2019 AMC 12B Problems/Problem 17
Contents
Problem
How many nonzero complex numbers have the property that and when represented by points in the complex plane, are the three distinct vertices of an equilateral triangle?
Solution 1
Convert and into form, giving and . Since the distance from to is , the distance from to must also be , so . Now we must find the requirements for being an equilateral triangle. From , we have and from , we see a monotonic increase of , from to , or equivalently, from to . Hence, there are 2 values that work for . But since the interval also consists of going from to , it also gives us 2 solutions. Our answer is
Here's a graph of how and move as increases- https://www.desmos.com/calculator/xtnpzoqkgs
Solution 2
To be equilateral triangle, we should have
Simplify left side:
That is,
We have two roots for both equations, therefore the total number of solution for is
(By Zhen Qin)
See Also
2019 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 16 |
Followed by Problem 18 |
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All AMC 12 Problems and Solutions |
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