## Task 5 - 330735

Classes 7a, 7b, and 7c held a football tournament. Each class played exactly one match against each other class. At the end, the following table was obtained:

| Class | Goal Ratio | Point Ratio |
| :---: | :---: | :---: |
| $7 \mathrm{~b}$ | $3: 2$ | $3: 1$ |
| $7 \mathrm{a}$ | $3: 1$ | $2: 2$ |
| $7 \mathrm{c}$ | $2: 5$ | $1: 3$ |

Investigate whether these data uniquely determine the number of goals each team scored in each match!

If this is the case, provide all these results!

Hint: As usual, the goal ratio $a: b$ for a team means that the team scored a total of $a$ goals and conceded $b$ goals in all matches.

Furthermore, the team receives 2 plus points for each win, 2 minus points for each loss, and 1 plus and 1 minus point for each draw. These points are added, and the point ratio $c: d$ means that the team has received the sum $c$ of plus points and the sum $d$ of minus points.
Class 7a, due to the point ratio of 2:2, has either played both or none of its two games to a draw. Since their goal ratio of 3:1 shows two different numbers, they cannot have played both games to a draw. Therefore:

Class 7a has not played any game to a draw. (1)

Since classes 7b and 7c have odd numbers in their point ratios but did not play to a draw against 7a according to (1), it follows:

The game between 7b and 7c ended in a draw. (2)

If this game had ended 0:0 or 1:1, then class 7b would have had to achieve their goal ratio of 3:2 such that their game against 7a would have resulted in 3:2 or 2:1. This is not possible, however, because class 7a, as evident from their goal ratio of 3:1, only conceded one goal in total.

Therefore, and because class 7c, due to their goal ratio of 2:5, only scored 2 goals in total, the only possibility for (2) is:

The game between 7b and 7c ended 2:2. (3)

From this, and from the goal ratios of 7b and 7c, it further follows:

The game between 7b and 7a ended 1:0, (4)

The game between 7a and 7c ended 3:0. (5)

Thus, the number of goals scored in each game is uniquely determined by the given information; they are as stated in (3), (4), and (5).