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Revision as of 08:49, 13 September 2021
Contents
Problem
In the complex plane, let be the set of solutions to
and let
be the set of solutions to
What is the greatest distance between a point of
and a point of
Solution 1
We solve each equation separately:
- We solve
by De Moivre's Theorem.
Let
where
is the magnitude of
such that
and
is the argument of
such that
We have
from which
so
so
or
is
In the complex plane, the solutions form the vertices of an equilateral triangle whose circumcircle has center
and radius
- We solve
by factoring by grouping.
We have
The set of solutions to
is
In the graph below, the points in set are shown in red, and the points in set
are shown in blue. The greatest distance between a point of
and a point of
is the distance between
to
as shown in the dashed line segments.
By the Distance Formula, the answer is
~lopkiloinm ~MRENTHUSIASM
Solution 2
Alternatively, we can solve by the difference of cubes:
- If
then
- If
then
by either completing the square or the quadratic formula.
The set of solutions to is
Following the rest of Solution 1 gives the answer
~MRENTHUSIASM
See Also
2020 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 14 |
Followed by Problem 16 |
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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