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1/(1 + a^2 + (b + 1)^2) +1/(1 + b^2 + (c + 1)^2)+1/(1 + c^2 +(a + 1)^2) <=1/2

Source: 2012 Romania JBMO TST 4.1

June 2, 2020

inequalitiesalgebra

Problem Statement

Show that, for all positive real numbers

a, b, c

such that

abc = 1

, the inequality

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