Difference between revisions of "2019 AMC 12B Problems/Problem 17"

(Fixed grammar and formatting, and added more detail to the solutions)
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''Note'': Here's a graph showing how <math>z</math> and <math>z^3</math> move as <math>\theta</math> increases: https://www.desmos.com/calculator/xtnpzoqkgs.
 
''Note'': Here's a graph showing how <math>z</math> and <math>z^3</math> move as <math>\theta</math> increases: https://www.desmos.com/calculator/xtnpzoqkgs.
  
==Solution 2==
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==Solution 2 (Quick Look)==
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As before, <math>r=1</math>. Represent <math>z</math> in polar form. By De Moivre's Theorem, <math>z^3=\text{cis}(3\theta)</math>. To form an equilateral triangle, their difference in angle must be <math>\frac{\pi}{3}</math>, so
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<cmath>\frac{\text{cis}(3\theta)}{\text{cis}(\theta)}=\text{cis}(2\theta)=\text{cis}(\pm\frac{\pi}{3})</cmath>
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From the polar form of <math>z</math>, we know that <math>0\geq\theta\leq2\pi</math>, so <math>\text{cis}(2\theta)</math> cycles in a circle twice. By contrast, <math>\pm\frac{\pi}{3}</math> represent <math>2</math> fixed, distinct points. Thus, <math>\text{cis}(2\theta)</math> intersects these points twice each<math>\implies\boxed{\textbf{(D) }4}</math>
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''Visual: https://www.desmos.com/calculator/rnpxzns0jn''
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To be more rigorous, you can find the <math>4</math> solutions. <math>\text{cis}(2\theta)</math> cycles twice, so <math>2\theta=\pm\frac{\pi}{3}+2n\pi</math>, where <math>n=0,1</math>. Then, <math>\theta=\frac{\pi}{6}</math>, <math>\frac{7\pi}{6}</math>, <math>\frac{5\pi}{6}</math>, <math>\frac{11\pi}{6}</math>. Substitute those values into <math>z</math> and check that they are valid. <math>\implies\boxed{\textbf{(D) }4}</math>
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(Solution by BJHHar)
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==Solution 3==
  
 
For the triangle to be equilateral, the vector from <math>z</math> to <math>z^3</math>, i.e <math>z^3 - z</math>, must be a <math>60^{\circ}</math> rotation of the vector from <math>0</math> to <math>z</math>, i.e. just <math>z</math>. Thus we must have
 
For the triangle to be equilateral, the vector from <math>z</math> to <math>z^3</math>, i.e <math>z^3 - z</math>, must be a <math>60^{\circ}</math> rotation of the vector from <math>0</math> to <math>z</math>, i.e. just <math>z</math>. Thus we must have
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Since any nonzero complex number will have two square roots, each equation gives two solutions. Thus, as before, the total number of possible values of <math>z</math> is <math>\boxed{\textbf{(D) }4}</math>.
 
Since any nonzero complex number will have two square roots, each equation gives two solutions. Thus, as before, the total number of possible values of <math>z</math> is <math>\boxed{\textbf{(D) }4}</math>.
  
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==Solution 4 (Quick and Easy) ==
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Since the complex numbers <math>0,z,</math> and <math>z^3</math> form an equilateral triangle in the complex plane, we note that either <math>z^3</math> is a 60 degrees counterclockwise rotation about the origin from <math>z</math> or <math>z</math> is a 60 degrees counterclockwise rotation about the origin from <math>z^3</math>.
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Therefore, we note that either <math>z^3 = z \text{cis} 60^\circ{}</math> or <math>z \text{cis}(-60^\circ{}) = z^3</math>
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The first equation in <math>z</math> (meaning <math>z^3 = z \text{cis} 60^\circ{}</math>) gives us: <math>z^2 = cis 60^\circ{}</math>, which gives 2 solutions in <math>z</math>.
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The second equation in <math>z</math> (which is <math>z \text{cis} (-60^\circ{}) = z^3</math>) gives us <math>z^2 = \text{cis} (-60^\circ{})</math>, which must give another 2 solutions in <math>z</math>.
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Therefore, there are <math>\boxed{(D) 4}</math> solutions for <math>z</math>. (Professor-Mom) (Minor edit: mathbw225
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Note: The motivation for this method came from an older AIME problem, namely https://artofproblemsolving.com/wiki/index.php/1994_AIME_Problems/Problem_8.
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==Video Solution==
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For those who prefer a video: https://www.youtube.com/watch?v=uBL80yd1ihc
 
==See Also==
 
==See Also==
 
{{AMC12 box|year=2019|ab=B|num-b=16|num-a=18}}
 
{{AMC12 box|year=2019|ab=B|num-b=16|num-a=18}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 23:27, 19 June 2021

Problem

How many nonzero complex numbers $z$ have the property that $0, z,$ and $z^3,$ when represented by points in the complex plane, are the three distinct vertices of an equilateral triangle?

