9.4. In the tournament, each of the six teams played against each other exactly once. In the end, the teams scored 12, 10, 9, 8, 7, and 6 points respectively. a) How many points were awarded for a win in a match, if 1 point was awarded for a draw, and 0 points for a loss? The answer, of course, should be a natural number. b) Find the number of wins, draws, and losses for each team and prove the uniqueness of these numbers. c) Provide an example of a corresponding tournament.
Answer. a) 4 points.

b) The first team had three wins, the second team had two wins and two draws, the third team had two wins and one draw, the fourth team had two wins, the fifth team had one win and three draws, the sixth team had one win and two draws. The rest of the matches were lost by the teams.

c) One example that meets the condition is as follows: the first team won against the fourth, fifth, and sixth teams, the second team won against the first and third teams, the third team won against the first and sixth teams, the fourth team won against the second and third teams, the fifth team won against the fourth team, and the sixth team won against the fourth team. In addition, the fifth team drew with the sixth, third, and second teams, and the sixth and second teams also drew with each other.

Solution. a) Let $n$ be the number of points awarded for a win in a match. There were a total of 15 matches played in the tournament, of which $x$ ended in a win for one of the teams, and the remaining $15-x$ ended in draws. In drawn matches, the participants collectively score 2 points, while in the others, they score $n$ points, so the total number of points scored by all teams in the tournament is $n x + 2(15-x) = (n-2)x + 30 = 12 + 10 + 9 + 8 + 7 + 6 = 52$ points, from which $(n-2)x = 22$. There were a total of 15 games, so $x$ can be 1, 2, or 11. In the first two cases, $n$ equals 24 or 13, which exceeds the maximum number of points scored by the teams, which is impossible because a win in a match would then be impossible in principle, meaning all matches would end in draws, which contradicts the assumption about $x$. In the remaining case, $x$ equals 11, then $n$ equals 4, which is the number of points awarded for a win.

b) We will find the number of wins, draws, and losses for each team. There were a total of four draws and 11 matches that ended in a win for one of the teams. Each team played five games, so it was impossible to score 6 or more points solely from draws, meaning the sixth and fifth teams each won once and drew two and three times, respectively.

If the fourth team had won no more than once, it would have had at least four draws, together with the fifth and sixth teams, there would already be at least $(2+3+4)/2 = 4.5$ draws, which exceeds the total of 4. Therefore, the fourth team won twice and lost three times.

For the same reasons, the third team had two wins and one draw, and the second team had two wins and two draws. For all teams except the first, there are at least $(2+3+1+2)/2 = 4$ draws, which equals their total number. Therefore, the first team had three wins and two losses.

c) Let's provide an example of a corresponding tournament. The fifth team drew three times, while the first and fourth teams had no draws. Therefore, the fifth team drew with the sixth, third, and second teams. In addition to these draws, the sixth and second teams each had one draw, so they drew with each other. All draws are thus uniquely determined. However, the decisive matches cannot be uniquely determined. One example that meets the condition is as follows: the first team won against the fourth, fifth, and sixth teams, the second team won against the first and third teams, the third team won against the first and sixth teams, the fourth team won against the second and third teams, the fifth team won against the fourth team, and the sixth team won against the fourth team.

Grading criteria. Correct and justified answer in part a): 3 points. Found the number of wins and draws for each team with justification: 2 points. Provided any valid example of a tournament in part c): 2 points.