A short while ago I came across this quite famous Russian problem which has a very intriguing result, but I myself have made little headway on it thus far:
In a chess tournament there are \( 2n+3 \) competitors. Every competitor plays every other competitor precisely once. No 2 matches can be played simultaneously, and after a competitor plays a match he can not play in any of the next n matches. Show that one of the competitors who plays in the opening match will also play in the closing match.