2020 AMC 12B 第 23 题

Check small cases: $n = 2$ forces antipodal points and $n = 3$ forces an equilateral triangle

For $n \ge 4,$ build a counterexample by combining a smaller balanced configuration with an antipodal pair

For $n = 2,$ $z_1 + z_2 = 0$ forces $z_2 = -z_1,$ which is equally spaced. For $n = 3,$ three unit vectors summing to zero must form an equilateral triangle, so they are equally spaced.

For every even $n\ge4,$ choose $\frac{n}{2}$ antipodal pairs at generic angles that do not form a regular $n$-gon. For every odd $n\ge5,$ choose the vertices of an equilateral triangle together with $\frac{n-3}{2}$ generic antipodal pairs. Each construction has sum $0$ but is not equally spaced.

Hence only $n = 2$ and $n = 3$ work, giving $2$ values.

Thus, the correct answer is B.