I just had an exam where I had to solve an induction problem, unlike any I'd previously done.
Let $f:$ $\mathbb{Z} \rightarrow \mathbb{R^+}$. Suppose $f(1)=1$ and $f(x-y) = \dfrac{f(x)}{f(y)}$ for all $x,y \in \mathbb{Z}$. Find $f(n)$ for all $n \in \mathbb{N}$. Prove by induction.
Additionally, I'm interested in developing a general strategy to solve problems I haven't experienced before. Any tips would be greatly appreciated.
First off all, we can notice that $f(0)=f(1-1)=\frac{f(1)}{f(1)}=1$. From that we can deduce the following: $f(-1)=f(0-1)=\frac{f(0)}{f(1)}=\frac{1}{1}=1$. Now we can easily see that $f(x+1)=f(x-(-1))=\frac{f(x)}{f(-1)}=\frac{f(x)}{1}=f(x)$ so we will guess that $f(x)=1,\;\; \forall x\in\mathbb{N}$ Let's prove it by induction:
Base Case $(n=0)$: We have proven that f(0)=1
Induction step $(n\Rightarrow n+1)$: we assume that $f(n)=1$, we saw that $f(-1)=1$, so $f(n+1)=f(n-(-1))=\frac{f(n)}{f(-1)}=\frac{1}{1}=1$
About how to develop a strategy to solve problems, my recommendation is to try to solve problems from past math olympiad, they often require a great deal of creativity and skills in problem solving, not just a vast knowledge in math.