Question 1
Let $n$ be a positive integer. Prove that for every positive real number $x$,
$$\frac{x^n(x^{n+1}+1)}{x^n+1}\le\left(\frac{x+1}{2}\right)^{2n+1}.$$
When does equality hold?
$$\frac{x^n(x^{n+1}+1)}{x^n+1}\le\left(\frac{x+1}{2}\right)^{2n+1}.$$
When does equality hold?