O inegalitate interesanta

[Marius Dragoi]

Fie \( a_1,...,a_n,b_1,...,b_n \in R \) si \( x \in {\left\[ 0,1 \right\]} \). Aratati ca:
\( (\sum_{i=1}^{n} {a_i^2} + 2x \sum_{i<j}^{} {a_ia_j})(\sum_{i=1}^{n} {b_i^2} + 2x \sum_{i<j}^{} {b_ib_j}) \geq {(\sum_{i=1}^{n} {a_ib_i} + x \sum_{i \leq j}^{} {a_ib_j})}^2 \)