NL-NWO 2009 Problem 3

A tennis tournament has at least three participants. Every participant plays exactly one match against every other participant. Moreover, every participant wins at least one of the matches he plays. (Draws do not occur in tennis matches.) Show that there are three participants $A, B$ and $C$ for which the following holds: $A$ wins against $B, B$ wins against $C$ , and $C$ wins against $A$ .