<s>[INST] Proving a reversed AM-HM inequality 
Problem:
Given $n \ge 3$, $a_i \ge 1$ for $i \in \{1, 2, \dots, n\}$. Prove the following inequality:
$$(a_1+a_2+\dots +a_n) (\frac{1}{a_1}+\frac{1}{a_2} + \dots +\frac{1}{a_n}) \le n^2+\sum_{1\le i<j\le n}|a_i-a_j|.$$

This inequality seems like a reversed form of AM-HM inequality (which states that $\operatorname{LHS}\ge n^2$). I know that Kantorovich inequality is of the same direction, but that is not helpful. I tried to apply Cauchy–Schwarz inequality, but failed.
Thanks for any help.
 [/INST] I will prove this via induction. Without loss of generality, assume $a_1 \le \cdots \le a_n$. It turns out this inequality remains true for $n = 1, 2$ as well, reducing to $1 \le 1$ in the first case. We actually need to prove the $n = 2$ case in order to elicit intuition for the inductive step.
Base Case $n = 2$
If $a_1 = a_2$, the inequality is trivially true, so assume $a_1 < a_2$. Then we seek to prove
\begin{align*}
(a_1 + a_2)(a_1^{-1} + a_2^{-1}) \le 2^2 + |a_2 - a_1| &\iff \frac{a_1}{a_2} + \frac{a_2}{a_1} \le 2 + (a_2 - a_1) \tag{*}\\
&\iff 0 \le \frac{a_1^2}{a_2 - a_1} + a_1 - 1
\end{align*}
which is true from the conditions $a_i \ge 1$.
Inductive Step
Define
\begin{align*}
S^A_n = \sum_{i=1}^{n}a_i \qquad \text{and} \qquad S^H_n = \sum_{i=1}^{n}a_i^{-1}
\end{align*}
We will use the relation
\begin{align*}
\sum_{1 \le i < j \le n}|a_i - a_j| = 2\sum_{i=1}^{n}ia_i - (n+1)S^A_n
\end{align*}
So we have
\begin{align*}
S^A_{n+1}S^H_{n+1} &= S^A_n S^H_n + a_{n+1}S^H_n + a^{-1}_{n+1}S^A_n + 1 \\
&\le n^2 + \left[2\sum_{i=1}^{n}ia_i - (n+1)S^A_n\right] + a_{n+1}S^H_n +  a^{-1}_{n+1}S^A_n + 1 & \text{(Inductive Hypothesis)} \\
&\overset{\text{def}}{=} M
\end{align*}
To conclude, we want to show that $M$ does not exceed
\begin{align*}
N &\overset{\text{def}}{=} (n+1)^2 + \left[2\sum_{i=1}^{n+1}ia_i - (n+2)S^A_{n+1}\right]
\end{align*}
Computing the difference,
\begin{align*}
N - M &= 2n + 1 + 2(n+1)a_{n+1} - S^A_n - (n+2)a_{n+1} - a_{n+1}S^H_n - a^{-1}_{n+1}S^A_n -1\\ 
&= 2n + na_{n+1} - S^A_n - a_{n+1}S^H_n - a^{-1}_{n+1}S^A_n 
\end{align*}
Finally, we have
\begin{align*}
na_{n+1} - S^A_n &= \sum_{i=1}^{n}(a_{n+1} - a_i) \\
&\ge\sum_{i=1}^{n}\left(\frac{a_i}{a_{n+1}}+\frac{a_{n+1}}{a_{i}}-2\right) & \text{From (*) in base step}\\
&=a_{n+1}S^H_n + a^{-1}_{n+1}S^A_n - 2n
\end{align*}
</s>