Problem:
The complex number z is equal to 9+bi, where b is a positive real number and i2=−1. Given that the imaginary parts of z2 and z3 are equal, find b.
Solution:
The complex number z=9+bi, so z2=(81−b2)+18bi and z3=(729−27b2)+(243b−b3)i. These two numbers have the same imaginary part, so 243b−b3=18b. Because b is not zero, 243−b2=18, and b=15​.
The problems on this page are the property of the MAA's American Mathematics Competitions