I have been trying this problem for a while.But somehow, my proof(I tried an inductive approach) appears to be break down at some point.Here it is:
There is a large pile of cards.On each card one of the numbers 1,2,..,n is written.It is known that the sum of all the numbers of all the cards is equal to $kn!$ for some integer $k$.Prove that it is possible to arrange the cards into $k$ stacks so that the sum of the numbers written on the cards in each stack is equal to $n!$.(Tournament of Towns,2002)
You may want to refer to the file of Tournament of the Towns Problems and Solutions. The problem is Problem 4, Fall 2002, Senior A-Level. If you prefer not to look, the proof given is by induction.
There are many very nice "Russian-flavoured" problems on this site, at various levels. A great resource!