# Tennis Tournament - Olympic training

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.

• Why does this have the projective space tag? – Jorge Fernández Hidalgo Mar 27 '13 at 19:05
• Are the games singles or doubles?? That is are the games 1vs1 or 2vs2 – Jorge Fernández Hidalgo Mar 27 '13 at 19:06
• Because this is an exercise that comes immediately after the definition of projective plane – leticia Mar 27 '13 at 19:11
• Probably you are meant to use the duality of points and lines in the projective plane. But the second condition (what you are calling ii) doesn't really make sense as it is written. – user641 Mar 27 '13 at 19:31
• @Steve: For a player $t$ let $O(t)$ be the set of opponents faced by $t$. Condition (ii) says that if $t_1$ and $t_2$ are distinct members of one team, then $O(t_1)\cup O(t_2)$ is not the entire opposing team. – Brian M. Scott Mar 27 '13 at 20:24