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sequence of functions on N

Source: Serbian Mathematical Olympiad 2007

June 9, 2007

functioninductionalgebra proposedalgebra

Problem Statement

Let

k

be a natural number. For each function

f : \mathbb{N}\to \mathbb{N}

define the sequence of functions

(f_{m})_{m\geq 1}

by

f_{1}= f

and

k

for

m \geq 1

. Function

f

is called

k

-nice if for each

n \in\mathbb{N}: f_{k}(n) = f (n)^{k}

. (a) For which

k

does there exist an injective

k

-nice function

f

? (b) For which

k

does there exist a surjective

k

-nice function

f

?

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