RO-NMO 2011 Problem 91

Let be a natural number $n$ and $n$ real numbers $a_1,a_2,\ldots ,a_n$ such that $$a_m+a_{m+1} +\cdots +a_n\ge \frac{(m+n)(n-m+1)}{2} , \forall m\in\{ 1,2,\ldots ,n \} .$$ Prove that $a_1^2+a_2^2+\cdots +a_n^2\ge\frac{n(n+1)(2n+1)}{6} .$