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Polynomials being a power of sth

Source: Iran MO 2023 3rd round , Algebra exam P3

August 21, 2023
algebrapolynomial

Problem Statement

For numbers a,b∈Ra,b \in \mathbb{R} we consider the sets: A={an∣n∈N},B={bn∣n∈N}A=\{a^n | n \in \mathbb{N}\} , B=\{b^n | n \in \mathbb{N}\} Find all a,b>1a,b > 1 for which there exists two real , non-constant polynomials P,QP,Q with positive leading coefficients st for each r∈Rr \in \mathbb{R}: P(r)∈A  ⟺  Q(r)∈B P(r) \in A \iff Q(r) \in B
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