Difference between revisions of "1997 AIME Problems/Problem 14"
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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: | ||
− | <math> | + | <math> |
− | e^{2\pi ik} | + | e^{2\pi ik}=\cos(2\pi k)+i\sin(2\pi k)\ |
− | + | =1+0i\ | |
− | + | =1+0\ | |
− | + | =1\ | |
− | z^{1997}-1 | + | z^{1997}-1=0\ |
− | z^{1997} | + | z^{1997}=1\ |
− | z^{1997} | + | z^{1997}=e^{2\pi ik}\ |
− | z | + | z=e^{\frac{2\pi ik}{1997}}<\math> |
− | |||
== See also == | == See also == | ||
− | * [[1997 AIME Problems]]</math> | + | * [[1997 AIME Problems]] |
+ | |||
+ | </math> |
Revision as of 19:12, 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: $e^{2\pi ik}=\cos(2\pi k)+i\sin(2\pi k)\ =1+0i\ =1+0\ =1\ z^{1997}-1=0\ z^{1997}=1\ z^{1997}=e^{2\pi ik}\ z=e^{\frac{2\pi ik}{1997}}<\math>
== See also ==
- [[1997 AIME Problems]]$ (Error compiling LaTeX. Unknown error_msg)