Inegalitate conditionata cu suma de radicali

[Claudiu Mindrila]

Fie \( a,b,c\in\left(0,1\right] \)si \( x,y,z\ge1 \) astfel incat \( \sqrt{x-a^{2}}+\sqrt{y-b^{2}}+\sqrt{z-c^{2}}=\frac{1}{2}\left(\frac{x}{a}+\frac{y}{b}+\frac{z}{c}\right) \).
a) Demonstrati ca \( x+y+z \le 6 \).
b) In ce caz avem egalitate la a) ?

Cecilia Deaconescu, lista scurta, 2005

[Marius Mainea]

a)

\( LHS=\sum{\frac{1}{a}\cdot\sqrt{a^2(x-a^2)}}\le\sum{\frac{1}{a}\frac{x}{2}}=RHS \)

Asadar avem egalitate la AM-GM deci \( a^2=x-a^2 \) si analoagele, etc.

b) Egalitatea are loc daca a=b=c=1 si x=y=z=2.