RO-TST 1978 Problem 9

A sequence $\left( x_n\right)_{n\ge 0}$ of real numbers satisfies $x_0>1=x_{n+1}\left( x_n-\left\lfloor x_n\right\rfloor\right) ,$ for each $n\ge 1.$ Prove that if $\left( x_n\right)_{n\ge 0}$ is periodic, then $x_0$ is a root of a quadratic equation. Study the converse.