Problem:
A function is defined by , where and is the complex conjugate of . How many values of satisfy both and ?
Answer Choices:
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B.
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Solution:
From the definition of ,
for all real numbers and , so the numbers that satisfy are the numbers of the form . The set of all such numbers is a line through the origin in the complex plane. The set of all numbers that satisfy is a circle centered at the origin of the complex plane. The numbers satisfying both equations correspond to the points of intersection of the line and circle, of which there are .
The problems on this page are the property of the MAA's American Mathematics Competitions