Doua inegalitati conditionate

[Andi Brojbeanu]

1. Fie \( a_1, a_2, ..., a_n \in \mathb{R}^{*}_{+} \), cu \( \frac{1}{a_1}+\frac{1}{a_2}+...+\frac{1}{a_n}=1 \). Aratati ca \( \frac{1}{a_1^3+a_2^2}+\frac{1}{a_2^3+a_3^2}+...+\frac{1}{a_n^3+a_1^2}<\frac{1}{2}. \)
Angela Tigaeru, Suceava, Recreatii Matermatice 1/2010

2. Fie \( k>0 \) si \( a, b, c\in [0, +\infty] \) astfel incat \( a+b+c=1 \). Demonstrati ca \( \frac{a}{a^2+a+k}+\frac{b}{b^2+b+k}+\frac{c}{c^2+c+k}\le \frac{9}{9k+4} \).
Titu Zvonaru, Comanesti, Recreatii Matematice 1/2010