IMC 2001 · Day 2 · P7

Multiply the left hand side polynomials. We obtain the following equalities: $$a_0 b_0 = 1, \quad a_0 b_1 + a_1 b_0 = 1, \quad \dots$$ Among them one can find equations $$a_0 + a_1 b_{s-1} + a_2 b_{s-2} + \dots = 1$$ and $$b_0 + b_1 a_{r-1} + b_2 a_{r-2} + \dots = 1.$$ From these equations it follows that $a_0, b_0 \le 1$. Taking into account that $a_0 b_0 = 1$ we can see that $a_0 = b_0 = 1$.

Now looking at the following equations we notice that all $a$'s must be less than or equal to 1. The same statement holds for the $b$'s. It follows from $a_0 b_1 + a_1 b_0 = 1$ that one of the numbers $a_1, b_1$ equals 0 while the other one must be 1. Follow by induction.