Problem

Source: China Southeast MO 2023 Grade11 Day1 Problem1

Tags: inequalities



Let $a, b>0$. Prove that:$$ (a^3+b^3+a^3b^3)(\frac{1}{a^3} + \frac{1}{b^3} + \frac{1}{a^3b^3} ) +27 \ge 6(a+b+\frac{1}{a} +\frac{1}{b} +\frac{a}{b} +\frac{b}{a}) $$