CN-TST 2012 Problem 8

Prove that there exists a positive real number $C$ with the following property: for any integer $n\ge 2$ and any subset $X$ of the set $\{1,2,\ldots,n\}$ such that $|X|\ge 2$ , there exist $x,y,z,w \in X$ (not necessarily distinct) such that $$0<|xy-zw|<C\alpha ^{-4}$$ where $\alpha =\frac{|X|}{n}$ .