Problem:
How many complex numbers satisfy the equation , where is the conjugate of the complex number ?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Taking the absolute value of each side of the equation gives
Because , this equation is satisfied only when or . If , then is the unique solution. If , then multiplying the original equation by gives , so must be a sixth root of unity. Each of the sixth roots of unity may be written as (for ), and each of these roots satisfies the original equation. Thus the total number of solutions is .
The problems on this page are the property of the MAA's American Mathematics Competitions