Question 1
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 $z, w \in S$.
i) each element of $S$ has modulus $1$,
ii) $\sum_{z \in S} z = 0$, and
iii) $z + w \neq 0$, for each $z, w \in S$.