Question 4
For $n \ge 2$, let $a_1,a_2,\ldots,a_n$ be positive real numbers such that
$$
\left( a_1 + a_2 + \ldots + a_n \right)
\left( \frac{1}{a_1} + \frac{1}{a_2} + \ldots + \frac{1}{a_n} \right)
\le \left( n + \tfrac12 \right)^2.
$$
Prove that $\max\left( a_1, \ldots, a_n \right) \le 4 \min\left( a_1, \ldots, a_n \right)$.
Proposed by Titu Andreescu (credit in the supplied third-party notes).
$$
\left( a_1 + a_2 + \ldots + a_n \right)
\left( \frac{1}{a_1} + \frac{1}{a_2} + \ldots + \frac{1}{a_n} \right)
\le \left( n + \tfrac12 \right)^2.
$$
Prove that $\max\left( a_1, \ldots, a_n \right) \le 4 \min\left( a_1, \ldots, a_n \right)$.
Proposed by Titu Andreescu (credit in the supplied third-party notes).