VN-TST 2002 Problem 3

Let $m$ be a given positive integer which has a prime divisor greater than $\sqrt {2m} +1$ . Find the minimal positive integer $n$ such that there exists a finite set $S$ of distinct positive integers satisfying the following two conditions:

I. $m\leq x\leq n$ for all $x\in S$ ;

II. the product of all elements in $S$ is the square of an integer.