Functional Equation's induction-Extending from $\mathbb{N}$ to $\mathbb{Q_+}$

Question

So I stumbled upon this problem from UkrMO97.

Find all functions $f:\mathbb{Q}_+\to\mathbb{Q}_+$ 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 $x\in\mathbb{Q}_+$.

And I can easily see that we can induct on $x \in \mathbb{N}$ using the condition (i) only. I want to extend this to $\mathbb{Q_+}$, by using (ii). And I do not know how to pull that off. Any help will be appreciated!!

Answer 1

Hint: Given $a,b\in\mathbb{Z}_+$, find an integer $n$ so that $(n+a/b)^2$ differs from $(a/b)^2$ by an integer and use that to solve for $f(a/b)$.

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Note that $(b+a/b)^2=b^2+2a+(a/b)^2$. So, we have $$(b+f(a/b))^2=f((b+a/b)^2)=b^2+2a+f(a/b)^2$$ so $2bf(a/b)=2a$ and thus $f(a/b)=a/b$.