KöMaL 2025 A921

Let $n \ge 2$ be an integer. Let $x_1,\ldots,x_n$ be positive real numbers whose product is $n-1$. Prove that $x_1+\ldots+x_n-\left(\frac{1}{x_1^{n-1}}+\ldots+\frac{1}{x_n^{n-1}}\right)<(n-1)^{1+\frac{1}{n}}.$