2. In volleyball competitions, where there are no ties, 5 teams participate. All teams played against each other. The team that took 1st place won all their matches, and the teams that took 2nd and 3rd place each won exactly two matches. In the case of equal points, the position is determined by the result of the match between the teams. How many victories did the team that took last place achieve? Determine who won against whom.
Solution. Let's denote a team by a point. If team A won against team B, we draw an arrow from A to B.

![](https://cdn.mathpix.com/cropped/2024_05_06_fce95d641fba8fae5984g-1.jpg?height=228&width=217&top_left_y=2488&top_left_x=811)

The number of outgoing arrows equals the number of wins. In total, 10 arrows can be drawn between five points, i.e., 10 matches were played. The first, second, and third teams won a total of 8 matches. This leaves 2 wins for the fourth and fifth teams. Since only the second and third teams won 2 matches each, the fourth and fifth teams won 1 match each. Since in the case of equal points, the higher position is determined by a win in the head-to-head match, the second team beat the third, and the fourth beat the fifth.
![](https://cdn.mathpix.com/cropped/2024_05_06_fce95d641fba8fae5984g-2.jpg?height=210&width=682&top_left_y=934&top_left_x=342)

The third team won 2 matches, so it beat the fourth and fifth teams.

![](https://cdn.mathpix.com/cropped/2024_05_06_fce95d641fba8fae5984g-2.jpg?height=223&width=211&top_left_y=1357&top_left_x=343)

The fifth team won its only match against the second team.

![](https://cdn.mathpix.com/cropped/2024_05_06_fce95d641fba8fae5984g-2.jpg?height=215&width=215&top_left_y=1732&top_left_x=338)

One arrow remains.

![](https://cdn.mathpix.com/cropped/2024_05_06_fce95d641fba8fae5984g-2.jpg?height=211&width=212&top_left_y=2077&top_left_x=314)

Comment. A correct answer alone is worth 3 points. Subsequent points are awarded based on the completeness of the justification of the answer.