3. A football invitational tournament involves sixteen cities, each sending two teams, Team A and Team B. According to the competition rules, after several days of matches, it was found that except for Team A of City A, all other teams had played a different number of matches. How many matches has Team B of City A played? Please prove your conclusion.
According to the competition rules, two teams from the same city do not play against each other, so the team with the most matches can play at most 30 matches. Apart from Team A from City A, there are 31 other teams, so the number of matches these teams have played must be from 0 to 30, and all different.

Consider the team with 30 matches, Team C. The teams not from the same city as C must have played at least 1 match against C, so the team from the same city as C can only be the team with 0 matches; Next, consider the team with 29 matches, Team D. The teams not from the same city as D must have played at least 2 matches against C and D, so the team from the same city as D can only be the team with 1 match; ..., similarly, the team from the same city as the team with \(30-k\) matches can only be the team with \(k\) matches. Thus, the team from the same city as the team with 15 matches can only be the team with 15 matches, so the team from City A, Team A, must have played 15 matches, and thus Team B from City A must have played 15 matches.