UKMT Mathematical Olympiad for Girls, 2014

The AM-GM inequality states that, for positive real numbers $x_1, x_2, \ldots, x_n$,

$$\frac{x_1 + x_2 + \cdots + x_n}{n} \geq \sqrt[n]{x_1 x_2 \cdots x_n}.$$

Moreover, equality holds if and only if $x_1 = x_2 = \cdots = x_n$.

Prove that

$$\frac{a}{b} + \frac{b}{c} + \frac{c}{a} \geq 3$$

for all positive real numbers $a$, $b$ and $c$, and determine when equality holds.