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Inequality with three variables

Source: Turkey JBMO Team Selection Test 2013, P4

May 31, 2013
inequalitiesquadraticsalgebrapolynomialinequalities proposed

Problem Statement

For all positive real numbers a,b,ca, b, c satisfying a+b+c=1a+b+c=1, prove that
a4+5b4a(a+2b)+b4+5c4b(b+2c)+c4+5a4c(c+2a)≥1−ab−bc−ca \frac{a^4+5b^4}{a(a+2b)} + \frac{b^4+5c^4}{b(b+2c)} + \frac{c^4+5a^4}{c(c+2a)} \geq 1- ab-bc-ca
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