Inegalitate cu trei variabile in [1,2]

[alex2008]

Fie \( a,b,c\in [1,2] \) . Sa se demonstreze ca :

\(
\frac{b\sqrt{a}}{4b\sqrt{c}-c\sqrt{a}}+\frac{c\sqrt{b}}{4c\sqrt{a}-a\sqrt{b}}+\frac{a\sqrt{c}}{4a\sqrt{b}-b\sqrt{c}}\ge 1 \)

[Marius Mainea]

Aplicam CBS

\( LHS=\sum {\frac{\sqrt{ab}^2}{4b\sqrt{ac}-ac}}\ge\frac{(\sqrt{ab}+\sqrt{bc}+\sqrt{ca})^2}{4\sqrt{abc}(\sqrt{a}+\sqrt{b}+\sqrt{c})-ab-bc-ca}\ge RHS \)