UA-UMO 1997 Problem 116

Find all functions $f: \mathbb{Q}^{+} \rightarrow \mathbb{Q}^{+}$ which satisfy the conditions:

$(i)$ $f(x+1)=f(x)+1$ for all $x \in \mathbb{Q}^{+}$

$(ii)$ $f(x^2)=f(x)^2$ for all $x \in \mathbb{Q}^{+}$ .