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Basic inequality involving a condition [Moldova TST 2017, D2, P2]

Source: Moldova TST 2017, Day 2, Problem 2

March 19, 2017
inequalitiesalgebra

Problem Statement

Let a,b,ca,b,c be positive real numbers that satisfy a+b+c=abca+b+c=abc. Prove that (1+a2)(1+b2)+(1+b2)(1+c2)+(1+a2)(1+c2)−(1+a2)(1+b2)(1+c2)≥4.\sqrt{(1+a^2)(1+b^2)}+\sqrt{(1+b^2)(1+c^2)}+\sqrt{(1+a^2)(1+c^2)}-\sqrt{(1+a^2)(1+b^2)(1+c^2)} \ge 4.
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