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

Solution

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See also

OIM Problems and Solutions