Problem:
Given f(z)=z2−19z, there are complex numbers z with the property that z,f(z), and f(f(z)) are the vertices of a right triangle in the complex plane with a right angle at f(z). There are positive integers m and n such that one such value of z is m+n​+11i. Find m+n.
Solution:
The arguments of two complex numbers differ by 90∘ if the ratio of the numbers is a pure imaginary number. Thus three distinct complex numbers A,B, and C form a right triangle in the complex plane with right angle at B if and only if B−AC−B​ has real part equal to 0 . Hence
must have real part equal to 0 . If z=x+11i, the real part of z2−18z−19 is x2−112−18x−19, which is 0 when x=9±221​. The requested sum is 9+221=230.