Let $a, b, c$ be different real numbers. prove that $$\left(\frac{2a-b}{a-b}\right)^2+ \left(\frac{2b- c}{b-c}\right)^2+ \left(\frac{2c-a}{c-a}\right)^2 \ge 5. $$
Source: 2006 Cuba MO 2.3
Tags: algebra, inequalities
Let $a, b, c$ be different real numbers. prove that $$\left(\frac{2a-b}{a-b}\right)^2+ \left(\frac{2b- c}{b-c}\right)^2+ \left(\frac{2c-a}{c-a}\right)^2 \ge 5. $$