Question 160
Let $k \gt 1$ be a positive integer and $n \gt 2018$ be an odd positive integer. The nonzero rational numbers $x_1, x_2, \dots, x_n$ are not all equal and satisfy
$$
x_1 + \frac{k}{x_2} = x_2 + \frac{k}{x_3} = x_3 + \frac{k}{x_4} = \dots = x_{n-1} + \frac{k}{x_n} = x_n + \frac{k}{x_1}.
$$
Find:
a) the product $x_1 \cdot x_2 \cdot \dots \cdot x_n$ as a function of $k$ and $n$;
b) the least value of $k$ such that there exist $n, x_1, x_2, \dots, x_n$ satisfying the given conditions.
$$
x_1 + \frac{k}{x_2} = x_2 + \frac{k}{x_3} = x_3 + \frac{k}{x_4} = \dots = x_{n-1} + \frac{k}{x_n} = x_n + \frac{k}{x_1}.
$$
Find:
a) the product $x_1 \cdot x_2 \cdot \dots \cdot x_n$ as a function of $k$ and $n$;
b) the least value of $k$ such that there exist $n, x_1, x_2, \dots, x_n$ satisfying the given conditions.