Difference between revisions of "2019 AMC 12B Problems/Problem 24"

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==Solution==
 
==Solution==
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Let <math>\omega=e^{\frac{2i\pi}{3}}</math> be the third root of unity. We wish to find the span of <math>a+b\omega+c\omega^2)</math> for reals <math>0\le a,b,c\le 1</math>.
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Note that if <math>a,b,c>0</math>, then <math>a-\min(a,b,c), b-\min(a,b,c), c-\min(a,b,c)</math> forms the same point as <math>a,b,c</math>. Therefore, assume that at least one of them is equal to <math>0</math>. If only one of them is equal to zero, we can form an equilateral triangle with the remaining two, of side length <math>1</math>. Similarly for if two are equal to zero. So the area of the six equilateral triangles is
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<cmath>\boxed{\text{(C) }\frac{3\sqrt{3}}{2}}</cmath>
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-programjames1
  
 
==See Also==
 
==See Also==
 
{{AMC12 box|year=2019|ab=B|num-b=23|num-a=25}}
 
{{AMC12 box|year=2019|ab=B|num-b=23|num-a=25}}

Revision as of 12:04, 14 February 2019

Problem

Solution

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

See Also

2019 AMC 12B (ProblemsAnswer KeyResources)
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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