Difference between revisions of "2023 AMC 12A Problems/Problem 14"
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Adding up the solutions, we get <math>1+6=</math> <math>\boxed{\textbf{(E)} 7}</math> | Adding up the solutions, we get <math>1+6=</math> <math>\boxed{\textbf{(E)} 7}</math> | ||
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+ | -SwordOfJustice | ||
==Video Solution by OmegaLearn== | ==Video Solution by OmegaLearn== |
Revision as of 23:15, 3 April 2024
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)
Solution 3 (Rectangular Form, similar to Solution 1)
Let .
Then, our equation becomes:
Note that since every single term in the expansion contains either an or
, simply setting
and
yields a solution.
Now, considering the other case that either or
does not equal
:
Multiplying both sides by (or
), we get:
(since
).
Substituting back into the left hand side, we get:
Note that this will have 6 distinct, non-zero solutions since in this case, we consider that either or
is not
.
Adding up the solutions, we get
-SwordOfJustice
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.