lean_workbook_plus_4987

1a) WLOG, let $a, b, c, \in \mathbb{R^+}$ s.t. $a \le b \le c$ . Then, $a = u, b = u+v, c = u+v+w$ , for some $u, v, w \ge 0$ . Then, if $a^4+b^4+c^4 \ge abc(a+b+c)$ , we have \n \n$3 u^4+8 u^3 v+4 u^3 w+12 u^2 v^2+12 u^2 v w+6 u^2 w^2+8 u v^3+12 u v^2 w+12 u v w^2+4 u w^3+2 v^4+4 v^3 w+6 v^2 w^2+4 v w^3+w^4 \ge 3 u^4+8 u^3 v+4 u^3 w+7 u^2 v^2+7 u^2 v w+u^2 w^2+2 u v^3+3 u v^2 w+u v w^2$ \n$\implies 5 u^2 v^2+5 u^2 v w+5 u^2 w^2+6 u v^3+9 u v^2 w+11 u v w^2+4 u w^3+2 v^4+4 v^3 w+6 v^2 w^2+4 v w^3+w^4 \ge 0$ \nwhich is true, since $u, v, w \ge 0$ .