Select Page

# Understand the problem

Find all functions $f : \mathbb{N}\rightarrow \mathbb{N}$ satisfying following condition:
$$f(n+1)>f(f(n)), \quad \forall n \in \mathbb{N}.$$

##### Source of the problem

IMO longlist 1977

##### Topic
Functional equations
Medium
##### Suggested Book
Functional Equations by B J Venkatachala

Do you really need a hint? Try it first!

Prove (by induction on $m$) that if $n\ge m$ then $f(n)\ge m$.
Note that hint 1 implies that $f(n+1)>f(n)$.
Note that hint 2 implies that $n+1>f(n)$, i.e. $f(n)\le n$.
Hint 3, combined with hint 1 gives $f(n)=n$ for all $n\in\mathbb{N}$.