2009 USAMO Problems/Problem 4

Problem

For $n \ge 2$ let $a_1$ , $a_2$ , ..., $a_n$ be positive real numbers such that

$(a_1+a_2+ ... +a_n)\left( {1 \over a_1} + {1 \over a_2} + ... +{1 \over a_n} \right) \le \left(n+ {1 \over 2} \right) ^2$

Assume without loss of generality that $a_1 \geq a_2 \geq \cdots \geq a_n$ . Now we seek to prove that $a_1 \le 4a_n$ .

as desired.

2009 USAMO (Problems • Resources: AoPS \| ML)
Preceded by Problem 3	1 • 2 • 3 • 4 • 5 • 6	Followed by Problem 5