Question 16
Find the greatest positive integer $k$ for which the following holds: For all positive real numbers $a$, $b$ and $c$ with $abc = 1$, we have the inequality
$$
\frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{k}{a+b+c+1} \ge \frac{k}{4} + 3.
$$
$$
\frac{1}{a} + \frac{1}{b} + \frac{1}{c} + \frac{k}{a+b+c+1} \ge \frac{k}{4} + 3.
$$