USAMO 1999 — Days 1 and 2 Question 4

Let $a_1,\ldots,a_n$ be $n>3$ real numbers such that $\sum_{i=1}^na_i\ge n$ and $\sum_{i=1}^na_i^2\ge n^2$. Prove that $\max(a_1,\ldots,a_n)\ge2$.