Question 33
Let $n$ be an integer number greater than $2$, let $x_1, x_2, \dots, x_n$ be $n$ positive real numbers such that
$$
\sum_{i=1}^{n} \frac{1}{x_i + 1} = 1,
$$
and let $\alpha$ be a real number greater than $1$. Show that
$$
\sum_{i=1}^{n} \frac{1}{x_i^{\alpha} + 1} \geq \frac{n}{(n-1)^{\alpha} + 1}
$$
$$
\sum_{i=1}^{n} \frac{1}{x_i + 1} = 1,
$$
and let $\alpha$ be a real number greater than $1$. Show that
$$
\sum_{i=1}^{n} \frac{1}{x_i^{\alpha} + 1} \geq \frac{n}{(n-1)^{\alpha} + 1}
$$