HMMT 2013 T7

There are $n$ children and $n$ toys such that each child has a strict preference ordering on the toys. We want to distribute the toys: say a distribution $A$ dominates a distribution $B \neq A$ if in $A$, each child receives at least as preferable of an toy as in $B$. Prove that if some distribution is not dominated by any other, then at least one child gets his/her favorite toy in that distribution.