Question 110
A sequence $(x_n)$ satisfies the following conditions: $x_1 = a$, $x_{n+1} = \frac{1}{2}\left(x_n - \frac{1}{x_n}\right)$, $n \in \mathbb{N}$. Prove that there exists a number $a$ such that the sequence $(x_n)$ has exactly 2018 pairwise distinct elements.
(If one of the elements of the sequence equals 0, then the sequence stops on that element.)
(If one of the elements of the sequence equals 0, then the sequence stops on that element.)