OLM Dambovita 2009, problema 1

[Claudiu Mindrila]

a) Daca \( x,y,z \in \mathbb{R} \) si \( a,b,c \in (0, \infty) \) demonstrati ca \( \frac{x^{2}}{a}+\frac{y^{2}}{b}+\frac{z^{2}}{c}\ge\frac{\left(x+y+z\right)^{2}}{a+b+c} \).

b) Folosind eventual rezultatul de la a) demonstrati ca daca \( a,b,c \in (0, \infty) \), astfel incat \( a^2+b^2+c^2=1 \) atunci \( \frac{a}{b}+\frac{b}{c}+\frac{c}{a}\ge\left(a+b+c\right)^{2} \).

[salazar]

a) Iese din inegalitatea C.B.S.;
b) \( \sum\frac{a}{b}=\sum\frac{a^2}{ab}\stackrel{(a)}\ge\frac{(a+b+c)^2}{ab+bc+ca}\ge(a+b+c)^2\Longleftrightarrow ab+bc+ca\le 1\Longleftrightarrow \) \( a^2+b^2+c^2\ge ab+bc+ca\Longleftrightarrow (a-b)^2+(b-c)^2+(c-a)^2\ge 0 \).