Difference between revisions of "2023 AMC 12A Problems/Problem 14"
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| − | + | ==Problem== | |
| − | + | How many complex numbers satisfy the equation <math>z^5=\overline{z}</math>, where <math>\overline{z}</math> is the conjugate of the complex number <math>z</math>? | |
| + | |||
| + | <math>\textbf{(A)} ~2\qquad\textbf{(B)} ~3\qquad\textbf{(C)} ~5\qquad\textbf{(D)} ~6\qquad\textbf{(E)} ~7</math> | ||
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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>. | ||
Revision as of 19:23, 9 November 2023
Problem
How many complex numbers satisfy the equation 
, where 
is the conjugate of the complex number 
?
Solution 1
When , there are two conditions: either 
or 
. When , since , 
. 
. Consider the 
form, when 
, there are 6 different solutions for . Therefore, the number of complex numbers satisfying 
is 
.
