Problem

Source: Federation of Bosnia, 1. Grades 2008.

Tags: inequalities



For arbitrary reals $ x$, $ y$ and $ z$ prove the following inequality: $ x^{2} + y^{2} + z^{2} - xy - yz - zx \geq \max \{\frac {3(x - y)^{2}}{4} , \frac {3(y - z)^{2}}{4} , \frac {3(y - z)^{2}}{4} \}$