Constanta 2006

[alex2008]

Fie \( a , b , c \in \mathb{R}_+^* \) . Aratati ca :
\( a\sqrt{{b+c}}+b\sqrt{c+a}+c\sqrt{a+b}\le\sqrt{2(a+b+c)(ab+bc+ca)}; \)

\( \frac{a}{sqrt{b+c}}+\frac{b}{sqrt{c+a}}+\frac{c}{sqrt{a+b}}\ge(a+b+c)sqrt{\frac{a+b+c}{2(ab+bc+ca}} \)


Cezar Lupu

[Marius Mainea]

\( LHS=\sum{\sqrt{a}\sqrt{a(b+c)}}\le\sqrt{(a+b+c)[a(b+c)+b(c+a)+c(a+b)]}=RHS \)

[Marius Mainea]

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