# Determine all functions $f : \mathbb{N} \rightarrow \mathbb{N}$ such that, for every positive integer $n$, we have: $2n+2001≤f(f(n))+f(n)≤2n+2002$.

Determine all functions $$f : \mathbb{N} \rightarrow \mathbb{N}$$ such that, for every positive integer $$n$$, we have: $$2n+2001≤f(f(n))+f(n)≤2n+2002\,.$$

I don't know where to start as in is there a function that I can get to the solution by slightly modifying it? Any ideas

• Please credit the original source. – Apass.Jack Nov 26 '18 at 3:59

Very old problem from the $$2002$$ Balkan Mathematical Olympiad. You have to keep on iterating $$f(f(f(...(f(n)...)))$$ and see what inequalities you get, see e.g.
How to start: You might start for example by checking what $$f(0)$$ might be. Is $$f(0) = 0$$ possible, and if not, why not? If $$f(0) = 1$$, what can we then say about $$f(1)$$? What can we then say about $$f(2000)$$ or $$f(2001)$$? Get a feeling for the problem. That's how you start.