2001 AIME II Problems/Problem 14

Problem

There are $2n$ complex numbers that satisfy both $z^{28} - z^{8} - 1 = 0$ and $\mid z \mid = 1$ . These numbers have the form $z_{m} = \cos\theta_{m} + i\sin\theta_{m}$ , where $0\leq\theta_{1} < \theta_{2} < \ldots < \theta_{2n} < 360$ and angles are measured in degrees. Find the value of $\theta_{2} + \theta_{4} + \ldots + \theta_{2n}$ .

Solution

Z can be written in the form $cis\theta$ . Rearranging, we see that $cis{28}\theta$ = $cis{8}\theta$ $+1$

Since the real part of $cis{28}\theta$ is one more than the real part of $cis {8}\theta$ and their imaginary parts are the same, it is clear that either $cis{28}\theta$ = ${1/2} + \sqrt{3}/{2}$ and $cis {8}\theta$ = ${-1/2} + \sqrt{3}/{2}$ , or $cis{28}\theta$ = ${1/2} - \sqrt{3}/{2}$ and $cis {8}\theta$ = ${-1/2} + \sqrt{3}/{2}$

Case One : $cis{28}\theta$ = ${1/2} + \sqrt{3}/{2}$ and $cis {8}\theta$ = ${-1/2} +\sqrt{3}/{2}$

Setting up and solving equations, $Z^{28}= cis{60^\circ$ (Error compiling LaTeX. Unknown error_msg) and $Z^8= cis{120^\circ$ (Error compiling LaTeX. Unknown error_msg), we see that the only common solutions are $15^\circ , 105^\circ, 195^\circ,$ and $\ 285^\circ$

Case 2 : $cis{28}\theta$ = ${1/2} - \sqrt{3}/{2}$ and $cis {8}\theta$ = ${-1/2} -\sqrt{3}/{2}$

Again setting up equations ($Z^{28}= cis{300^\circ$ (Error compiling LaTeX. Unknown error_msg) and $Z^{8} = cis{240^\circ$ (Error compiling LaTeX. Unknown error_msg)) we see that the only common solutions are $75^\circ, 165^\circ, 255^\circ,$ and $345^\circ$

Listing all of these values, it is seen that $\theta_{2} + \theta_{4} + \ldots + \theta_{2n}$ is equal to $(75 + 165 + 255 + 345) ^\circ$ which is equal to $\boxed {840}$ degrees

2001 AIME II (Problems • Answer Key • Resources)
Preceded by Problem 13	Followed by Problem 15
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2001 AIME II Problems/Problem 14

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