Question

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

$1$

$2$

$3$

$4$

$5$

Answer: B

Concepts:roots of unity complex number counterexample

Difficulty rating: 2100

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 even $n\ge4,$ choose $n/2$ 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 $(n-3)/2$ 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.

2020 AMC 12B Problem 23

Answer: B

Difficulty rating: 2100

Solution:

Problem 23 in Other Years