Problem

Source: Stars of Mathematics 2017, Juniors, Problem 2

Tags: inequalities



Let $ x,y,z $ be three positive real numbers such that $ x^2+y^2+z^2+3=2(xy+yz+zx) . $ Show that $$ \sqrt{xy}+\sqrt{yz}+\sqrt{zx}\ge 3, $$and determine in which circumstances equality happens. Vlad Robu