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

Question

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

Answer 1

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.

Answer 2

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.

https://artofproblemsolving.com/community/c6h76767