Six teams take part in a football tournament. Each team plays exactly one game with any other team. A team receives 3 points for a win, 1 point for a draw, and 0 point for a loss. After the tournament is over, the teams have 10, 9, 6, 6, 4, and 2 points.
a) Prove that the team taking the second place (i.e. having 9 points) does not lose the game with the team winning the first place (i.e. having 10 points).
b) Is it possible uniquely to determine the result of the game between the teams taking the second and the first places?
Solution
The following tables contain all possible results of the tournament.
Wins
0
0
0
0
0
1
1
1
1
1
Draws
0
1
2
3
4
5
0
1
2
3
Losses
5
4
3
2
1
0
4
3
2
1
Points
0
1
2
3
4
5
3
4
5
6
Wins
2
2
2
2
3
3
3
4
4
5
Draws
0
1
2
3
0
1
2
0
1
0
Losses
3
2
1
0
2
1
0
1
0
0
Points
6
7
8
9
9
10
11
12
13
15
We see that the first team getting 10 points and having the first place wins 3 games, loses 1 game, and ends 1 game in a draw; so this team has 4 effective games. If the second team getting 9 points and having the second place loses the game with the first team, then it wins 3 games and loses 2 games, i.e. 5 of its games are effective. Therefore the total number of effective games of the first and the second teams is equal to 9, but only one of these games is common. So there are at least 8 effective games in the tournament, which contradicts the equality x=7. Hence, the second team does not lose the game with the first team.
b) There are tournaments with distinct results (see the tables).
Nº1
Nº2
Nº3
Nº4
Nº5
Nº6
Σ
Nº1
•
0
3
3
3
1
10
Nº2
3
•
1
1
1
3
9
Nº3
0
1
•
1
1
3
6
Nº4
0
1
1
•
1
3
6
Nº5
0
1
1
1
•
1
4
Nº6
1
0
0
0
1
•
2
Table 1
Nº1
Nº2
Nº3
Nº4
Nº5
Nº6
Σ
Nº1
•
1
0
3
3
3
10
Nº2
1
•
3
1
1
3
9
Nº3
3
0
•
1
1
1
6
Nº4
0
1
1
•
1
3
6
Nº5
0
1
1
1
•
1
4
Nº6
0
0
1
0
1
•
2
Table 2
Want a route through all this instead of an archive? The track
puts 2,604 problems in a working order, from Junior Challenge level to the IMO shortlist.
Source: MathNet,
licensed CC-BY-4.0.
Statement reproduced verbatim; metadata (topic, difficulty) added by this project.