TSTST 2011 Problem 8

Let $x_{0}, x_{1}, \ldots, x_{n_{0}-1}$ be integers, and let $d_{1}, d_{2}, \ldots, d_{k}$ be positive integers with $n_{0}=d_{1}>d_{2}>\cdots>d_{k}$ and $\operatorname{gcd}\left(d_{1}, d_{2}, \ldots, d_{k}\right)=1$. For every integer $n \geq n_{0}$, define $$x_{n}=\left\lfloor\frac{x_{n-d_{1}}+x_{n-d_{2}}+\cdots+x_{n-d_{k}}}{k}\right\rfloor .$$ Show that the sequence $\left(x_{n}\right)$ is eventually constant.