Problem 2. In a football tournament, seven teams played: each team played once with each other. In the next round, teams that scored thirteen or more points qualify. Three points are awarded for a win, one point for a draw, and zero points for a loss. What is the maximum number of teams that can advance to the next round?
Answer: 4.

Solution. In total, the teams played $\frac{7 \cdot 6}{2}=21$ games, each of which awarded 2 or 3 points. Therefore, the maximum total number of points that all teams can have is $21 \cdot 3=63$. Hence, the number of teams $n$ advancing to the next stage satisfies the inequality $n \cdot 13 \leqslant 63$, from which $n \leqslant 4$.

On the other hand, an example of a tournament table can be provided where 4 teams advance to the next round:

|  | 1 | 2 | 3 | 4 | 5 | 6 | 7 | Total |
| :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- |
| 1 | $\mathrm{X}$ | 0 | 1 | 3 | 3 | 3 | 3 | 13 |
| 2 | 3 | $\mathrm{X}$ | 0 | 1 | 3 | 3 | 3 | 13 |
| 3 | 1 | 3 | $\mathrm{X}$ | 0 | 3 | 3 | 3 | 13 |
| 4 | 0 | 1 | 3 | $\mathrm{X}$ | 3 | 3 | 3 | 13 |
| 5 | 0 | 0 | 0 | 0 | $\mathrm{X}$ | 1 | 1 | 2 |
| 6 | 0 | 0 | 0 | 0 | 1 | $\mathrm{X}$ | 1 | 2 |
| 7 | 0 | 0 | 0 | 0 | 1 | 1 | $\mathrm{X}$ | 2 |