Problem

Source: Turkey JBMO Team Selection Test 2013, P4

Tags: inequalities, quadratics, algebra, polynomial, inequalities proposed



For all positive real numbers $a, b, c$ satisfying $a+b+c=1$, prove that \[ \frac{a^4+5b^4}{a(a+2b)} + \frac{b^4+5c^4}{b(b+2c)} + \frac{c^4+5a^4}{c(c+2a)} \geq 1- ab-bc-ca \]