$(a)$ Let $a_1,a_2,a_3$ be three real numbers. Prove that $(a_1-a_2)(a_1-a_3)+(a_2-a_1)(a_2-a_3)+(a_3-a_1)(a_2-a_2)\geq 0$. $(b)$ Prove that the inequality above doesn't hold if we use four number instead of three.
Source: Kosovo MO 2009 Grade 12, Problem 4
Tags: algebra
$(a)$ Let $a_1,a_2,a_3$ be three real numbers. Prove that $(a_1-a_2)(a_1-a_3)+(a_2-a_1)(a_2-a_3)+(a_3-a_1)(a_2-a_2)\geq 0$. $(b)$ Prove that the inequality above doesn't hold if we use four number instead of three.