Difference between revisions of "2005 AMC 12B Problems/Problem 22"

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== Problem ==
 
== Problem ==
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A sequence of complex numbers <math>z_{0}, z_{1}, z_{2}, ...</math> is defined by the rule
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<cmath>z_{n+1} = \frac {iz_{n}}{\overline {z_{n}}},</cmath>
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where <math>\overline {z_{n}}</math> is the [[complex conjugate]] of <math>z_{n}</math> and <math>i^{2}=-1</math>. Suppose that <math>|z_{0}|=1</math> and <math>z_{2005}=1</math>. How many possible values are there for <math>z_{0}</math>?
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<math>
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\textbf{(A)}\ 1 \qquad
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\textbf{(B)}\ 2 \qquad
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\textbf{(C)}\ 4 \qquad
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\textbf{(D)}\ 2005 \qquad
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\textbf{(E)}\ 2^{2005}
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</math>
  
 
== Solution ==
 
== Solution ==

Revision as of 17:40, 22 February 2010

Problem

A sequence of complex numbers $z_{0}, z_{1}, z_{2}, ...$ is defined by the rule

\[z_{n+1} = \frac {iz_{n}}{\overline {z_{n}}},\]

where $\overline {z_{n}}$ is the complex conjugate of $z_{n}$ and $i^{2}=-1$. Suppose that $|z_{0}|=1$ and $z_{2005}=1$. How many possible values are there for $z_{0}$?

$\textbf{(A)}\ 1 \qquad  \textbf{(B)}\ 2 \qquad  \textbf{(C)}\ 4 \qquad  \textbf{(D)}\ 2005 \qquad  \textbf{(E)}\ 2^{2005}$

Solution

See also