Question 72
Find the largest subsets $A_1, A_2 \subset (0, \infty)$ such that:
$$
ab + cd \ge \sqrt{a^2 + b^2} + \sqrt{c^2 + d^2}, \quad \forall a, b, c, d \in A_1, \quad (1)
$$
$$
ab + cd \ge \sqrt{a^2 + c^2} + \sqrt{b^2 + d^2}, \quad \forall a, b, c, d \in A_2. \quad (2)
$$
$$
ab + cd \ge \sqrt{a^2 + b^2} + \sqrt{c^2 + d^2}, \quad \forall a, b, c, d \in A_1, \quad (1)
$$
$$
ab + cd \ge \sqrt{a^2 + c^2} + \sqrt{b^2 + d^2}, \quad \forall a, b, c, d \in A_2. \quad (2)
$$