Let $a, b, c$ be positive real number such that $a + b + c \ge \frac{1}{a}+ \frac{1}{b}+ \frac{1}{c}$ . Prove that $ \frac{a}{b}+ \frac{b}{c}+ \frac{c}{a}\ge \frac{1}{ab}+ \frac{1}{bc}+ \frac{1}{ca}$ .
Source: 2009 Romania JBMO TST 3.1
Tags: inequalities, algebra
Let $a, b, c$ be positive real number such that $a + b + c \ge \frac{1}{a}+ \frac{1}{b}+ \frac{1}{c}$ . Prove that $ \frac{a}{b}+ \frac{b}{c}+ \frac{c}{a}\ge \frac{1}{ab}+ \frac{1}{bc}+ \frac{1}{ca}$ .