1997 AIME Problems/Problem 14: Difference between revisions
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z^{1997}=1\\ | z^{1997}=1\\ | ||
z^{1997}=e^{2\pi ik}\\ | z^{1997}=e^{2\pi ik}\\ | ||
z=e^{\frac{2\pi ik}{1997}}< | z=e^{\frac{2\pi ik}{1997}}</math> | ||
== See also == | == See also == | ||
* [[1997 AIME Problems]] | * [[1997 AIME Problems]] | ||
Revision as of 19:13, 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:
