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Difference between revisions of "2023 AMC 12A Problems/Problem 14"
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==Solution 1==
 
==Solution 1==
 
When <math>z^5=\overline{z}</math>, there are two conditions: either <math>z=0</math> or <math>z\neq 0</math>. When <math>z\neq 0</math>, since <math>z^5=\overline{z}</math>, <math>|z|=1</math>. <math>z^5\cdot z=z^6=\overline{z}\cdot z=|z|^2=1</math>. Consider the <math>r(\cos \theta +i\sin \theta)</math> form, when <math>z^6=1</math>, there are 6 different solutions for <math>z</math>. Therefore, the number of complex numbers satisfying <math>z^5=\bar{z}</math> is <math>\boxed{\textbf{(E)} 7}</math>.
 
When <math>z^5=\overline{z}</math>, there are two conditions: either <math>z=0</math> or <math>z\neq 0</math>. When <math>z\neq 0</math>, since <math>z^5=\overline{z}</math>, <math>|z|=1</math>. <math>z^5\cdot z=z^6=\overline{z}\cdot z=|z|^2=1</math>. Consider the <math>r(\cos \theta +i\sin \theta)</math> form, when <math>z^6=1</math>, there are 6 different solutions for <math>z</math>. Therefore, the number of complex numbers satisfying <math>z^5=\bar{z}</math> is <math>\boxed{\textbf{(E)} 7}</math>.
 +
 +
~plasta
 +
 +
==See also==
 +
{{AMC12 box|year=2023|ab=A|num-b=13|num-a=15}}
 +
{{MAA Notice}}

Revision as of 19:23, 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

See also

2023 AMC 12A (Problems • Answer Key • Resources)
Preceded by
Problem 13 		Followed by
Problem 15
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 12 Problems and Solutions

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