ILL 1983 Problem 32

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 non-negative integers $n$ and taking real values such that $f(0) = 0$ and $$f(n) \leq an + f([bn]) + f([cn]), \text{ for all } n \geq 1.$$ Prove that if $b + c < 1$ , there is a real number $k$ such that $$f(n) \leq kn \text{ for all } n (1)$$ while if $b + c = 1$ , there is a real number $K$ such that $f(n) \leq K n \log_2 n$ for all $n \geq 2$ . Show that if $b + c = 1$ , there may not be a real number $k$ that satisfies $(1).$