South African Mathematics Olympiad Second Round

It is easy to see that if $a = b$, then the expression is equal to $2$, and a few trials will suggest that $2$ is the smallest value. A proof is that

Let $x = \frac{a}{b}$, so $x > 0$. Then $\frac{a}{b} + \frac{b}{a} = x + \frac{1}{x}$.

By the AM-GM inequality:

$$x + \frac{1}{x} \geq 2\sqrt{x \cdot \frac{1}{x}} = 2\sqrt{1} = 2.$$

Equality holds when $x = 1$, i.e., $a = b$.

Therefore, the smallest possible value is $2$.