Mongolian Mathematical Olympiad

It suffices to prove $$F(a,b,c)=2(a^4+b^4+c^4)-(a+b+c)(a^3+b^3+c^3-abc)\ge 0$$ for all $a,b,c\ge 0$. We may assume $b\ge a$ and $c\ge a$ and then we have $$F(a,b,c)=(b-c{)}^2(b(b-a)+bc+c(c-a))+a^2(c-a)(b-a)\ge 0.$$ Equality holds for $(a,b,c)=(0,1,1),(1,0,1),(1,1,0),(1,1,1)$.