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# 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 $n\in\mathbb{N}$.

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#### Math Olympiad Program

Math Olympiad is the greatest and most challenging academic contest for school students. Brilliant school students from over 100 countries participate in it every year. Cheenta works with small groups of gifted students through an intense training program. It is a deeply personalized journey toward intellectual prowess and technical sophistication.

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