a^2+b^2+c^2=3...

[Marius Mainea]

Fie a, b, c, x, y,z numere reale pozitive astfel incat \( a^2+b^2+c^2=3. \) Atunci:

\( \frac{x}{a}+\frac{y}{b}+\frac{z}{c}\geq sqrt {xy}+sqrt {yz}+\sqrt {zx} \)

Marius Damian SHL 2006

[Marius Dragoi]

\( \sum_{cyc}{} {\frac {x}{a}} \) \( \frac {Cauchy}{\geq} \) \( \frac {{(\sqrt {x} + \sqrt {y} +\sqrt {z} )}^2}{a+b+c} \) \( \geq \) \( \frac {{(\sqrt {x} +\sqrt {y} + \sqrt {z} )}^2}{3} \) \( \geq \) \( \sqrt {xy} +\sqrt {yz} + \sqrt {zx} \) \( \Leftrightarrow \)

\( \Leftrightarrow \) \( \sum_{cyc}{} {{(\sqrt {x} - \sqrt {y})}^2} \) \( \geq 0 \) adevarat