5. In the tournament, 15 volleyball teams are playing, and each team plays against all other teams only once. Since there are no draws in volleyball, there is a winner in each match. A team is considered to have performed well if it loses no more than two matches. Find the maximum possible number of teams that performed well.
Answer: 5.

Solution.

Evaluation. If the number of teams that played well is not less than 6, then consider six of them. They could have lost no more than $6 \times 2=12$ matches. But the games between them amounted to $6 \times 5 / 2=15$. Thus, they lost no fewer than 15 matches in total. Contradiction.

Example. Place the captains of 5 teams in a circle facing the center. Let each team lose to the two teams whose captains are to the right of their captain in the circle, and win against the two teams whose captains are to the left, and also win against the remaining 10 teams. The games between the 10 teams can end in any way. The five teams whose captains stand in the circle performed well. The rest lost no fewer than five matches.
Criteria. Any correct solution: 7 points.

The evaluation that 6 or more teams cannot be good: 4 points.

An example for 5 good teams is provided: 2 points.

Only the answer: 0 points.

## Grading Criteria:

| Points | Correctness (incorrectness) of the solution |
| :---: | :---: |
| 7 | Complete correct solution. |
| 6-7 | Correct solution. There are minor flaws that do not significantly affect the solution. |
| 5-6 | The solution is generally correct. However, it contains several errors or the omission of individual cases, but can become correct with minor corrections or additions. |
| 4 | One of the two (more complex) significant cases is correctly considered, or in a problem of the type "evaluation + example," the evaluation is correctly obtained. |
| 2-3 | Auxiliary statements that help in solving the problem are proven, or in a problem of the type "evaluation + example," a correct example is constructed. |
| 1 | Individual important cases are considered in the absence of a solution (or in the case of an incorrect solution). |
| 0 | The solution is incorrect, and there is no progress. |
| 0 | The solution is absent. |