Gh. Titeica 2009, echipe IX-X, problema 1

[mumble]

Daca \( x_1,x_2,...,x_n\in\(0,\infty\) \) si \( n\in\mathbb{N},n\geq 2 \) demonstrati ca:
\( \sqrt{x_1}+\sqrt[4]{x_2}+\sqrt[6]{x_3}+...+\sqrt[2n]{x_n}>\sqrt[n(n+1)]{x_1x_2...x_n}. \)

GM 2/2003

[Marius Mainea]

Folosind AM-GM avem :

\( LHS =\sum 2k\frac{\sqrt[2k]{x_k}}{2k}\ge n(n+1)\sqrt[n(n+1)]{\frac{x_1x_2...x_n}{2^24^4...(2n)^{2n}}}>n(n+1)\frac{\sqrt{x_1x_2...x_n}}{\frac{2^2+4^2+...+(2n)^2}{n(n+1)}}>RHS \)