WakeUp wrote: Let $a_1,a_2,\ldots ,a_n$ be positive real numbers, and let $S=a_1+a_2 +\ldots +a_n$ . Prove that \[(2S+n)(2S+a_1a_2+a_2a_3+\ldots +a_na_1)\ge 9(\sqrt{a_1a_2}+\sqrt{a_2a_3}+\ldots +\sqrt{a_na_1})^2 \] $LHS \geq (2S+ \sqrt{a_{1}a_{2}} +...+\sqrt{a_{n}a_{1}})^2 \geq 9 ( \sqrt{a_{1}a_{2}} +...+\sqrt{a_{n}a_{1}})^2$

WakeUp wrote: Let $a_1,a_2,\ldots ,a_n$ be positive real numbers, and let $S=a_1+a_2 +\ldots +a_n$ . Prove that \[(2S+n)(2S+a_1a_2+a_2a_3+\ldots +a_na_1)\ge 9(\sqrt{a_1a_2}+\sqrt{a_2a_3}+\ldots +\sqrt{a_na_1})^2 \] Let $\sqrt{a_1a_2}+\sqrt{a_2a_3}+\ldots +\sqrt{a_na_1}=x$. Hence, $S\geq x$ and $a_1a_2+a_2a_3+\ldots +a_na_1\geq\frac{x^2}{n}$. Thus, it remains to prove that $(2x+n)\left(2x+\frac{x^2}{n}\right)\geq9x^2$, which is C-S.