IMC 2001 · Problem 1

Let $r, s \ge 1$ be integers and $a_0, a_1, \dots, a_{r-1}, b_0, b_1, \dots, b_{s-1}$ be real non-negative numbers such that

$$(a_0 + a_1 x + a_2 x^2 + \dots + a_{r-1}x^{r-1} + x^r)(b_0 + b_1 x + b_2 x^2 + \dots + b_{s-1}x^{s-1} + x^s) =$$ $$= 1 + x + x^2 + \dots + x^{r+s-1} + x^{r+s}.$$

Prove that each $a_i$ and each $b_j$ equals either $0$ or $1$.