sqing wrote: Let $a,b,c$ be positive real numbers such that $(a+2b)(b+2c)=9$. Prove that\[\sqrt{\frac{a^2+b^2}{2}}+2\sqrt[3]{\frac{a^3+b^3}{2}}\geq 3.\] It's wrong for $a=b$ and $c\rightarrow+\infty$.