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1/3 (f(a+b-c)+f(b+c-a)+f(c+a-b))=f(\sqrt{(ab+bc+ca)/3) for a,b,c sidelengths

Source: 2006 VMEO III Shortlist SL A5 Vietnamese Mathematics e - Olympiad https://artofproblemsolving.com/community/c2461015_vmeo__viet

October 3, 2021
functional equationfunctionalalgebra

Problem Statement

Find all continuous functions f:(0,+∞)→(0,+∞)f : (0,+\infty) \to (0,+\infty) such that if a,b,ca, b, c are the lengths of the sides of any triangle then it is satisfied that f(a+b−c)+f(b+c−a)+f(c+a−b)3=f(ab+bc+ca3)\frac{f(a+b-c)+f(b+c-a)+f(c+a-b)}{3}=f\left(\sqrt{\frac{ab+bc+ca}{3}}\right)
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