2020 AMC 12B Problems/Problem 23

Problem 23

How many integers $n \geq 2$ are there such that whenever $z_1, z_2, ..., z_n$ are complex numbers such that

$|z_1| = |z_2| = ... = |z_n| = 1 \text{ and } z_1 + z_2 + ... + z_n = 0,$ then the numbers $z_1, z_2, ..., z_n$ are equally spaced on the unit circle in the complex plane?

$\textbf{(A)}\ 1 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ 3 \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ 5$

Solution

For $n=2$ , we see that if $z_{1}+z_{2}=0$ , then $z_{1}=-z_{2}$ , so they are evenly spaced along the unit circle.

For $n=3$ , WLOG, we can set $z_{1}=1$ . Notice that now $\Re(z_{2}+z{3})=-1$ and $\Im\{z_{2}\}=-\Im\{z_{3}\}$ . This forces $z_{2}$ and $z_{3}$ to be equal to $e^{i\frac{2\pi}{3}}$ and $e^{-i\frac{2\pi}{3}}$ , meaning that all three are equally spaced along the unit circle.

We can now show that we can construct complex numbers when $n\geq 4$ that do not satisfy the conditions in the problem.

Consider $n=4$ . We can set $z_{1}$ and $z_{3}$ arbitrarily and let $z_{2}=-z_{1}$ and $z_{4}=-z_{3}$ . We constructed two pairs of points that cancel each other out. For example, the complex numbers $1,-1,e^{i\frac{\pi}{4}},e^{-i\frac{3\pi}{4}}$

2020 AMC 12B (Problems • Answer Key • Resources)
Preceded by Problem 22	Followed by Problem 24
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All AMC 12 Problems and Solutions

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2020 AMC 12B Problems/Problem 23

Problem 23

Solution

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