Question 106
For positive real numbers $a$, $b$, $c$ and $d$ such that $\sum_{cyc} \frac{1}{ab} = 1$, prove that
$$
abcd + 16 \geq 8\sqrt{(a+c)\left(\frac{1}{a} + \frac{1}{c}\right)} + 8\sqrt{(b+d)\left(\frac{1}{b} + \frac{1}{d}\right)}.$$
$$
abcd + 16 \geq 8\sqrt{(a+c)\left(\frac{1}{a} + \frac{1}{c}\right)} + 8\sqrt{(b+d)\left(\frac{1}{b} + \frac{1}{d}\right)}.$$