Problem

Source: Chinese TST 2009 P4

Tags: combinatorics proposed, combinatorics



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 {\displaystyle\prod_{x\in S}}{mn}$ is square of a rational number.