CN-TST 2012 Problem 3

Let $x_n=\binom{2n}{n}$ for all $n\in\mathbb{Z}^+$ . Prove there exist infinitely many finite sets $A,B$ of positive integers, satisfying $A \cap B = \emptyset$ , and $$\frac{{\prod\limits_{i \in A} {{x_i}} }}{{\prod\limits_{j\in B}{{x_j}} }}=2012.$$