Problem

Source: 2018 Romania JBMO TST 6.2

Tags: inequalities, algebra



Let $k > 2$ be a real number. a) Prove that for all positive real numbers $x,y$ and $z$ the following inequality holds: $$\sqrt{x + y }+\sqrt{y + z }+\sqrt{z + x} > 2\sqrt{\frac{(x + y)(y + z)(z + x)}{xy + yz + zx}}$$b) Prove that there exist positive real numbers $x, y$ and $z$ such that $$\sqrt{x + y }+\sqrt{y + z}+\sqrt{z + x} <k\sqrt{\frac{(x + y)(y + z)(z + x)}{xy + yz + zx}}$$Leonard Giugiuc