2018 AMC 12A Problems/Problem 22
Contents
Problem
The solutions to the equations and
where
form the vertices of a parallelogram in the complex plane. The area of this parallelogram can be written in the form
where
and
are positive integers and neither
nor
is divisible by the square of any prime number. What is
Solution 1
The roots are (easily derivable by using DeMoivre and half-angle). From there, shoelace on
and multiplying by
gives the area of
, so the answer is
. (trumpeter)
Solution 2 (No DeMoivre's)
Write as
. For the first equation,
Setting the real parts equal and imaginary parts equal, we have:
Squaring the second equation gives
. We now need two numbers that have a difference of
and a product of
. By inspection,
and
work, so
and
. Since
is positive,
and
must have the same sign. Thus we have two solutions for
:
Repeating the process for the second equation, we have two solutions:
In a clockwise direction, the points are
. Now we can use the shoelace theorem. The area is
, so the answer is
.
Video Solution by Richard Rusczyk
https://artofproblemsolving.com/videos/amc/2018amc12a/472
~ dolphin7
See Also
2018 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 21 |
Followed by Problem 23 |
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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