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(a^3+b^3+c^3)<(a+b+c)(a^2+b^2+c^2)<=3(a^3+b^3+c^3) where a,b,c sidelengths

Source: Irmo 2018 p2 q7

September 16, 2018
geometric inequalityinequalitiesalgebra

Problem Statement

Let a,b,ca, b, c be the side lengths of a triangle. Prove that 2(a3+b3+c3)<(a+b+c)(a2+b2+c2)≤3(a3+b3+c3)2 (a^3 + b^3 + c^3) < (a + b + c) (a^2 + b^2 + c^2) \le 3 (a^3 + b^3 + c^3)
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