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Bashy min/max algebra

Source: Romania TST 2024 Day 1 P2

July 31, 2024
algebrainequalities

Problem Statement

Let n⩾2n\geqslant 2 be a fixed integer. Consider nn real numbers a1,a2,…,ana_1,a_2,\ldots,a_n not all equal and letd:=max⁡1⩽i<j⩽n∣ai−aj∣;s=∑1⩽i<j⩽n∣ai−aj∣.d:=\max_{1\leqslant i<j\leqslant n}|a_i-a_j|;\qquad s=\sum_{1\leqslant i<j\leqslant n}|a_i-a_j|.Determine in terms of nn{} the smalest and largest values the quotient s/ds/d may achieve.
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