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Rotating in the complex plane

Source: 2019 AMC 12B 17

February 14, 2019
complex numbersAMC 12B 17

Problem Statement

How many nonzero complex numbers zz have the property that 0,z,0, z, and z3,z^3, when represented by points in the complex plane, are the three distinct vertices of an equilateral triangle?
<spanclass=latexbold>(A)</span>0<spanclass=latexbold>(B)</span>1<spanclass=latexbold>(C)</span>2<spanclass=latexbold>(D)</span>4<spanclass=latexbold>(E)</span>infinitely many<span class='latex-bold'>(A) </span>0\qquad<span class='latex-bold'>(B) </span>1\qquad<span class='latex-bold'>(C) </span>2\qquad<span class='latex-bold'>(D) </span>4\qquad<span class='latex-bold'>(E) </span>\text{infinitely many}