2019 AMC 12B Problems/Problem 24
Contents
[hide]Problem
Let Let denote all points in the complex plane of the form where and What is the area of ?
Solution
Let be the third root of unity. We wish to find the span of for reals . Note that if , then forms the same point as . Therefore, assume that at least one of them is equal to . If only one of them is equal to zero, we can form an equilateral triangle with the remaining two, of side length . Similarly for if two are equal to zero. So the area of the six equilateral triangles is
Here is a diagram:
-programjames1
Solution 2
We can add on each term one at a time. First off, the possible values of lie on the following graph:
For each point on the line, we can add . This means that we can extend the area to
by "moving" the blue line along the red line. Finally, we can add to every point
by "moving" the previous area along the green line. This leaves us with a regular hexagon with side length , so the total area is .
~~IYN~~
See Also
2019 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 23 |
Followed by Problem 25 |
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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