A tennis tournament is played between two teams. Each member of a team plays with one or more members of the other team, so that i) Two members of the same team have exactly one opponent in common. ii) No two members of a team facing together all members of the other team.
Prove that each player must play the same number of matches.
Call the members of one team points and the members of the other team lines. If a point plays a line, say that the point is incident with the line and vice versa. Condition (i) says that for any two points, there is exactly one line incident with both points, and for any two lines, there is exactly one point incident with both lines. Condition (ii) says that for any two points there is a line that is not incident with either point, and for any two lines there is a point that is not incident with either line. You can check that you now have a finite projective plane and use whatever theorems you already know about projective planes. The relevant ones can also be found here.