Difference between revisions of "1997 AIME Problems/Problem 14"
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== Solution == | == Solution == | ||
| − | The solution requires use of Euler's formula: | + | The solution requires the use of Euler's formula: |
| + | |||
<math>\displaystyle e^{i\theta}=\cos(\theta)+i\sin(\theta)</math> | <math>\displaystyle e^{i\theta}=\cos(\theta)+i\sin(\theta)</math> | ||
| + | |||
| + | If <math>\displaystyle \theta=2\pi ik</math>, 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 | ||
| + | \end{eqnarray*} | ||
== See also == | == See also == | ||
* [[1997 AIME Problems]] | * [[1997 AIME Problems]] | ||
Revision as of 20:06, 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
\end{eqnarray*}
