Question 140
For $a, b, c \gt 0$, denote $m = \frac{a}{b} + \frac{b}{c} + \frac{c}{a}$, $n = \frac{a}{c} + \frac{c}{b} + \frac{b}{a}$. Find all possible values of $k$ such that the following inequality is true for all $m, n$ defined as above:
$$
(m - n)k^2 + (2m - n - 3)k - 8m + 6n + 6 \le 0.
$$
$$
(m - n)k^2 + (2m - n - 3)k - 8m + 6n + 6 \le 0.
$$