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
Taking the magnitude of both sides tells us that , so
Dividing by we have
so
. Each of the
th roots of unity satisfy this condition.
In total, there are
numbers that work.
-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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