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1/(4+(a+b)^2)+1/(4+(b+c)^2)+1/(4+(c+a)^2)<= 3/8 for ab + bc+ ca = 3

Source: Balkan MO Shortlist 2013 A1 BMO

March 8, 2020

inequalitiesalgebra

Problem Statement

Positive real numbers

a, b,c

satisfy

ab + bc+ ca = 3

. Prove the inequality

\frac{1}{4+(a+b)^2}+\frac{1}{4+(b+c)^2}+\frac{1}{4+(c+a)^2}\le \frac{3}{8}