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

Solution: Assume $a_1$ is the largest and $a_2$ is the smallest.
Let $M = a_1/a_2$.
We wish to show $M \le 4$.

In left-hand side of given, write as $a_2+a_1 + \dots + a_n$.
By Cauchy Schwarz, one obtains
\begin{align*}
  \left( n+\half \right)^2 &\ge
    \left( a_2 + a_1 + a_3 + \dots + a_n  \right)
    \left( \frac{1}{a_1} + \frac{1}{a_2} + \frac{1}{a_3}
      + \dots + \frac{1}{a_n} \right) \\
  &\ge \left(\sqrt{\frac{a_2}{a_1}} + \sqrt{\frac{a_1}{a_2}}
    + 1 + \dots + 1 \right)^2 \\
  &\ge \left(\sqrt{1/M} + \sqrt{M} + (n-2) \right)^2.
\end{align*}
Expanding and solving for $M$ gives $1/4 \le M \le 4$ as needed.