CMO 2006 Problem 1

Let $a_1,a_2,\ldots,a_k$ be real numbers and $a_1+a_2+\ldots+a_k=0$ . Prove that $$\max_{1\leq i \leq k} a_i^2 \leq \frac{k}{3} \left( (a_1-a_2)^2+(a_2-a_3)^2+\cdots +(a_{k-1}-a_k)^2\right).$$