IMO 1989 LL KOR60

Origin: KOR

Problem

A real-valued function f on Q satisfies the following conditions for arbitrary \alpha, \beta \inQ: (i) f(0) = 0, (ii) f(\alpha) > 0 if \alpha ̸= 0, (iii) f(\alpha\beta) = f(\alpha)f(\beta), (iv) f(\alpha + \beta) \leqf(\alpha) + f(\beta), (v) f(m) \leq1989 for all m \inZ. Prove that f(\alpha + \beta) = max{f(\alpha), f(\beta)} if f(\alpha) ̸= f(\beta). Here, Z, Q denote the sets of integers and rational numbers, respectively.