Zaslavsky A.A.

In a volleyball tournament, each team met every other team once. Each match consisted of several sets - until one of the teams won three sets. If a match ended with a score of $3: 0$ or $3: 1$, the winning team received 3 points, and the losing team received 0. If the set score was

$3: 2$, the winner received 2 points, and the loser received 1 point. By the end of the tournament, it turned out that the team "Tricksters" scored the most points, while the team "Simples" scored the least. However, the "Tricksters" won fewer matches than they lost, while the "Simples" had more wins than losses. What is the smallest number of teams for which this is possible?
Evaluation. If there are no more than three teams, then the "Simpletons" won all the matches, which means they have the most points. Contradiction.

If there are four or five teams, then each team will play three or four matches. This means that the "Cunning" won no more than one match and scored a maximum of $5=3+1+1$ or $6=3+1+1+1$ points, respectively. The "Simpletons" won at least two matches with four participants and at least three matches with five participants, that is, they scored at least $4=2+2+0$ or $6=2+2+2+0$ points, respectively. Considering that the "Cunning" must score at least 2 points more than the "Simpletons," we get a contradiction.

Therefore, there are no fewer than six teams.

Example. See the table.

| Teams | $\mathbf{1}$ | $\mathbf{2}$ | $\mathbf{3}$ | $\mathbf{4}$ | $\mathbf{5}$ | $\mathbf{6}$ | Points |
| :--- | :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| "Cunning" | $\bullet$ | 1 | 1 | 1 | 3 | 3 | $\mathbf{9}$ |
| A | 2 | $\bullet$ | 2 | 2 | 1 | 1 | $\mathbf{8}$ |
| $\mathbf{\text { B }}$ | 2 | 1 | $\bullet$ | 2 | 2 | 1 | $\mathbf{8}$ |
| $\mathbf{B}$ | 2 | 1 | 1 | $\bullet$ | 2 | 1 | 7 |
| $\Gamma$ | 0 | 2 | 1 | 1 | $\bullet$ | 3 | 7 |
| "Simpletons" | 0 | 2 | 2 | 2 | 0 | $\bullet$ | $\mathbf{6}$ |

## Answer

With six teams.