MathDB
Misplaced Complex Numbers

Source: 2020 AMC 12B #23

February 6, 2020
AMCAMC 12AMC 12 Bcomplex numbers2020 AMC 12B2020 AMC

Problem Statement

How many integers n2n \geq 2 are there such that whenever z1,z2,...,znz_1, z_2, ..., z_n are complex numbers such that z1=z2=...=zn=1 and z1+z2+...+zn=0,|z_1| = |z_2| = ... = |z_n| = 1 \text{ and } z_1 + z_2 + ... + z_n = 0, then the numbers z1,z2,...,znz_1, z_2, ..., z_n are equally spaced on the unit circle in the complex plane?
<spanclass=latexbold>(A)</span> 1<spanclass=latexbold>(B)</span> 2<spanclass=latexbold>(C)</span> 3<spanclass=latexbold>(D)</span> 4<spanclass=latexbold>(E)</span> 5<span class='latex-bold'>(A)</span>\ 1 \qquad<span class='latex-bold'>(B)</span>\ 2 \qquad<span class='latex-bold'>(C)</span>\ 3 \qquad<span class='latex-bold'>(D)</span>\ 4 \qquad<span class='latex-bold'>(E)</span>\ 5