Problem

Source: ELMO Shortlist 2011, A5

Tags: inequalities, geometry, inequalities proposed



Given positive reals $x,y,z$ such that $xy+yz+zx=1$, show that \[\sum_{\text{cyc}}\sqrt{(xy+kx+ky)(xz+kx+kz)}\ge k^2,\]where $k=2+\sqrt{3}$. Victor Wang.