USAMO 2014 Problem 6

Prove that there is a constant $c>0$ with the following property: If $a, b, n$ are positive integers such that $\gcd(a+i, b+j)>1$ for all $i, j\in\{0, 1, \ldots n\}$ , then$$\min\{a, b\}>c^n\cdot n^{\frac{n}{2}}.$$