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Sum of some nth roots of unity is 0

Source: Romanian NMO 2021 grade 10 P3

April 15, 2023
number theoryprime numbers

Problem Statement

Let n≥2n\ge 2 be a positive integer such that the set of nnth roots of unity has less than 2⌊n⌋−12^{\lfloor\sqrt n\rfloor}-1 subsets with the sum 00. Show that nn is a prime number.
Cristi Săvescu
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