Question 63
Fix an integer $n \ge 2$. Consider $n$ real numbers $a_1, a_2, \dots, a_n$, not all equal, and let
$$
d = d(a_1, a_2, \dots, a_n) = \max_{1 \le i \lt j \le n} |a_i - a_j|
$$
and
$$
s = s(a_1, a_2, \dots, a_n) = \sum_{1 \le i \lt j \le n} |a_i - a_j|.
$$
Determine, in terms of $n$, the smallest and the largest values the quotient $s/d$ may achieve.
$$
d = d(a_1, a_2, \dots, a_n) = \max_{1 \le i \lt j \le n} |a_i - a_j|
$$
and
$$
s = s(a_1, a_2, \dots, a_n) = \sum_{1 \le i \lt j \le n} |a_i - a_j|.
$$
Determine, in terms of $n$, the smallest and the largest values the quotient $s/d$ may achieve.