Mehedinti 2006

[alex2008]

Fie \( a,b,c \) numere reale pozitive . Sa se arate ca :

\( \sum_{cyc}\frac{4a}{2a^2+b^2+c^2}\le \frac{1}{a}+\frac{1}{b}+\frac{1}{c} \)







Manuela Prajea , Etapa locala Mehedinti 2006

[Marius Mainea]

Fie \( a,b,c \) numere reale pozitive . Sa se arate ca :

\( \sum_{cyc}\frac{4a}{2a^2+b^2+c^2}\le \frac{1}{a}+\frac{1}{b}+\frac{1}{c} \)

\( LHS\le\sum{\frac{4a}{4\sqrt[4]{a^2\cdot a^2\cdot b^2\cdot c^2}}}=\sum{\frac{4a}{4a\sqrt{bc}}=\sum{\frac{1}{\sqrt{b}\sqrt{c}}\le\sum{\frac{1}{a}} \)

[alex2008]

Altfel \( LHS \le \sum_{cyc}\frac{2}{b+c}\le \sum_{cyc}\frac{b+c}{2bc}=\sum_{cyc}\frac{1}{2}(\frac{1}{b}+\frac{1}{c}) \)