Problem:
The complex numbers and satisfy , and the imaginary part of is for relatively prime positive integers and with . Find .
Solution:
It follows from the given conditions that which implies that . All solutions of this equation satisfying the given conditions can be represented as , where for some positive integer . It is straightforward to verify that for all of this form, and satisfy the original equations. The number is prime, so and are relatively prime, and it follows that .
The problems on this page are the property of the MAA's American Mathematics Competitions