Question 33
Let $k \gt 1$ be a real number, $n \ge 3$ be an integer, and $x_1 \ge x_2 \ge x_3 \ge \dots \ge x_n \gt 0$ be real numbers. Prove the inequality:
$$
\frac{x_1 + kx_2}{x_2 + x_3} + \frac{x_2 + kx_3}{x_3 + x_4} + \dots + \frac{x_{n-1} + kx_n}{x_n + x_1} + \frac{x_n + kx_1}{x_1 + x_2} \ge \frac{n(k+1)}{2}.
$$
$$
\frac{x_1 + kx_2}{x_2 + x_3} + \frac{x_2 + kx_3}{x_3 + x_4} + \dots + \frac{x_{n-1} + kx_n}{x_n + x_1} + \frac{x_n + kx_1}{x_1 + x_2} \ge \frac{n(k+1)}{2}.
$$