2005 AMC 12B Problems/Problem 22

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

Since $|z_0|=1$ , let $z_0=e^{i\theta_0}$ , where $\theta_0$ is an argument of $z_0$ . I will prove by induction that $z_n=e^{i\theta_n}$ , where $\theta_n=2^n(\theta_0+\frac{\pi}{2})-\frac{\pi}{2}$ .

Base Case: trivial

Inductive Step: Suppose the formula is correct for $z_k$ , then $z_{k+1}=\frac{iz_k}{\overline {z_k}}=i e^{i\theta_k} e^{i\theta_k}=e^{i(2\theta_k+\pi/2)}$ Since $2\theta_k+\frac{\pi}{2}=2\cdot 2^n(\theta_0+\frac{\pi}{2})-\pi+\frac{\pi}{2}=2^{n+1}(\theta_0+\frac{\pi}{2})-\frac{\pi}{2}=\theta_{n+1}$ the formula is proven

$z_{2005}=1\Rightarrow \theta_{2005}=2k\pi$ , where $k$ is an integer. Therefore, $2^{2005}(\theta_0+\frac{\pi}{2})=(2k+\frac{1}{2})\pi\\ \theta_0=\frac{k}{2^{2004}}\pi+\left(\frac{1}{2^{2006}}-\frac{1}{2}\right)\pi$ The value of $\theta_0$ only matters modulo $2\pi$ . Since $\frac{k+2^{2005}}{2^{2004}}\pi\equiv\frac{k}{2^{2004}}\pi\mod 2\pi$ , k only needs to take values from 0 to $2^{2005}-1$ , so the answer is $2^{2005}\Rightarrow\boxed{E}$

2005 AMC 12B (Problems • Answer Key • Resources)
Preceded by Problem 21	Followed by Problem 23
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2005 AMC 12B Problems/Problem 22

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