MathDB

P

is a monic polynomial of odd degree greater than one such that there exists a function

f : \mathbb{R} \rightarrow \mathbb{N}

such that for each

x \in \mathbb{R}

f(P(x))=P(f(x))

(a) Prove that there are a finite number of natural numbers in range of

f

. (b) Prove that if

f

is not constant then the equation

P(x)-x=0

has at least two real solutions. (c) For each natural

n>1

prove that there exists a function

f : \mathbb{R} \rightarrow \mathbb{N}

and a monic polynomial of odd degree greater than one

P

such that for each

x \in \mathbb{R}

f(P(x))=P(f(x))

and range of

f

contains exactly

n

different numbers.
Time allowed for this problem was 105 minutes.

Problem Statement