Middle European Mathematical Olympiad 2011 Problem T-7

Let $A$ and $B$ be disjoint nonempty sets with $A\cup B=\{1,2,3,\ldots,10\}$. Show that there exist elements $a\in A$ and $b\in B$ such that the number $a^{3}+ab^{2}+b^{3}$ is divisible by $11$.