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Found problems: 2

2024 TASIMO, 5
Tags: algebraic number theory , algebra , functional equation
Find all functions $f: \mathbb{Z^+} \to \mathbb{Z^+}$ such that for all integers $a, b, c$ we have $$ af(bc)+bf(ac)+cf(ab)=(a+b+c)f(ab+bc+ac). $$ [i]Note. The set $\mathbb{Z^+}$ refers to the set of positive integers.[/i] [i]Proposed by Mojtaba Zare, Iran[/i]

STEMS 2021 Math Cat C, Q2
Tags: algebraic number theory , number theory
Does there exist a nonzero algebraic number $\alpha$ with $|\alpha| \neq 1$ such that there exists infinitely many positive integers $n$ for which there's $\beta_n \in \mathbb{C}$ with $\beta_n \in \mathbb{Q}(\alpha)$ and $\beta_n^n = \alpha$?