## Task 23/77

In a football tournament where four teams A, B, C, and D participated, and each team played exactly once against each other, the following standings were recorded after the completion of the tournament:

| Rank | Team | Won | Drawn | Lost | Goals |
| :---: | :---: | :---: | :---: | :---: | :---: |
| 1. | A | 2 | 1 | 0 | $4: 1$ |
| 2. | B | 2 | 0 | 1 | $4: 1$ |
| 3. | C | 0 | 2 | 1 | $1: 2$ |
| 4. | D | 0 | 1 | 2 | $0: 5$ |

Determine the results of the six matches!
The following considerations lead to the solution:

1. Team B has lost one game. Since C and D have not won any games, the winner of this game must have been Team A.

Since B only had 1 goal against, it is certain that the game AB ended with 1:0 or BA with 0:1.

2. Team C has two draws; the opponents in these games can only have been A and D. Since D did not score any goals, the game CD ended with 0:0. It follows that the result of the game AC was 1:1; if it had been 0:0, C would have scored its "goal difference" in the game against B, but B only conceded one goal against in the game against A.
3. The lost game of C was therefore the game against B; due to the goal difference of C, the result was 0:1.
4. The games AD and BD remain, which D lost. From the goal differences, it follows unambiguously: AD 2:0, BD 3:0.

This results in the following table of results:

$$
\begin{array}{lll}
\text { AB 1:0 } & \text { AC 1:1 } & \text { AD } 2: 0 \\
\text { BC 1:0 } & \text { BD } 3: 0 & \text { CD } 0: 0
\end{array}
$$