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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.

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  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.

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