Problem 857

32. (GBR 5) 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
$$f(n) \leq a n+f([b n])+f([c n]), \quad \text { for all } n \geq 1 .$$

Prove that if $b+c<1$, there is a real number $k$ such that
$$f(n) \leq k n \quad \text { for all } n,$$
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).