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Unit Modulus Set Zero Sum Condition

2175ExpertComplex NumbersCombinatorics

SHORTLISTED PROBLEMS FOR THE 62nd NMO

Find all positive integers $n$ for which there exists a set $S \subset \mathbb{C}$ with $n$ elements, such that i) each element of $S$ has modulus $1$, ii) $\sum_{z \in S} z = 0$, and iii) $z + w \neq 0$, for each $z, w \in S$.
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