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geometry in complex numbers

Source: 2019 AIME I #12

March 14, 2019
geometryAMCAIMEAIME Icomplex numbers2019 AIME I2019 AMC

Problem Statement

Given f(z)=z2−19zf(z) = z^2-19z, there are complex numbers zz with the property that zz, f(z)f(z), and f(f(z))f(f(z)) are the vertices of a right triangle in the complex plane with a right angle at f(z)f(z). There are positive integers mm and nn such that one such value of zz is m+n+11im+\sqrt{n}+11i. Find m+nm+n.
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