Suppose that $a_1,a_2,...,a_n$ are positive real numbers.Also assume that $b_1,b_2,...b_n$ are an arbitrary permutation of $a_i$'s. Prove that: $$\sum_{i=1}^n a_i^2\geq \sum_{i=1}^n a_ib_i.$$
We know that $\sum\limits_{i=1}^n a_i=\sum\limits_{i=1}^n b_i$ and $\sum\limits_{i=1}^n a_i^2=\sum\limits_{i=1}^nb_i^2$ . I think the rearrangement inequality can't be used to prove it.Maybe some other classic inequality should be used...