CZ-SK-MO 2001 Problem 6

Let be given natural numbers $a_1,a_2,...,a_n$ and a function $f : Z \to R$ such that $f(x) = 1$ for all integers $x < 0$ and $f(x) = 1- f(x-a_1)f(x-a_2)... f(x-a_n)$ for all integers $x \ge 0$ . Prove that there exist natural numbers $s$ and $t$ such that for all integers $x > s$ it holds that $f(x+t) = f(x)$ .