Question 5
Let $n \in \mathbb{N}$, $n \ge 2$, and let $a_1, a_2, \dots, a_n$ be real numbers. Denote
$$
S = \sum_{1 \le i \lt j \le n} |a_j - a_i| \quad \text{and} \quad d = \max\{|a_j - a_i| \mid 1 \le i, j \le n\}.
$$
Prove that
$$
(n-1)d \le S \le \frac{n^2 d}{4}
$$
$$
S = \sum_{1 \le i \lt j \le n} |a_j - a_i| \quad \text{and} \quad d = \max\{|a_j - a_i| \mid 1 \le i, j \le n\}.
$$
Prove that
$$
(n-1)d \le S \le \frac{n^2 d}{4}
$$