Question 57
Determine the largest real number $k$ such that the inequality
$$
(k + \frac{a}{b}) (k + \frac{b}{c}) (k + \frac{c}{a}) \le \left(\frac{a}{b} + \frac{b}{c} + \frac{c}{a}\right) \left(\frac{b}{a} + \frac{c}{b} + \frac{a}{c}\right)
$$
holds for all positive real numbers $a$, $b$, and $c$.
$$
(k + \frac{a}{b}) (k + \frac{b}{c}) (k + \frac{c}{a}) \le \left(\frac{a}{b} + \frac{b}{c} + \frac{c}{a}\right) \left(\frac{b}{a} + \frac{c}{b} + \frac{a}{c}\right)
$$
holds for all positive real numbers $a$, $b$, and $c$.