament situation could have occurred. Suppose Andrey won twice against Sasha, drew twice with Vanya, and drew twice with Dima. Sasha, on the other hand, won twice against Vanya and lost twice to Dima. Finally, Dima lost once to Vanya and drew once with him. It is easy to see that all the conditions of the problem are satisfied.