Using Cauchy inequality we get $\forall i=1, n$:
$$\begin{align*} x_1 x_2 \dots x_n &= x_1^{n-1} + x_2^{n-1} + \dots + x_n^{n-1} \ge x_1^{n-1} + (n-1) x_1 x_2 \dots x_{n-1} x_{n+1} \dots x_n \\ &\Rightarrow x_1 x_2 \dots x_{n-1} x_{n+1} \dots x_n (x_i - n + 1) \ge x_i^{n-1} \Rightarrow x_i - n + 1 \ge \frac{x_i^{n-1}}{x_1 x_2 \dots x_{n-1} x_{n+1} \dots x_n} \\ &\Rightarrow \prod_{i=1}^n (x_i - n + 1) \ge \prod_{i=1}^n \frac{x_i^{n-1}}{x_1 x_2 \dots x_{n-1} x_{n+1} \dots x_n} = 1, \end{align*}$$
as needed.