Difference between revisions of "1997 AIME Problems/Problem 14"
Ninja glace (talk | contribs) (→Solution) |
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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:
$
== See also ==
- [[1997 AIME Problems]]$ (Error compiling LaTeX. Unknown error_msg)