2000 AIME II Problems/Problem 9

Problem

Given that $z$ is a complex number such that $z+\frac 1z=2\cos 3^\circ$ , find the least integer that is greater than $z^{2000}+\frac 1{z^{2000}}$ .

Solution

Note that if z is on the unit circle in the complex plane, then $z = e^{i\theta} = cos \theta + isin \theta$ and $\frac 1z= e^{-i\theta} = cos \theta - isin \theta$

Let $z = a + bi$

2000 AIME II (Problems • Answer Key • Resources)
Preceded by Problem 8	Followed by Problem 10
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
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2000 AIME II Problems/Problem 9

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