Question 39
Prove that for all positive integers $n$ and all real numbers $x_1, x_2, \dots, x_n$ the inequality
$$
\sqrt{x_1^2 + 1} + 2\sqrt{x_2^2 + 1} + \dots + n\sqrt{x_n^2 + 1} \ge \sqrt{(x_1 + 2x_2 + \dots + nx_n)^2 + \frac{n^2(n+1)^2}{4}}
$$
holds.
$$
\sqrt{x_1^2 + 1} + 2\sqrt{x_2^2 + 1} + \dots + n\sqrt{x_n^2 + 1} \ge \sqrt{(x_1 + 2x_2 + \dots + nx_n)^2 + \frac{n^2(n+1)^2}{4}}
$$
holds.