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Find the largest subset

Source: 2002 Austrian-Polish, problem 4

September 23, 2006

inequalitiesalgebra unsolvedalgebra

Problem Statement

For each positive integer

n

find the largest subset

M(n)

of real numbers possessing the property: n+\sum_{i=1}^{n}x_{i}^{n+1}\geq n \prod_{i=1}^{n}x_{i}+\sum_{i=1}^{n}x_{i} \text{for all}\; x_{1},x_{2},\cdots,x_{n}\in M(n) When does the inequality become an equality ?

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