Question 1
Let $n \ge 2$ be a positive integer. Determine all possible values of the sum
$$
S = \lfloor x_2 - x_1 \rfloor + \lfloor x_3 - x_2 \rfloor + \dots + \lfloor x_n - x_{n-1} \rfloor,
$$
where $x_1, x_2, \dots, x_n$ are real numbers whose integer parts are $1, 2, \dots, n$, respectively.
$$
S = \lfloor x_2 - x_1 \rfloor + \lfloor x_3 - x_2 \rfloor + \dots + \lfloor x_n - x_{n-1} \rfloor,
$$
where $x_1, x_2, \dots, x_n$ are real numbers whose integer parts are $1, 2, \dots, n$, respectively.