Question 6
1. In a soccer tournament between four teams, $A$, $B$, $C$ and $D$, each team plays each of the others only once.
a) Decide if it is possible that, at the end of the tournament, the numbers of goals scored and received by the teams are:
| | A | B | C | D |
|-------------------|---|---|---|---|
| Goals scored | 1 | 3 | 6 | 7 |
| Goals received | 4 | 4 | 4 | 5 |
If the answer is affirmative, give an example for the results of the six matches; If not, justify why.
b) Decide if it is possible that, at the end of the tournament, the numbers of goals scored and received by the teams are:
| | A | B | C | D |
|-------------------|---|---|---|---|
| Goals scored | 1 | 3 | 6 | 13 |
| Goals received | 4 | 4 | 4 | 11 |
If the answer is affirmative, give an example for the results of the six matches; If not, justify why.
a) Decide if it is possible that, at the end of the tournament, the numbers of goals scored and received by the teams are:
| | A | B | C | D |
|-------------------|---|---|---|---|
| Goals scored | 1 | 3 | 6 | 7 |
| Goals received | 4 | 4 | 4 | 5 |
If the answer is affirmative, give an example for the results of the six matches; If not, justify why.
b) Decide if it is possible that, at the end of the tournament, the numbers of goals scored and received by the teams are:
| | A | B | C | D |
|-------------------|---|---|---|---|
| Goals scored | 1 | 3 | 6 | 13 |
| Goals received | 4 | 4 | 4 | 11 |
If the answer is affirmative, give an example for the results of the six matches; If not, justify why.