1997 AIME Problems/Problem 14
Problem
Let and
be distinct, randomly chosen roots of the equation
. Let
be the probability that
, where
and
are relatively prime positive integers. Find
.
Solution
The solution requires the use of Euler's formula:
If , where k is any constant, the equation reduces to:
$e^{2\pi ik}=\cos(2\pi k)+i\sin(2\pi k)\\
=1+0i\\
=1+0\\
=1\\
z^{1997}-1=0\\
z^{1997}=1\\
z^{1997}=e^{2\pi ik}\\
z=e^{\frac{2\pi ik}{1997}}<\math>
== See also ==
- [[1997 AIME Problems]]$ (Error compiling LaTeX. Unknown error_msg)