Difference between revisions of "2005 AMC 12B Problems/Problem 22"
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== Problem == | == Problem == | ||
| + | |||
| + | A sequence of complex numbers <math>z_{0}, z_{1}, z_{2}, ...</math> is defined by the rule | ||
| + | |||
| + | <cmath>z_{n+1} = \frac {iz_{n}}{\overline {z_{n}}},</cmath> | ||
| + | |||
| + | 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>? | ||
| + | |||
| + | <math> | ||
| + | \textbf{(A)}\ 1 \qquad | ||
| + | \textbf{(B)}\ 2 \qquad | ||
| + | \textbf{(C)}\ 4 \qquad | ||
| + | \textbf{(D)}\ 2005 \qquad | ||
| + | \textbf{(E)}\ 2^{2005} | ||
| + | </math> | ||
== Solution == | == Solution == | ||
Revision as of 16:40, 22 February 2010
Problem
A sequence of complex numbers 
is defined by the rule
![\[z_{n+1} = \frac {iz_{n}}{\overline {z_{n}}},\]](http://latex.artofproblemsolving.com/5/6/1/5615d4f35b70105ed92f07cc305ec89d9a9cdd96.png)
where 
is the complex conjugate of 
and 
. Suppose that 
and 
. How many possible values are there for 
?
