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.

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