2014 USAMO 第 6 题

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}}.

$$

证明存在一个常数 $c>0$，其性质如下：如果 $a, b, n$ 是正整数，且对于所有 $i, j\in\{0, 1, \ldots n\}$，$\gcd(a+i, b+j)>1$，则

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