Problem 2. In a school tic-tac-toe tournament, 16 students participated, each playing one game against every other student. A win was worth 5 points, a draw -2 points, and a loss -0 points. After the tournament, it was found that the participants collectively scored 550 points. What is the maximum number of participants who could have never played a draw in this tournament?

## Answer:
Solution. A total of $\frac{16 \cdot 15}{2}=120$ games were played in the tournament. In each game, either 5 points were awarded (in case of a win-loss) or 4 points (in case of a draw). If all games had ended in a draw, the total number of points for all participants would have been $120 \cdot 4=480$, which is 70 points less than the actual total points of all participants. In a non-draw game, the two participants collectively receive 1 point more than in a draw game. This means that exactly 70 games ended with one participant winning, and the remaining 50 games ended in a draw.

Suppose that at least 6 participants never played a draw. Then the draw games could only have been played between the remaining 10 participants, and they played a total of $\frac{10 \cdot 9}{2}=45$ games, which is less than 50. This is a contradiction. Therefore, no more than 5 participants never played a draw.

It is not difficult to describe an example for 5 participants. Fix 11 participants, they played $\frac{11 \cdot 10}{2}=55$ games among themselves. Choose any 50 of these games, let them be draws (it is clear then that each of the fixed 11 participants will play at least one draw), and all other games in the tournament ended with one participant winning. Therefore, $16-11=5$ people never played a draw. It is clear that all conditions of the problem are satisfied.

## Criteria

## The following criteria are cumulative:

2 6. A correct example is described where exactly 5 people never played a draw.

5 p. It is proven that no more than 5 people never played a draw.

If the proof that no more than 5 people never played a draw is missing:

2 p. It is proven that there were exactly 50 draws (or that there were exactly 70 decisive games).