"Why didn't you solve this problem?"
Translation:Почему ты не решил эту задачу?
«Почему́» is a question about the reason (Why?), and «заче́м» is a question about the purpose (What for?). They often overlap. «Почему́» is used much more often, because «заче́м» implies there is some purpose behind the situation, and a lot of things happen without any prior planning.
For example, «Заче́м ты разби́ла ча́шку?» 'What did you break the cup for?' implies that the cup was broken on purpose (and a possible answer could be «Что́бы тебя́ разозли́ть» 'To make you angry', but not «Потому́ что она́ была́ горя́чая» 'Because it was hot'), while «Почему́ ты разби́ла ча́шку?» doesn't have such implications.
We generally don't ask зачем in negative sentences. Not doing something is the natural state of things, so it makes little sense to ask about the purpose of not doing something. Therefore, «Заче́м ты не реши́л эту зада́чу» is not used.
I can't think of any. I think I use «зачем» more often than «для чего».
«Почему» asks for the cause, «зачем» asks for the goal. Sometimes «почему» is used as «зачем», but not vice versa.
More strictly without the context задача - "a task" or "objective, exercise".
Проблема - a problem, or at least an issue, or even :) pain in the ass.
Hello! Solve is part of the verb form didn't solve in this sentence. The combined form didn't solve is the Past tense.