Question 40
Let $x_1$, $x_2$, $x_3$, $\dots$ be a sequence of nonzero real numbers satisfying
$$
x_n = \frac{x_{n-2} x_{n-1}}{2x_{n-2} - x_{n-1}} \quad \text{for } n = 3, 4, 5, \dots
$$
Find all pairs $(x_1, x_2)$ such that $x_n$ is an integer for infinitely many $n$.
$$
x_n = \frac{x_{n-2} x_{n-1}}{2x_{n-2} - x_{n-1}} \quad \text{for } n = 3, 4, 5, \dots
$$
Find all pairs $(x_1, x_2)$ such that $x_n$ is an integer for infinitely many $n$.