Difference between revisions of "2019 AMC 12B Problems/Problem 17"
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Simplify left side: | Simplify left side: | ||
− | <cmath>z^2-1= \text{cis}(\pi/3) \text{ or } z^2-1= \text{cis}( | + | <cmath>z^2-1= \text{cis}(\pi/3) \text{ or } z^2-1= \text{cis}(5\pi/3)</cmath> |
That is, | That is, | ||
− | <cmath>z^2=1+\text{cis}(\pi/3) \text{ or } z^2=1+\text{cis}( | + | <cmath>z^2=1+\text{cis}(\pi/3) \text{ or } z^2=1+\text{cis}(5\pi/3)</cmath> |
We have two roots for both equations, therefore the total number of solution for <math>z</math> is <math>\boxed{\textbf{(D) }4}</math> | We have two roots for both equations, therefore the total number of solution for <math>z</math> is <math>\boxed{\textbf{(D) }4}</math> |
Revision as of 22:11, 15 February 2019
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
Someone pls help with LaTeX formatting, thanks -FlatSquare , I did, -Dodgers66
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 |
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.