Question 190
Let $a, b, c$ be positive real numbers such that
$$
a + b + c \ge abc.
$$
Prove that at least two of the inequalities
$$
\frac{2}{a} + \frac{3}{b} + \frac{6}{c} \ge 6, \quad \frac{2}{c} + \frac{3}{a} + \frac{6}{b} \ge 6,
$$
are true.
$$
a + b + c \ge abc.
$$
Prove that at least two of the inequalities
$$
\frac{2}{a} + \frac{3}{b} + \frac{6}{c} \ge 6, \quad \frac{2}{c} + \frac{3}{a} + \frac{6}{b} \ge 6,
$$
are true.