MathDB

A6

Part of 2020-IMOC

Problems(1)

f(P(x)) = P(f(x)) - range of polynomial and f

Source: A6 IMOC 2020

9/5/2020
\definecolor{A}{RGB}{255,0,0}\color{A}\fbox{A6.} Let P(x) P (x) be a polynomial with real coefficients such that deg⁡P≥3\deg P \ge 3 is an odd integer. Let f:R→Zf : \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 ff is finite. \definecolor{A}{RGB}{255,150,0}\color{A}\fbox{(b)} Show that for any positive integer nn, there exist PP, ff that satisfies the above condition and also that the range of ff has cardinality nn.
Proposed by [color=#419DAB]ltf0501. [color=#3D9186]#1735
algebrapolynomialIMOC