IMO 1983 LL GBR32

Origin: GBR

Problem

Let a, b, c be positive real numbers and let [x] denote the greatest integer that does not exceed the real number x. Suppose that f is a function defined on the set of nonnegative integers n and taking real values such that f(0) = 0 and f(n) \leqan + f([bn]) + f([cn]), for all n \geq1. Prove that if b + c < 1, there is a real number k such that f(n) \leqkn for all n, (1) while if b + c = 1, there is a real number K such that f(n) \leqKn log2 n for all n \geq2. Show that if b + c = 1, there may not be a real number k that satisfies (1).