2006 OIM Problems/Problem 2

Problem

We consider $n$ real numbers $a_1, a_2,\cdots a_n$ that are not necessarily different. Let $d$ be the difference between the largest and the smallest of them and:

$s=\sum_{i < j}^{}\left| a_i-a_j \right|$

Prove that:

$(n-1)d \le s \le \frac{n^2d}{4}$

and find the conditions that these $n$ numbers must meet for each one of the equalities to be verified.

~translated into English by Tomas Diaz. ~orders@tomasdiaz.com

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2006 OIM Problems/Problem 2

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