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## Functional equation on positive integers

Let $p$ be a fixed positive integer. Find all functions $f : \Bbb{N} \to \Bbb{N}$ such that $\forall n \in \Bbb{N}$ we have $f(n+1) > \underbrace{f\circ f \circ...\circ f}_{p \text{ times}}(n)$.

$\Bbb{N}=\{0,1,2,3,...\}$.

Answer: $f(n)=n$ for every $n\geq 0$.

1. March 3, 2012 at 3:27 pm

What is $p$?

• March 3, 2012 at 3:32 pm

$p$ is a positive integer.

• March 3, 2012 at 3:37 pm

Is it fixed or for arbitrary $p$? How about $p=1$?

2. March 3, 2012 at 4:27 pm

Yes, it is the same $p$ for all $n$. The particular case $p=1$ was given to an old IMO; I do not know exactly the year. I’ll edit the post of the problem, so that there is no confusion. And I’ll add a solution if you want.