ILL 1990 Problem 73

Let $\mathbb Q$ be the set of all rational numbers and $\mathbb R$ be the set of real numbers. Function $f: \mathbb Q \to \mathbb R$ satisfies the following conditions:

(i) $f(0) = 0$ , and for any nonzero $a \in Q, f(a) > 0.$ (ii) $f(x + y) = f(x)f(y) \forall x,y \in \mathbb Q.$ (iii) $f(x + y) \leq \max\{f(x), f(y)\} \forall x,y \in \mathbb Q , x,y \neq 0.$ Let $x$ be an integer and $f(x) \neq 1$ . Prove that $f(1 + x + x^2+ \cdots + x^n) = 1$ for any positive integer $n.$