2000 AIME II Problems/Problem 9: Difference between revisions

Revision as of 23:09, 14 March 2008

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 $z = e^{i\theta} = \cos \theta + i\sin \theta$ and $\frac 1z= e^{-i\theta} = \cos \theta - i\sin \theta$

We have $z+\frac 1z = 2\cos \theta = 2\cos 3^\circ$ and $\theta = 3^\circ$ Alternatively, we could let $z = a + bi$ and solve to get $z=\cos 3^\circ + i\sin 3^\circ$

Using De Moivre's Theorem we have $z^{2000} = \cos 6000^\circ + i\sin 6000^\circ$ , $6000 = 16(360) + 240$ , so $z^{2000} = \cos 240^\circ + i\sin 240^\circ$

We want $z^{2000}+\frac 1{z^{2000}} = 2\cos 240^\circ = -1$

Finally, the least integer greater than $-1$ is $\boxed{000}$ .

@@ Line 13: / Line 13: @@
 We want <math>z^{2000}+\frac 1{z^{2000}} = 2\cos 240^\circ = -1</math>
-Of course, we cannot have -1 as an answer on the AIME, but they asked for the smallest integer greater than this value, which is 0.
+Finally, the least integer greater than <math>-1</math> is <math>\boxed{000}</math>.
 == See also ==
 {{AIME box|year=2000|n=II|num-b=8|num-a=10}}

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2000 AIME II Problems/Problem 9: Difference between revisions

Revision as of 23:09, 14 March 2008

Problem

Solution

See also