Problem

Source: V International Festival of Young Mathematicians Sozopol 2014, Theme for 10-12 grade

Tags: algebra, Inequality, triangle inequality, inequalities



Prove that, if $a,b,c$ are sides of a triangle, then we have the following inequality: $3(a^3 b+b^3 c+c^3 a)+2(ab^3+bc^3+ca^3 )\geq 5(a^2 b^2+a^2 c^2+b^2 c^2 )$.