Problem

Source: 2010 Romania JBMO TST 5.3

Tags: inequalities, algebra



Let $a, b, c$ be real numbers with the property as $ab + bc + ca = 1$. Show that: $$\frac {(a + b) ^ 2 + 1} {c ^ 2 + 2} + \frac {(b + c) ^ 2 + 1} {a ^ 2 + 2} + \frac {(c + a) ^ 2 + 1} {b ^ 2 + 2} \ge 3 $$.