ISL 1989 Problem 81

A real-valued function $f$ on $\mathbb{Q}$ satisfies the following conditions for arbitrary $\alpha, \beta \in \mathbb{Q}:$ (i) $f(0) = 0,$ (ii) $f(\alpha) > 0 \text{ if } \alpha \neq 0,$ (iii) $f(\alpha \cdot \beta) = f(\alpha)f(\beta),$ (iv) $f(\alpha + \beta) \leq f(\alpha) + f(\beta),$ (v) $f(m) \leq 1989$ $\forall m \in \mathbb{Z}.$ Prove that $$f(\alpha + \beta) = \max\{f(\alpha), f(\beta)\} \text{ if } f(\alpha) \neq f(\beta).$$