Learn Spanish (Latin America): Necesitamos una pista para este problema.

Even though it ends in -a, problema is a masculine word (el problema). Many Spanish words ending in -ma (like el sistema or el clima) are masculine. Since it is masculine, we use the masculine word for "this", which is este.

Could I say esto problema?

No, you can't use esto right before a noun. Esto is a neutral word for "this" used when you don't know what an object is (like asking "What is this?") or when talking about a general situation. When pointing out a specific masculine noun, you must use este.

Why do we use para instead of por for "for"?

In Spanish, para is used to talk about purpose, destination, or what something is intended for. Here, the clue is intended for the problem, so para is the right choice.

Why isn't the word for "we" (nosotros) in this sentence?

In Spanish, the ending of the verb tells you who is doing the action. The -amos ending on necesitamos already means "we need." You can add nosotros at the beginning if you want to be emphatic, but it is completely normal to leave it out!