Question 44
Find the minimal possible value of $\frac{1}{a} + \frac{1}{b} + \frac{1}{c}$ over all positive real numbers $a, b, c$ satisfying
$$
abc = 1, \quad a+b+c = 5 \text{ and}
$$
$$
(ab+2a+2b-9)(bc+2b+2c-9)(ca+2c+2a-9) \geq 0.
$$
$$
abc = 1, \quad a+b+c = 5 \text{ and}
$$
$$
(ab+2a+2b-9)(bc+2b+2c-9)(ca+2c+2a-9) \geq 0.
$$