USAMO 1999 Problem 4

Let $a_{1}, a_{2}, \ldots, a_{n}$ be a sequence of $n>3$ real numbers such that $$a_{1}+\cdots+a_{n} \geq n \text { and } a_{1}^{2}+\cdots+a_{n}^{2} \geq n^{2}$$ Prove that $\max \left(a_{1}, \ldots, a_{n}\right) \geq 2$.