1997 AIME Problems/Problem 14: Difference between revisions
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If <math>\displaystyle \theta=2\pi ik</math>, where k is any constant, the equation reduces to: | If <math>\displaystyle \theta=2\pi ik</math>, where k is any constant, the equation reduces to: | ||
\begin{eqnarray*} | <math>\begin{eqnarray*} | ||
e^{2\pi ik}&=&\cos(2\pi k)+i\sin(2\pi k) | e^{2\pi ik}&=&\cos(2\pi k)+i\sin(2\pi k)\\ | ||
&=&1+0i | &=&1+0i\\ | ||
&=&1+0 | &=&1+0\\ | ||
&=&1 | &=&1\\ | ||
\end{eqnarray*} | z^{1997}-1&=&0\\ | ||
z^{1997}&=&1\\ | |||
z^{1997}&=&e^{2\pi ik}\\ | |||
z&=&e^{\frac{2\pi ik}{1997}} | |||
\end{eqnarray*}<math> | |||
== See also == | == See also == | ||
* [[1997 AIME Problems]] | * [[1997 AIME Problems]]</math> | ||
Revision as of 19:09, 7 March 2007
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:
$\begin{eqnarray*}
e^{2\pi ik}&=&\cos(2\pi k)+i\sin(2\pi k)\\
&=&1+0i\\
&=&1+0\\
&=&1\\
z^{1997}-1&=&0\\
z^{1997}&=&1\\
z^{1997}&=&e^{2\pi ik}\\
z&=&e^{\frac{2\pi ik}{1997}}
\end{eqnarray*}<math>
== See also ==
- [[1997 AIME Problems]]$ (Error compiling LaTeX. Unknown error_msg)
