2020 AMC 12A Problems/Problem 15
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
Realize that will create an equilateral triangle on the complex plane with the first point at
and two other points with equal magnitude at
.
Also, realize that can be factored through grouping:
will create points at
and
Plotting the points and looking at the graph will make you realize that and
are the farthest apart and through Pythagorean Theorem, the answer is revealed to be
~lopkiloinm
Remark
In the graph below, the solutions to are shown in red, and the solutions to
are shown in blue. The greatest distance between one red point and one blue point is shown in a black dashed line segment.
~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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