2019 AMC 12B Problem 17

Problem 17 of 25IntermediateAlgebra Geometry

How many nonzero complex numbers

z

have the property that

0,

z,

and

z^3,

when represented by points in the complex plane, are the three distinct vertices of an equilateral triangle?

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Solution

The three points form an equilateral triangle iff

|z|=|z^3|=|z^3-z|.

From

|z|=|z^3|=|z|^3

we get

|z|=1.

Then

|z^3-z|=|z|\,|z^2-1|=|z^2-1|,

so we need

|z^2-1|=1.

Writing

z=e^{i\theta},

|z^2-1|=2|\sin\theta|=1,

|\sin\theta|=\dfrac12.

This gives

\theta=30^\circ,150^\circ,210^\circ,330^\circ,

four values of

z,

all yielding distinct vertices. Thus, D is the correct answer.