$\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }4\qquad\textbf{(E) }\text{infinitely many}$

Solution 1

Convert $z$ and $z^3$ into modulus-argument (polar) form, giving $z=r\text{cis}(\theta)$ for some $r$ and $\theta$. Thus, by De Moivre's Theorem, $z^3=r^3\text{cis}(3\theta)$. Since the distance from $0$ to $z$ is $r$, and the triangle is equilateral, the distance from $0$ to $z^3$ must also be $r$, so $r^3=r$, giving $r=1$. (We know $r \neq 0$ since the problem statement specifies that $z$ must be nonzero.)

Now, to get from $z$ to $z^3$, which should be a rotation of $120^{\circ}$ if the triangle is equilateral, we multiply by $z^2 = r^2\text{cis}(2\theta)$, again using De Moivre's Theorem. Thus we require $2\theta=\pm\frac{\pi}{3} + 2\pi k$ (where $k$ can be any integer). If $0 < \theta < \frac{\pi}{2}$, we must have $\theta=\frac{\pi}{6}$, while if $\frac{\pi}{2} \leq \theta < \pi$, we must have $\theta = \frac{5\pi}{6}$. Hence there are $2$ values that work for $0 < \theta < \pi$. By symmetry, the interval $\pi \leq \theta < 2\pi$ will also give $2$ solutions. The answer is thus $2 + 2 = \boxed{\textbf{(D) }4}$.

Note: Here's a graph showing how $z$ and $z^3$ move as $\theta$ increases: https://www.desmos.com/calculator/xtnpzoqkgs.

Solution 2 (Quick Look)

As before, $r=1$. Represent $z$ in polar form. By De Moivre's Theorem, $z^3=\text{cis}(3\theta)$. To form an equilateral triangle, their difference in angle must be $\frac{\pi}{3}$, so \[\frac{\text{cis}(3\theta)}{\text{cis}(\theta)}=\text{cis}(2\theta)=\text{cis}(\pm\frac{\pi}{3})\] From the polar form of $z$, we know that $0\geq\theta\leq2\pi$, so $\text{cis}(2\theta)$ cycles in a circle twice. By contrast, $\pm\frac{\pi}{3}$ represent $2$ fixed, distinct points. Thus, $\text{cis}(2\theta)$ intersects these points twice each$\implies\boxed{\textbf{(D) }4}$

Visual: https://www.desmos.com/calculator/rnpxzns0jn


To be more rigorous, you can find the $4$ solutions. $\text{cis}(2\theta)$ cycles twice, so $2\theta=\pm\frac{\pi}{3}+2n\pi$, where $n=0,1$. Then, $\theta=\frac{\pi}{6}$, $\frac{7\pi}{6}$, $\frac{5\pi}{6}$, $\frac{11\pi}{6}$. Substitute those values into $z$ and check that they are valid. $\implies\boxed{\textbf{(D) }4}$

(Solution by BJHHar)

Solution 3

For the triangle to be equilateral, the vector from $z$ to $z^3$, i.e $z^3 - z$, must be a $60^{\circ}$ rotation of the vector from $0$ to $z$, i.e. just $z$. Thus we must have

\[\frac{(z^3-z)}{(z-0)}=\text{cis}{(\pi/3)} \text{ or } \text{cis}(5\pi/3)\]

Simplifying gives \[z^2-1= \text{cis}(\pi/3) \text{ or } z^2-1= \text{cis}(5\pi/3)\] so \[z^2=1+\text{cis}(\pi/3) \text{ or } z^2=1+\text{cis}(5\pi/3)\]

Since any nonzero complex number will have two square roots, each equation gives two solutions. Thus, as before, the total number of possible values of $z$ is $\boxed{\textbf{(D) }4}$.

Solution 4 (Quick and Easy)

Since the complex numbers $0,z,$ and $z^3$ form an equilateral triangle in the complex plane, we note that either $z^3$ is a 60 degrees counterclockwise rotation about the origin from $z$ or $z$ is a 60 degrees counterclockwise rotation about the origin from $z^3$.

Therefore, we note that either $z^3 = z \text{cis} 60^\circ{}$ or $z \text{cis}(-60^\circ{}) = z^3$

The first equation in $z$ (meaning $z^3 = z \text{cis} 60^\circ{}$) gives us: $z^2 = cis 60^\circ{}$, which gives 2 solutions in $z$.

The second equation in $z$ (which is $z \text{cis} (-60^\circ{}) = z^3$) gives us $z^2 = \text{cis} (-60^\circ{})$, which must give another 2 solutions in $z$.

Therefore, there are $\boxed{(D) 4}$ solutions for $z$. (Professor-Mom) (Minor edit: mathbw225

Note: The motivation for this method came from an older AIME problem, namely https://artofproblemsolving.com/wiki/index.php/1994_AIME_Problems/Problem_8.

Video Solution

For those who prefer a video: https://www.youtube.com/watch?v=uBL80yd1ihc

See Also

2019 AMC 12B (ProblemsAnswer KeyResources)
Preceded by
Problem 16
Followed by
Problem 18
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All AMC 12 Problems and Solutions

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