Question 124
Find the largest real number $a$ such that for all $n \ge 1$ and all real numbers $x_0, x_1, \dots, x_n$ satisfying
$$
0 = x_0 \lt x_1 \lt x_2 \lt \dots \lt x_n,
$$
we have
$$
\frac{1}{x_1 - x_0} + \frac{1}{x_2 - x_1} + \dots + \frac{1}{x_n - x_{n-1}} \ge a \left( \frac{2}{x_1} + \frac{3}{x_2} + \dots + \frac{n+1}{x_n} \right).
$$
$$
0 = x_0 \lt x_1 \lt x_2 \lt \dots \lt x_n,
$$
we have
$$
\frac{1}{x_1 - x_0} + \frac{1}{x_2 - x_1} + \dots + \frac{1}{x_n - x_{n-1}} \ge a \left( \frac{2}{x_1} + \frac{3}{x_2} + \dots + \frac{n+1}{x_n} \right).
$$