TSTST 2011 Problem 9

Let $n$ be a positive integer. Suppose we are given $2^{n}+1$ distinct sets, each containing finitely many objects. Place each set into one of two categories, the red sets and the blue sets, so that there is at least one set in each category. We define the symmetric difference of two sets as the set of objects belonging to exactly one of the two sets. Prove that there are at least $2^{n}$ different sets which can be obtained as the symmetric difference of a red set and a blue set.