lean_workbook_plus_39152

Here is my approach: \n\nIt is easy to notice that: \n $(a^2 + b^2)(b^2 + c^2)(c^2 + a^2)\ge \frac{8}{9}(a^2+b^2+c^2)(a^2b^2+b^2c^2+c^2a^2)$ \nNow by AM-GM: \n $a^2+b^2+c^2\ge ab+bc+ca$ \n $a^2b^2+b^2c^2+c^2a^2\ge \frac{1}{3}(ab+bc+ca)^2$ \nTherefore, $27LHS\ge 27.\frac{8}{9}(ab+bc+ca).\frac{1}{3}(ab+bc+ca)^2=RHS$ \nOur proof is completed.