2000 AIME II Problems/Problem 9
Contents
Problem
Given that is a complex number such that
, find the least integer that is greater than
.
Solution
Using the quadratic equation on , we have
.
There are other ways we can come to this conclusion. Note that if is on the unit circle in the complex plane, then
and
. We have
and
. Alternatively, we could let
and solve to get
.
Using De Moivre's Theorem we have ,
, so
.
We want .
Finally, the least integer greater than is
.
Solution 2
Let . Notice that we have
Without loss of generality, suppose
. Therefore,
and plugging into the desired expression, we get $e^{i\frac{100\pi}{3}}+e^{-i\frac{100\pi}{3}}=2\cos{\frac{100\pi}{3}=0$ (Error compiling LaTeX. Unknown error_msg). Therefore, the least integer greater is
See also
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 | ||
All AIME Problems and Solutions |
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