2014 USAMO Problems/Problem 6

Problem

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 $\min\{a, b\}>c^n\cdot n^{\frac{n}{2}}.$

Solution

Without loss of generality, let $a < b$ .

Q.E.D.

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2014 USAMO Problems/Problem 6

Problem

Solution