AUSTRIAN-POLISH 2002 Problem 4

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 ?