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2(a^4+b^4+c^4) < (a^2+b^2+c^2)^2 iff a,b,c>0 are sidelengths

Source: 2002 Estonia National Olympiad Final Round grade 12 p3

March 14, 2020

inequalitiesGeometric Inequalitiessidelenghts

Problem Statement

Prove that for positive real numbers

a, b

and

c

the inequality

2(a^4+b^4+c^4) < (a^2+b^2+c^2)^2

holds if and only if

a,b,c

are the sides of a triangle.