Question 119
Let $a$, $b$, $c$ and $d$ be positive real numbers with $a b c d = 1$. Prove that
$$
\frac{a b + 1}{a + 1} + \frac{b c + 1}{b + 1} + \frac{c d + 1}{c + 1} + \frac{d a + 1}{d + 1} \geq 4
$$
and determine all quadruples $(a, b, c, d)$ for which equality holds.
$$
\frac{a b + 1}{a + 1} + \frac{b c + 1}{b + 1} + \frac{c d + 1}{c + 1} + \frac{d a + 1}{d + 1} \geq 4
$$
and determine all quadruples $(a, b, c, d)$ for which equality holds.