<s>[INST] Prove that $ \max(a_1, a_2, \ldots, a_n)\leq 4\min(a_1, a_2, \ldots, a_n)$ For $ n\geq2$ let $ a_1, a_2, \ldots a_n$ be positive real numbers such that
$$ (a_1 + a_2 + \cdots + a_n)\left(\frac {1}{a_1} + \frac {1}{a_2} + \cdots + \frac {1}{a_n}\right) \leq \left(n + \frac {1}{2}\right)^2.
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Prove that $ \max(a_1, a_2, \ldots, a_n)\leq 4\min(a_1, a_2, \ldots, a_n)$.
Can someone help me [rove this? It's USAMO 2009 .I thought of using cauchy.
 [/INST] Update:
Just realized we can prove directly without taking derivative.
WLOG assume $a_1 \ge a_2 \ge \cdots \ge a_n$, and $a_1 > 4a_n$ then
$$\begin{split} & & \left(\sum a_i \right) \left( \sum \frac{1}{a_i} \right) \\ & = & n + \left(\frac{a_1}{a_n}+\frac{a_n}{a_1}\right) + \sum_{1 < i<j<n} \left( \frac{a_i}{a_j} + \frac{a_j}{a_i}\right) + \sum_{i=2}^{n-1} \left( \frac{a_i}{a_1} + \frac{a_n}{a_i} \right) + 
\sum_{i=2}^{n-1} \left( \frac{a_i}{a_n} + \frac{a_1}{a_i} \right)\\
& > & n+\left(4+\frac 14\right) + 2 \binom{n-2}{2} + (n-2)\left(2\sqrt{\frac{a_n}{a_1}}+ 2\sqrt{\frac{a_1}{a_n}}\right)\\
& > & n + 4 + \frac 14 + (n-2)(n-3) + 2(n-2) \left( 2 + \frac 12 \right)\\ 
& = & n + 4 + \frac 14 + (n-2)(n+2) = n^2 + n + \frac 14 \left(n+\frac 12\right)^2 \\ 
\end{split}$$

Original proof:
We prove by induction that if $\max(a_i) > 4 \min (a_i)$ then $\sum a_i \sum \frac{1}{a_i} > \left(n+\frac 12\right)^2$
When $n=2$ it's true.
Suppose it's true for $n$. WLOG we assume $a_1 \ge a_2 \ge \cdots \ge a_n$ and $a_{n+1}$ is between $a_n$ and $a_1$.  Denote
$$f(x)=\frac{a_1}{x} + \frac{x}{a_1} + \frac{a_n}{x}+\frac{x}{a_n}$$
$$f^{(2)}(x) > 0,f'(x)=\frac{1}{a_1} + \frac{1}{a_n} - \frac{a_1+a_n}{x^2}$$
It's easy to show that $f(x)$ obtains its minimum at $x=\sqrt{a_1 a_n}$, and $$f(\sqrt{a_1a_n})=2 \left(\sqrt{\frac{a_1}{a_n}}+\sqrt{\frac{a_n}{a_1}} \right) > 2(2+\frac 12)=5$$
Define $S_n = \sum_{i=1}^n a_n, T_n = \sum_{i=1}^n 1/a_n$. Then
$$
\sum_{i=1}^{n+1} a_i \sum_{i=1}^{n+1} \frac{1}{a_i} = (S_n+a_{n+1})\left(T_n+\frac{1}{a_{n+1}}\right) \\
= S_n T_n + 1 + \sum_{i=2}^{n-1} \left(\frac{a_{n+1}}{a_i} + \frac{a_i}{a_{n+1}}\right)+f(a_{n+1})\\
> \left(n+\frac 12\right)^2 + 1 + 2(n-2)+5=\left(n+1+\frac 12\right)^2.\blacksquare
$$
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