AT-OEMO-REG 2009 Problem 1

Find the largest interval $M \subseteq \mathbb{R^ + }$ , such that for all $a$ , $b$ , $c$ , $d \in M$ the inequality $$\sqrt {ab} + \sqrt {cd} \ge \sqrt {a + b} + \sqrt {c + d}$$ holds. Does the inequality $$\sqrt {ab} + \sqrt {cd} \ge \sqrt {a + c} + \sqrt {b + d}$$ hold too for all $a$ , $b$ , $c$ , $d \in M$ ? ( $\mathbb{R^ + }$ denotes the set of positive reals.)