16. At the quiz in the Museum of Entertaining Sciences of SFU, 10 schoolchildren are participating. In each round, the students are divided into pairs. Each participant meets every other participant exactly once. A win in a match earns 1 point, a draw earns 0.5 points, and a loss earns 0 points. What is the minimum number of rounds after which an early winner can appear?

## Answers and solutions:

## Translation of the question and answers into English:

16. At the quiz in the Museum of Entertaining Sciences of SFU, 10 schoolchildren are participating. In each round, the students are divided into pairs. Each participant meets every other participant exactly once. A win in a match earns 1 point, a draw earns 0.5 points, and a loss earns 0 points. What is the minimum number of rounds after which an early winner can appear?

## Answers and solutions:
16. 7. Evaluation. After the sixth round, 30 points have been played, and the leader has no more than 6 points, while the other nine participants have collectively scored no less than 24 points. Therefore, among them, there is at least one participant with more than three points. Since there are still 3 rounds ahead, the winner is still unknown.

Example. Suppose in the first 7 rounds, the leader won all their matches, and all other matches ended in draws. Then two students who have not yet faced the leader have 3.5 points each, while the others have 3 points each. Since only 2 rounds remain until the end of the tournament, the winner is already determined.