Putnam 1979 A3

Problem A-3 Let $x_{1}, x_{2}, x_{3}, \ldots$ be a sequence of nonzero real numbers satisfying $$x_{n}=\frac{x_{n-2} x_{n-1}}{2 x_{n-2}-x_{n-1}} \text { for } n=3,4,5, \ldots$$

Establish necessary and sufficient conditions on $x_{1}$ and $x_{2}$ for $x_{n}$ to be an integer for infinitely many values of $n$.