2023 AMC 12A Problems/Problem 14: Difference between revisions

Revision as of 22:50, 9 November 2023

Problem

How many complex numbers satisfy the equation $z^5=\overline{z}$ , where $\overline{z}$ is the conjugate of the complex number $z$ ?

$\textbf{(A)} ~2\qquad\textbf{(B)} ~3\qquad\textbf{(C)} ~5\qquad\textbf{(D)} ~6\qquad\textbf{(E)} ~7$

Solution 1

When $z^5=\overline{z}$ , there are two conditions: either $z=0$ or $z\neq 0$ . When $z\neq 0$ , since $z^5=\overline{z}$ , $|z|=1$ . $z^5\cdot z=z^6=\overline{z}\cdot z=|z|^2=1$ . Consider the $r(\cos \theta +i\sin \theta)$ form, when $z^6=1$ , there are 6 different solutions for $z$ . Therefore, the number of complex numbers satisfying $z^5=\bar{z}$ is $\boxed{\textbf{(E)} 7}$ .

~plasta

Video Solution by OmegaLearn

https://youtu.be/rbdIrmOyczk

2023 AMC 12A (Problems • Answer Key • Resources)
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All AMC 12 Problems and Solutions

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 ==Video Solution by OmegaLearn==
 https://youtu.be/rbdIrmOyczk
 ==See also==
 {{AMC12 box|year=2023|ab=A|num-b=13|num-a=15}}
 {{MAA Notice}}

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2023 AMC 12A Problems/Problem 14: Difference between revisions

Revision as of 22:50, 9 November 2023

Contents

Problem

Solution 1

Video Solution by OmegaLearn

See also