Inegalitate proprie cu radicali intr-un triunghi.

[Virgil Nicula]

Sa se arate ca in orice triunghi \( ABC \) exista inegalitatea \( \sum\sqrt {\frac {b + c - a}{a}} + 2\sum\frac {a}{b + c}\ \ge \ 6. \)

[Marius Mainea]

Aplicand ineg \( M_g\ge M_h \)

\( LHS=\sum {\sqrt{\frac{b+c-a}{a}}+2\sum{\frac{a}{b+c}}\ge\sum {\frac{2}{\frac{a}{b+c-a}+1}}+2\sum{\frac{a}{b+c}}=2\sum{\frac{b+c-a}{b+c}}+2\sum{\frac{a}{b+c}}=6=RHS \)