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a^2+b^2+c^2 +4abc<= \frac12, for sidelengths with a+b+c=1 (likely classic)

Source: 1990 ITAMO p4

January 31, 2020
inequalitiesalgebrageometric inequality

Problem Statement

Let a,b,ca,b,c be side lengths of a triangle with a+b+c=1a+b+c = 1. Prove that a2+b2+c2+4abc≤12a^2 +b^2 +c^2 +4abc \le \frac12 .
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