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Found problems: 1

2015 HMIC, 5
Tags: diophantine , cyclotomic field , roots of unity
Let $\omega = e^{2\pi i /5}$ be a primitive fifth root of unity. Prove that there do not exist integers $a, b, c, d, k$ with $k > 1$ such that \[(a + b \omega + c \omega^2 + d \omega^3)^{k}=1+\omega.\] [i]Carl Lian[/i]