KMO 2017 Problem 4

Let $f: \mathbb{R} \rightarrow \mathbb{R}$ be the function as $$f(x) = \begin{cases} \frac{1}{x-1}& (x > 1) 1& (x=1) \frac{x}{1-x} & (x<1) \end{cases}$$ Let $x_1$ be a positive irrational number which is a zero of a quadratic polynomial with integer coefficients. For every positive integer $n$ , let $x_{n+1} = f(x_n)$ . Prove that there exists different positive integers $k$ and $\ell$ such that $x_k = x_\ell$ .