Inegalitate conditionata

[DrAGos Calinescu]

Fie \( a,b,c\in\mathbb{R}_+ \)astfel incat \( a^2+b^2+c^2+2abc=1. \)Demonstrati ca:
\( ab+bc+ac\le\frac{a+b+c}{2} \)

Shortlist 2003

[Marius Mainea]

Se noteaza \( a=\sin \frac{A}{2} \) \( b=\sin \frac{B}{2} \) \( c=\sin \frac{C}{2} \) cu A, B, C unghiurile unui triunghi.

Deasemenea se poate si \( a=\cos A \), \( b=\cos B \), \( c=\cos C \).