Difference between revisions of "2019 AMC 12B Problems/Problem 24"
(→Solution) |
(→Problem) |
||
Line 1: | Line 1: | ||
==Problem== | ==Problem== | ||
+ | Let <math>\omega=-\tfrac{1}{2}+\tfrac{1}{2}i\sqrt3.</math> Let <math>S</math> denote all points in the complex plane of the form <math>a+b\omega+c\omega^2,</math> where <math>0\leq a \leq 1,0\leq b\leq 1,</math> and <math>0\leq c\leq 1.</math> What is the area of <math>S</math>? | ||
+ | <cmath>\textbf{(A) } \frac{1}{2}\sqrt3 \qquad\textbf{(B) } \frac{3}{4}\sqrt3 \qquad\textbf{(C) } \frac{3}{2}\sqrt3\qquad\textbf{(D) } \frac{1}{2}\pi\sqrt3 \qquad\textbf{(E) } \pi</cmath> | ||
==Solution== | ==Solution== |
Revision as of 12:10, 14 February 2019
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
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 |