Problem

Source: Jack Garfunkel, P1490, CRUX 1989/9, p.270, Chinese TST 2007 5th quiz P3

Tags: inequalities, 3-variable inequality, cyclic inequality, square root inequality



Find the smallest constant $ k$ such that $ \frac {x}{\sqrt {x + y}} + \frac {y}{\sqrt {y + z}} + \frac {z}{\sqrt {z + x}}\leq k\sqrt {x + y + z}$ for all positive $ x$, $ y$, $ z$.