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tan A, tan B, tan C are roots of an equation

Source: Vietnam NMO 1988 Problem 5

February 2, 2009
trigonometryalgebra unsolvedalgebra

Problem Statement

Suppose that ABC ABC is an acute triangle such that tan⁡A \tan A, tan⁡B \tan B, tan⁡C \tan C are the three roots of the equation x^3 \plus{} px^2 \plus{} qx \plus{} p \equal{} 0, where q≠1 q\neq 1. Show that p \le \minus{} 3\sqrt 3 and q>1 q > 1.
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