Question 172
Let $x_0, x_1, \dots, x_{2017}$ be a non-decreasing sequence of positive integers. Suppose that $x_0 = 1$ and the subsequence $x_1, x_2, \dots, x_{2017}$ contains exactly 25 distinct positive integers. Show that
$$
\sum_{i=2}^{2017} x_i (x_i - x_{i-2}) \ge 623.
$$
Find the total number of such sequences in the case of equality.
$$
\sum_{i=2}^{2017} x_i (x_i - x_{i-2}) \ge 623.
$$
Find the total number of such sequences in the case of equality.