Question 139
Find the maximum integer $k$ such that
$$
\frac{a}{b} + \frac{b}{c} + \frac{c}{a} + k \frac{(a-b)(b-c)(c-a)}{(a+b)(b+c)(c+a)} \ge 3
$$
for all positive real numbers $a, b, c$.
$$
\frac{a}{b} + \frac{b}{c} + \frac{c}{a} + k \frac{(a-b)(b-c)(c-a)}{(a+b)(b+c)(c+a)} \ge 3
$$
for all positive real numbers $a, b, c$.