2019 AMC 12B Problems/Problem 24

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Let $\omega=e^{\frac{2i\pi}{3}}$ be the third root of unity. We wish to find the span of $a+b\omega+c\omega^2)$ for reals $0\le a,b,c\le 1$ . Note that if $a,b,c>0$ , then $a-\min(a,b,c), b-\min(a,b,c), c-\min(a,b,c)$ forms the same point as $a,b,c$ . Therefore, assume that at least one of them is equal to $0$ . If only one of them is equal to zero, we can form an equilateral triangle with the remaining two, of side length $1$ . Similarly for if two are equal to zero. So the area of the six equilateral triangles is $\boxed{\text{(C) }\frac{3\sqrt{3}}{2}}$ -programjames1

2019 AMC 12B (Problems • Answer Key • Resources)
Preceded by Problem 23	Followed by Problem 25
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2019 AMC 12B Problems/Problem 24

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