Binkov A, A:

In a football championship, 16 teams participated. Each team played against each of the others once, with 3 points awarded for a win, 1 point for a draw, and 0 points for a loss. We will call a team successful if it scored at least half of the maximum possible number of points. What is the maximum number of successful teams that could have been in the tournament?
Each team played 15 games and therefore could have earned a maximum of $15 \cdot 3=45$ points. Thus, a team is successful if it has at least 23 points.

But one of the teams scored no more than the average possible number of points. Even if all matches were successful, the average is

$15 \cdot 1.5=22.5$

We will show that there could have been 15 successful teams in the championship. Number the teams. Let team number 16 lose to all the others. Arrange the numbers of the other teams (numbers from 1 to 15) in a circle. Let each of these teams win against the next seven teams in the circle (and lose to the rest). Then 15 teams will win 8 games each and score 24 points each.

## Answer

15 teams.