At the tournament, teams Akouska, Bovenska, Colska, and Demecka met. Each team played against each other exactly once. The winning team received three points, the losing team received no points, and in the event of a draw, each of the drawing teams received one point. After all six matches were played, Akouska had 7 points, Bovenska had 4 points, Colska had 3 points, and Demecka had 2 points.

a) How many matches ended in a draw?

b) What was the result of the match between Bovenska and Colska?

(J. Tkadlec)
The team names are abbreviated to A, B, C, and D. Each team played three matches against the remaining teams, and could earn 0, 1, or 3 points per match. The distribution of points for teams A, B, and D is therefore uniquely determined, while for team C there are two possibilities:

$$
7=3+3+1, \quad 4=3+1+0, \quad 3=3+0+0=1+1+1, \quad 2=1+1+0 .
$$

One point is always awarded to two teams, so the total number of ones in the above decompositions must be even. In the decompositions of 7, 3, and 2 (i.e., the points of teams A, B, and D), there are a total of four ones. Therefore, team C must have earned its 3 points as follows:

$$
3=3+0+0 \text{. }
$$

From this, it also follows that two matches ended in a draw.

Team D drew twice, and teams A and B each drew once. Therefore, team D drew with teams A and B.

Team A drew once and won twice. Since it drew with team D, it must have won against teams B and C.

Team B lost once, drew once, and won once. Since it lost to team A and drew with team D, it must have won against team C.

Note. From the above, we can construct a table of all matches:

|  | A | B | C | D | total |
| :---: | :---: | :---: | :---: | :---: | :---: |
| A | - | 3 | 3 | 1 | 7 |
| B | 0 | - | 3 | 1 | 4 |
| C | 0 | 0 | - | 3 | 3 |
| D | 1 | 1 | 0 | - | 2 |

The answer to question a) can also be derived as follows: In total, 16 points were awarded in the tournament: \(7 + 4 + 3 + 2 = 16\). If no match had ended in a draw, a total of \(6 \times 3 = 18\) points would have been awarded. Each draw contributes two points to the total (i.e., one point less than a win), so two matches ended in a draw.

Evaluation. 2 points for the answer to each of the questions a) and b); 2 points for the quality of the commentary.