HMIC 2015 Problem 4

Prove that there exists a positive integer $N$ such that for any positive integer $n \geq N$, there are at least 2015 non-empty subsets $S$ of $\left\{n^{2}+1, n^{2}+2, \ldots, n^{2}+3 n\right\}$ with the property that the product of the elements of $S$ is a perfect square.