Problem 2. In a football tournament, eight teams played: each team played once with each other. In the next round, teams that scored fifteen 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: 5.

Solution. In total, the teams played $\frac{8 \cdot 7}{2}=28$ games, in each of which 2 or 3 points were at stake. Therefore, the maximum total number of points that all teams can have is $28 \cdot 3=84$. Hence, the number of teams $n$ that advance to the next stage satisfies the inequality $n \cdot 15 \leqslant 84$, from which $n \leqslant 5$.

On the other hand, an example of a tournament table can be provided in which 5 teams qualify for the next round:

|  | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | Sum |
| :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- | :--- |
| 1 | $\mathrm{X}$ | 0 | 0 | 3 | 3 | 3 | 3 | 3 | 15 |
| 2 | 3 | $\mathrm{X}$ | 0 | 0 | 3 | 3 | 3 | 3 | 15 |
| 3 | 3 | 3 | $\mathrm{X}$ | 0 | 0 | 3 | 3 | 3 | 15 |
| 4 | 0 | 3 | 3 | $\mathrm{X}$ | 0 | 3 | 3 | 3 | 15 |
| 5 | 0 | 0 | 3 | 3 | $\mathrm{X}$ | 3 | 3 | 3 | 15 |
| 6 | 0 | 0 | 0 | 0 | 0 | $\mathrm{X}$ | 1 | 1 | 2 |
| 7 | 0 | 0 | 0 | 0 | 0 | 1 | $\mathrm{X}$ | 1 | 2 |
| 8 | 0 | 0 | 0 | 0 | 0 | 1 | 1 | $\mathrm{X}$ | 2 |