MathDB

\definecolor{A}{RGB}{255,0,0}\color{A}\fbox{A6.} Let

P (x)

be a polynomial with real coefficients such that

\deg P \ge 3

is an odd integer. Let

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

be a function such that \definecolor{A}{RGB}{0,0,200}\color{A}\forall_{x\in\mathbb{R}}\ f(P(x)) = P(f(x)). \definecolor{A}{RGB}{255,150,0}\color{A}\fbox{(a)} Prove that the range of

f

is finite. \definecolor{A}{RGB}{255,150,0}\color{A}\fbox{(b)} Show that for any positive integer

n

, there exist

P

f

that satisfies the above condition and also that the range of

f

has cardinality

n

.
Proposed by [color=#419DAB]ltf0501. [color=#3D9186]#1735

Problem Statement