Question 96
Given an integer number $n \ge 2$, determine the minimum value the sum
$$
\sum_{i=1}^{n} x_i^2 \left( 1 + \frac{x_i^{n-2}}{x_1 \cdots x_{i-1} x_{i+1} \cdots x_n} \right)
$$
may achieve, when $x_1, x_2, \dots, x_n$ run through the positive real numbers subject to
$$
\sum_{i=1}^{n} \frac{1}{x_i + 1} = 1.
$$
$$
\sum_{i=1}^{n} x_i^2 \left( 1 + \frac{x_i^{n-2}}{x_1 \cdots x_{i-1} x_{i+1} \cdots x_n} \right)
$$
may achieve, when $x_1, x_2, \dots, x_n$ run through the positive real numbers subject to
$$
\sum_{i=1}^{n} \frac{1}{x_i + 1} = 1.
$$