FKMO 2005 Problem 2

Let $(a_{n})_{n=1}^{\infty}$ be a sequence of positive real numbers and let $\alpha_{n}$ be the arithmetic mean of $a_{1},..., a_{n}$ . Prove that for all positive integers $N$ , $$\sum_{n=1}^{N}\alpha_{n}^{2}\leq 4\sum_{n=1}^{N}a_{n}^{2}.$$