Problema Shortlist ONM 2008

[DrAGos Calinescu]

Fie \( n\in \mathbb{N}, n\ge{2} \) si \( a_1,a_2,...,a_{2n} \) numere reale strict pozitive astfel incat \( a_1+a_2+...+a_{2n}=s \). Demonstrati inegalitatea:

\( \frac{a_1}{s+a_{n+1} - a_1}+...+\frac{a_n}{s+a_{2n}-a_n}+\frac{a_{n+1}}{s+a_1-a_{n+1}}+...+\frac{a_{2n}}{s+a_n-a_{2n}}\ge1 \)

[Marius Mainea]

Folosind CBS,

\( LHS=\sum{\frac{a_i^2}{sa_i+a_{n+i}a_i-a_i^2}}\ge \frac{(a_1+a_2+..+a_{2n})^2}{s^2+2a_1a_{n+1}+2a_2a_{n+2}+...+2a_na_{2n}-a_1^2-a_2^2-...-a_{2n}^2}\ge 1 \)