Hanoi Open Mathematics Competition 2007 Senior Question 9

Let $a_1,a_2,\ldots,a_{2007}$ be real numbers such that $a_1+a_2+\cdots+a_{2007}\ge2007^2$ and $a_1^2+a_2^2+\cdots+a_{2007}^2\le2007^3-1$. Prove that $a_k\in[2006;2008]$ for all $k\in\{1,2,\ldots,2007\}$.