Difference between revisions of "2012 AIME I Problems/Problem 6"

(Solution 1)
(Omitted Solution 2, which was the same as Solution 1 but with worse wording.)
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The complex numbers <math>z</math> and <math>w</math> satisfy <math>z^{13} = w,</math> <math>w^{11} = z,</math> and the imaginary part of <math>z</math> is <math>\sin{\frac{m\pi}{n}}</math>, for relatively prime positive integers <math>m</math> and <math>n</math> with <math>m<n.</math> Find <math>n.</math>
 
The complex numbers <math>z</math> and <math>w</math> satisfy <math>z^{13} = w,</math> <math>w^{11} = z,</math> and the imaginary part of <math>z</math> is <math>\sin{\frac{m\pi}{n}}</math>, for relatively prime positive integers <math>m</math> and <math>n</math> with <math>m<n.</math> Find <math>n.</math>
  
==Solutions==
+
==Solution==
  
===Solution 1===
 
 
Substituting the first equation into the second, we find that <math>(z^{13})^{11} = z</math> and thus <math>z^{142} = 1.</math> So <math>z</math> must be a <math>142</math>nd root of unity, and thus the imaginary part of <math>z</math> will be <math>\sin{\frac{2k\pi}{142}} = \sin{\frac{k\pi}{71}}</math> for some <math>k</math> with <math>0 \le k < 142.</math> But note that <math>71</math> is prime and <math>k<71</math> by the conditions of the problem, so the denominator in the argument of this value will always be <math>71</math> and thus <math>n = \boxed{071.}</math>
 
Substituting the first equation into the second, we find that <math>(z^{13})^{11} = z</math> and thus <math>z^{142} = 1.</math> So <math>z</math> must be a <math>142</math>nd root of unity, and thus the imaginary part of <math>z</math> will be <math>\sin{\frac{2k\pi}{142}} = \sin{\frac{k\pi}{71}}</math> for some <math>k</math> with <math>0 \le k < 142.</math> But note that <math>71</math> is prime and <math>k<71</math> by the conditions of the problem, so the denominator in the argument of this value will always be <math>71</math> and thus <math>n = \boxed{071.}</math>
 
===Solution 2===
 
Note that <math>w^{143}=w</math> and similar for <math>z</math>, and they are not equal to <math>0</math> because the question implies the imaginary part is positive.  Thus <math>w^{142}=z^{142}=1</math>, so the imaginary part of each is of the form <math>\sin\left(\frac{2 \pi k}{142}\right)</math> where <math>k</math> is a positive integer between <math>1</math> and <math>141</math> inclusive.  This simplifies to <math>\sin\left(\frac{\pi k}{71}\right)</math>. Therefore, the imaginary part of <math>z</math> is <math>\sin\left(\frac{\pi m}{71}\right)</math>, where <math>0 \le m < 71</math>. Since <math>071</math> is prime, it is the only possible denominator, so <math>\boxed{n = 71}</math>.
 
  
 
== See also ==
 
== See also ==
 
{{AIME box|year=2012|n=I|num-b=5|num-a=7}}
 
{{AIME box|year=2012|n=I|num-b=5|num-a=7}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 00:53, 26 February 2014

Problem 6

The complex numbers $z$ and $w$ satisfy $z^{13} = w,$ $w^{11} = z,$ and the imaginary part of $z$ is $\sin{\frac{m\pi}{n}}$, for relatively prime positive integers $m$ and $n$ with $m<n.$ Find $n.$

Solution

Substituting the first equation into the second, we find that $(z^{13})^{11} = z$ and thus $z^{142} = 1.$ So $z$ must be a $142$nd root of unity, and thus the imaginary part of $z$ will be $\sin{\frac{2k\pi}{142}} = \sin{\frac{k\pi}{71}}$ for some $k$ with $0 \le k < 142.$ But note that $71$ is prime and $k<71$ by the conditions of the problem, so the denominator in the argument of this value will always be $71$ and thus $n = \boxed{071.}$

See also

2012 AIME I (ProblemsAnswer KeyResources)
Preceded by
Problem 5
Followed by
Problem 7
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
All AIME Problems and Solutions

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