Question 166
Let $a$, $b$, $c$, $d$ be positive real numbers lying in the interval $[1, 2]$ that also satisfy the equation $(a + c)(b + d) = 8$. Prove that the inequality
$$
\frac{1}{a^2 + b^2 - 1} + \frac{1}{b^2 + c^2 - 1} + \frac{1}{c^2 + d^2 - 1} + \frac{1}{d^2 + a^2 - 1} \geq 1,
$$
is satisfied for all such quadruples and determine all the cases when the equality holds.
$$
\frac{1}{a^2 + b^2 - 1} + \frac{1}{b^2 + c^2 - 1} + \frac{1}{c^2 + d^2 - 1} + \frac{1}{d^2 + a^2 - 1} \geq 1,
$$
is satisfied for all such quadruples and determine all the cases when the equality holds.