Show that the following problem is decidable: Given a context-free grammar $G$, does $G$ generate any odd-length, nonempty strings?
The answer would be yes, but showing how the problem is decidable? I am not sure, I think I should list a series of steps involving multiple FSMs but I feel like I should get an idea of how others would solve this problem first. I assume there are a wide array of possible ways to show decidability.
The set of odd-length strings is a regular language, say $R$. Now, if $L$ is the language generated by your grammar, your question amounts to ask whether $L \cap R$ is nonempty. This question is decidable. Indeed, $L \cap R$ is context-free (and there is an algorithm to generate a grammar for it) and one can decide whether a context-free language is empty or not.