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 ? inequalitiesalgebra unsolvedalgebra