Past exam of the mathematics course of the University of Cambridge / 2007 / ia / Paper 1 / 7C / i / Solution

Apply the Cauchy-Schwarz inequality to $(x,y,z)$ and $(1,1,1)$ in the Euclidean inner product on ${\mathbb{R}}^3$. It givesAfter expansion and division by two,The equality criterion from the introductory proof says $x=y=z$. One can also see both the inequality and equality case from $x^2+y^2+z^2-xy-yz-zx=\frac{1}{2}[(x-y{)}^2+(y-z{)}^2+(z-x{)}^2]$.