2019 AMC 12B Problems/Problem 17

Problem

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?

$\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }4\qquad\textbf{(E) }\text{infinitely many}$

Solution

Convert $z$ and $z^3$ into $r\text{cis}\theta$ form, giving $z=r\text{cis}\theta$ and $z^3=r^3\text{cis}(3\theta)$ . Since the distance from $0$ to $z$ is $r$ , the distance from $0$ to $z^3$ must also be $r$ , so $r=1$ . Now we must find $\text{cis}(2\theta)=60$ . From $0 < \theta < \pi/2$ , we have $\theta=\frac{\pi}{6}$ and from $\pi/2 < \theta < \pi$ , we see a monotonic decrease of $\text{cis}(2\theta)$ , from $180$ to $0$ . Hence, there are 2 values that work for $0 < \theta < \pi$ . But since the interval $\pi < \theta < 2\pi$ is identical, because $3\theta=\theta$ at pi, we have 4 solutions. There are not infinitely many solutions since the same four solutions are duplicated. $\boxed{D}$

Here's a graph of how the points move as \theta increases: https://www.desmos.com/calculator/xtnpzoqkgs -FlatSquare

Someone pls help with LaTeX formatting, thanks , I did, -Dodgers66

2019 AMC 12B (Problems • Answer Key • Resources)
Preceded by Problem 16	Followed by Problem 18
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2019 AMC 12B Problems/Problem 17

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