Question 89
Suppose $x_1, x_2, \dots, x_n$, with $n \ge 4$, are positive integers arranged in order around a circle so that the sum of the neighbors of each $x_i$ is a multiple of $x_i$ itself, i.e., the fraction
$$
\frac{x_{i-1} + x_{i+1}}{x_i} = k_i.
$$
is an integer, where $x_0 = x_n$ and $x_{n+1} = x_1$. Prove that the sum of these multiples $k_1 + k_2 + \dots + k_n$ satisfies the inequalities
$$
2n \le k_1 + k_2 + \dots + k_n \lt 3n.
$$
$$
\frac{x_{i-1} + x_{i+1}}{x_i} = k_i.
$$
is an integer, where $x_0 = x_n$ and $x_{n+1} = x_1$. Prove that the sum of these multiples $k_1 + k_2 + \dots + k_n$ satisfies the inequalities
$$
2n \le k_1 + k_2 + \dots + k_n \lt 3n.
$$