8. In a commercial football tournament, five teams participated. Each was supposed to play exactly one match against each other. Due to financial difficulties, the organizers canceled some games. In the end, it turned out that all teams had scored a different number of points, and no team had a zero in the points scored column. What is the minimum number of games that could have been played in the tournament, if three points were awarded for a win, one for a draw, and zero for a loss?
Answer: 6.

Solution. The minimum possible total score: $1+2+3+4+5=15$. In one game, a maximum of 3 points (in total) can be scored. Therefore, there were at least 5 games. However, if there were exactly 5 games, then all games would have to end with one of the teams winning, and then no team would have scored exactly 1 point.

Thus, there were at least 6 games. An example of the tournament results can be given by the table.

| $*$ | 1 | 2 | 3 | 4 | 5 | $\sum$ |
| :---: | :---: | :---: | :---: | :---: | :---: | :---: |
| 1 | $*$ | $*$ | $*$ | 3 | 3 | 6 |
| 2 | $*$ | $*$ | $*$ | $*$ | 3 | 3 |
| 3 | $*$ | $*$ | $*$ | 1 | 3 | 4 |
| 4 | 0 | $*$ | 1 | $*$ | 1 | 2 |
| 5 | 0 | 0 | 0 | 1 | $*$ | 1 |