Find largest interval
Source: 40th Austrian Mathematical Olympiad 2009, round 1, problem 1
August 5, 2009
inequalitiesinequalities proposed
Problem Statement
Find the largest interval M \subseteq \mathbb{R^ \plus{} }, such that for all , , , the inequality
\sqrt {ab} \plus{} \sqrt {cd} \ge \sqrt {a \plus{} b} \plus{} \sqrt {c \plus{} d}
holds. Does the inequality
\sqrt {ab} \plus{} \sqrt {cd} \ge \sqrt {a \plus{} c} \plus{} \sqrt {b \plus{} d}
hold too for all , , , ?
( \mathbb{R^ \plus{} } denotes the set of positive reals.)