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

Revision as of 11:36, 11 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

Solution 2

Let $z = re^{i\theta}.$ We now have $\overline{z} = re^{-i\theta},$ and want to solve

$re^{5i\theta} = re^{-i\theta}.$

From this, we have $r = 0$ as a solution, which gives $z = 0$ . If $r\neq 0$ , then we divide by it, yielding

$e^{5i\theta} = e^{-i\theta}.$

Multiplying by $e^{i\theta}$ cancels the righthand side's exponents:

$e^{6i\theta} = 1.$

Hence, $z^6 = 1$ and each of the $6$ th roots of unity satisfies this case.

In total, there are $1 + 6 = \boxed{\textbf{(E)} 7}$ numbers that work.

-Benedict T (countmath 1)

Video Solution by OmegaLearn

https://youtu.be/rbdIrmOyczk

Video Solution

https://youtu.be/m627Mjp3PkM

~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)

2023 AMC 12A (Problems • Answer Key • Resources)
Preceded by Problem 13	Followed by Problem 15
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All AMC 12 Problems and Solutions

These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. Error creating thumbnail: File missing

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 ==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

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