7. (15 points) Five football teams compete, with each pair of teams playing one match: the winner gets 3 points, the loser gets 0 points, and in the case of a draw, each team gets 1 point. After all the matches are completed, it is found that no team has more than 9 points, and exactly two teams have the same score. Let the scores of the five teams, from highest to lowest, be (two of these letters represent the same number). If is exactly a multiple of 15, how many matches ended in a draw?
Problem 875
Official solution
【Solution】Solution: (matches)
There are a total of 10 matches, with the total score ranging from 20 to 30 points.
The five-digit number is exactly a multiple of 15. Using divisibility rules, can be 0 or 5, considering is the smallest. If the minimum total score is points, it does not hold, so the fifth place must have lost all four matches, accumulating 0 points.
If the fifth place loses four matches, then the maximum number of draws is 6, and the minimum total score is 24 points. Considering the total score is a multiple of 3, the possible total scores are . We will discuss each case separately:
(1) Total score 30 points:
In this case, there are no draws, so the scores of the top four teams can only be points. It is impossible to sum up to 30 with only two repetitions. Therefore, the total score of 30 points does not exist.
(2) Total score 27 points:
After testing, it is found to exist, meeting the score requirements of the problem, and the four teams win 7 matches and lose 3 matches, which exactly satisfies the fifth team's 4 losses, so this is one solution, with 3 draws in the matches.
(3) Total score 24 points:
In the case of 24 points, the top four teams can only each win 1 match and draw 2 matches, but this does not satisfy only two teams having the same score. Therefore, the total score of 24 points does not exist.
In summary, the only existing case is when the total score is 27 points, with a total of 3 draws in the matches.