Maximizing Reciprocals

Algebra Level 4

Let \(x_1, x_2, \dots, x_n\) be positive reals satisfying \(x_1x_2 \cdots x_n = 1\).

Find the maximum value of \[ \dfrac{1}{n-1+x_1} + \dfrac{1}{n-1+x_2} + \dots + \dfrac{1}{n-1+x_n} . \]