2023 AMC 12A Problems/Problem 14
Contents
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 .
~plasta
Solution 2
Let We now have and want to solve
From this, we have as a solution, which gives . If , then we divide by it, yielding
Dividing both sides by yields . Taking the magnitude of both sides tells us that , so . However, if , then , but must be real. Therefore, .
Multiplying both sides by ,
Each of the th roots of unity is a solution to this, so there are solutions.
-Benedict T (countmath 1)
Video Solution by OmegaLearn
Video Solution
~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)
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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