Question 20
Let $a$, $b$, $c$, and $d$ be positive real numbers satisfying $a b c d = 1$. Prove that
$$
\frac{1}{\sqrt{\frac{1}{2} + a + a b + a b c}} + \frac{1}{\sqrt{\frac{1}{2} + b + b c + b c d}} + \frac{1}{\sqrt{\frac{1}{2} + c + c d + c d a}} + \frac{1}{\sqrt{\frac{1}{2} + d + d a + d a b}} \geq \sqrt{2}.
$$
$$
\frac{1}{\sqrt{\frac{1}{2} + a + a b + a b c}} + \frac{1}{\sqrt{\frac{1}{2} + b + b c + b c d}} + \frac{1}{\sqrt{\frac{1}{2} + c + c d + c d a}} + \frac{1}{\sqrt{\frac{1}{2} + d + d a + d a b}} \geq \sqrt{2}.
$$