Problem:
Consider the set of complex numbers z satisfying ∣∣∣1+z+z2∣∣∣=4. The maximum value of the imaginary part of z can be written in the form nm, where m and n are relatively prime positive integers. What is m+n?
Answer Choices:
A. 20
B. 21
C. 22
D. 23
E. 24
Solution:
Completing the square gives
1+z+z2=43+(21+z)2
Let w=21+z. Then ∣∣∣w2+43∣∣∣=4. Because Imw=Imz, it suffices to maximize Imw. By the Triangle Inequality,
4=∣∣∣∣∣w2+43∣∣∣∣∣≥∣w∣2−43≥(Imw)2−43
Solving yields Imw≤219, and the requested sum is 19+2=(B)21. Equality is achieved when z=−21+219i.