# Minimal value

**Algebra**Level 5

Given an integer \(\displaystyle n \ge 2\) .The minimal value of \[ \displaystyle \frac{x_{1}^{5}}{x_{2}+{x_{3}}+…+x_{n}} + \frac{x_{2}^{5}}{x_{1}+x_{3}+…+x_{n}} +…+ \frac{x_{n}^{5}}{x_{1}+…x_{n-1}} \] ,for positive real numbers \( x_{1},x_{2},…,x_{n}\) subject to the condition that sum of their squares is 1 , can be expressed as \(\displaystyle \frac{\alpha}{n(n-\beta)}\) where \(\displaystyle \alpha\) and \(\displaystyle \beta \) are positive integers then what is the value of \(\displaystyle \alpha+\beta\)