Question

How many integers $n \ge 2$ are there such that whenever $z_1, z_2, \ldots, z_n$ are complex numbers such that $|z_1| = |z_2| = \cdots = |z_n| = 1 \quad\text{and}\quad z_1 + z_2 + \cdots + z_n = 0,$ then the numbers $z_1, z_2, \ldots, z_n$ are equally spaced on the unit circle in the complex plane?

$1$

$2$

$3$

$4$

$5$

Solution:

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 $n \ge 4,$ a counterexample exists. For instance, take an antipodal pair $\{1, -1\}$ together with any other balanced set (for $n = 4,$ use two antipodal pairs at different angles; for $n = 5,$ use an equilateral triangle plus an antipodal pair). These sum to zero but are not equally spaced.

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

Thus, the correct answer is B.

2020 AMC 12B Problem 23

Solution:

Problem 23 in Other Years