USAMO 2009 Problem 4

For $n \ge 2$ let $a_1, a_2, \dots, a_n$ be positive real numbers such that $$(a_1 + a_2 + \dots + a_n) \left( \frac{1}{a_1} + \frac{1}{a_2} + \dots + \frac{1}{a_n} \right) \le \left( n + \frac{1}{2} \right)^2.$$ Prove that $\max(a_1, a_2, \dots, a_n) \le 4 \min(a_1, a_2, \dots, a_n)$.