8.5. Ten football teams each played one game against each of the others. 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 points.)
Answer: 13.

Evaluation. Let's prove that $x$ cannot be greater than 13. Indeed, in each match, either 3 points are awarded (if one of the teams wins) or 2 points (if there is a draw). In total, $\frac{10 \cdot 9}{2}=45$ matches were played, meaning no more than 135 points were awarded, that is, the total points scored by all teams is no more than 135. Thus, $10 x \leqslant 135$, which means $x \leqslant 13.5$. Since $x$ is an integer, $x \leqslant 13$.

Example. Let's show that teams could have scored 13 points each. Arrange the teams in a circle and divide them

|  | $\mathbf{1}$ | $\mathbf{2}$ | $\mathbf{3}$ | $\mathbf{4}$ | $\mathbf{5}$ | $\mathbf{6}$ | $\mathbf{7}$ | $\mathbf{8}$ | $\mathbf{9}$ | $\mathbf{1 0}$ |  |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| $\mathbf{A}$ |  | 1 | 3 | 3 | 3 | 3 | $\mathbf{0}$ | $\mathbf{0}$ | $\mathbf{0}$ | $\mathbf{0}$ | $\mathbf{1 3}$ |
| $\mathbf{B}$ | 1 |  | 3 | 3 | 3 | 3 | $\mathbf{0}$ | $\mathbf{0}$ | $\mathbf{0}$ | $\mathbf{0}$ | $\mathbf{1 3}$ |
| $\mathbf{C}$ | $\mathbf{0}$ | 0 |  | $\mathbf{1}$ | 3 | 3 | 3 | 3 | $\mathbf{0}$ | $\mathbf{0}$ | $\mathbf{1 3}$ |
| $\mathbf{D}$ | $\mathbf{0}$ | $\mathbf{0}$ | $\mathbf{1}$ |  | 3 | 3 | 3 | 3 | $\mathbf{0}$ | $\mathbf{0}$ | $\mathbf{1 3}$ |
| $\mathbf{E}$ | $\mathbf{0}$ | $\mathbf{0}$ | $\mathbf{0}$ | 0 |  | $\mathbf{1}$ | 3 | 3 | 3 | 3 | $\mathbf{1 3}$ |
| $\mathbf{F}$ | 0 | 0 | 0 | 0 | 1 |  | 3 | 3 | 3 | 3 | $\mathbf{1 3}$ |
| $\mathbf{G}$ | 3 | 3 | 0 | 0 | 0 | 0 |  | 1 | 3 | 3 | $\mathbf{1 3}$ |
| $\mathbf{H}$ | 3 | 3 | 0 | 0 | 0 | 0 | 1 |  | 3 | 3 | $\mathbf{1 3}$ |
| $\mathbf{I}$ | 3 | 3 | 3 | 3 | 0 | 0 | 0 | 0 |  | $\mathbf{1}$ | $\mathbf{1 3}$ |
| $\mathbf{K}$ | 3 | 3 | 3 | 3 | $\mathbf{0}$ | $\mathbf{0}$ | $\mathbf{0}$ | $\mathbf{0}$ | $\mathbf{1}$ |  | $\mathbf{1 3}$ |

sequentially into 5 pairs. Let the teams of each pair play a draw against each other, each of them win against four teams following the given pair clockwise, and lose to the remaining teams. Then each team scored exactly 13 points.

A similar example can be provided in the form of a table.

+ full justified solution

干 correct answer and example provided

$\mp$ only proved that $x \leqslant 13$.

- only the answer is provided

Note: The symbol "干" in the last part seems to be a typo or a non-standard symbol. It has been left as is in the translation.