CN-TST 2009 Problem 4

Let positive real numbers $a,b$ satisfy $b - a > 2.$ Prove that for any two distinct integers $m,n$ belonging to $[a,b),$ there always exists non-empty set $S$ consisting of certain integers belonging to $[ab,(a + 1)(b + 1))$ such that $\frac { \prod_{x\in S}}{mn}$ is square of a rational number.