1. In a tournament, each team plays exactly one match against each other team. A won match earns the winner 3 points and 0 for the loser. In case of a draw, both teams receive 1 point each. At the end of the tournament, it turns out that all teams together have scored 15 points. The last team on the final list has scored exactly 1 point. The team that is second to last has not lost any match.

a) How many teams participated in the tournament?

b) How many points did the team that stands in second place on the final ranking score?
1. a) In every match of the tournament, 2 or 3 points are scored. In a tournament with four teams, $\left(\begin{array}{l}4 \\ 2\end{array}\right)$, so six matches are played, resulting in a total of 12 to 18 points scored. For three teams, the maximum is $\left(\begin{array}{l}3 \\ 2\end{array}\right) \times 3$, thus 9 points. For five teams, the minimum is $\left(\begin{array}{l}5 \\ 2\end{array}\right) \times 2=20$.

b) Order the teams A, B, C, D from highest to lowest: then team D has lost two matches and drawn one. Team C has not lost any matches. Assume that C only drew matches. Then the competition results would look like those in schema 1.

There are still 3 points to be distributed in the two remaining boxes. Then A must have won against B, and team B has scored a total of 4 points.

If team C has drawn twice and won once, then C would have already scored 5 points, see schema 2. B must have won against A to rank above C, which is contradictory to the ranking (and also to the total points scored).

The answer is thus 4 points for B.

![](https://cdn.mathpix.com/cropped/2024_04_17_29d173b1c3c8d37aa9abg-1.jpg?height=303&width=1245&top_left_y=1182&top_left_x=331)