I came across an interesting question in a competition (it is long over by now, but I haven't found a solution yet):
In a chess tournament $2^n$ ($n \in \mathbb{N}$) players play each other round-robin (everybody against everybody). If a game draws it is repeated until one of the player wins. Show that it is possible to pick a sequence of $n+1$ players in which each player won over his predecessors.
Obviously this is true for $n = 1$. But I'm stuck even to show this for any higher number.