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Integer cubics with all roots of magnitude 20 and 13

Source: AIME II 2013, Problem 12

April 4, 2013
algebrapolynomialtrigonometryAMCAIMEVietacomplex numbers

Problem Statement

Let SS be the set of all polynomials of the form z3+az2+bz+cz^3+az^2+bz+c, where aa, bb, and cc are integers. Find the number of polynomials in SS such that each of its roots zz satisfies either ∣z∣=20\left\lvert z \right\rvert = 20 or ∣z∣=13\left\lvert z \right\rvert = 13.
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