8,9 [Examples and counterexamples. Constructions]

Ten football teams each played against each other once. As a result, each team ended up with exactly $x$ points.

What is the greatest possible value of $x$? (Win - 3 points, draw - 1 point, loss - 0.)

#
Evaluation. In each match, either 3 or 2 points are awarded. A total of $10 \cdot 9: 2=45$ matches were played, so no more than 135 points were awarded. Thus, $10 x \leq 135$, which means $x \leq 13.5$. Since $x$ is an integer, $x \leq$ 13.

Example. Arrange the teams in a circle and sequentially divide them into 5 pairs. Let the teams in each pair play to a draw, each of them winning against the four teams following the given pair in a clockwise direction, and losing to the remaining teams. Then each team scored exactly 13 points.

## Answer

13.