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arithmetic means given, find max/min(max)

Source: Serbia & Montenegro 2001 2nd Grade P4

June 2, 2021
inequalitiesalgebra

Problem Statement

Let SS be the set of all nn-tuples of real numbers, with the property that among the numbers x1,x1+x22,…,x1+x2+…+xnnx_1,\frac{x_1+x_2}2,\ldots,\frac{x_1+x_2+\ldots+x_n}n the least is equal to 00, and the greatest is equal to 11. Determine max⁡(x1,x2,…,xn)∈Smax⁡1≤i,j≤n(xi−xj)andmin⁡(x1,x2,…,xn)∈Smax⁡1≤i,j≤n(xi−xj).\max_{(x_1,x_2,\ldots,x_n)\in S}\max_{1\le i,j\le n}(x_i-x_j)\qquad\text{and}\min_{(x_1,x_2,\ldots,x_n)\in S}\max_{1\le i,j\le n}(x_i-x_j).
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