Questions & Answers about Niech ja rozwiążę ten problem.
Wait, rozwiążę is the future tense. Why does Niech ja rozwiążę translate to an imperative "Let me solve"?
To create a command or strong suggestion for the first-person singular ("let me..."), Polish pairs the particle niech with the standard first-person verb form. Since solving the problem is a completed action here, we use the perfective verb rozwiązać. Its first-person form is technically future tense (rozwiążę), but when paired with niech, it creates the imperative mood.
Polish usually drops pronouns. Why is ja explicitly included here?
While you can drop it, including ja adds emphasis. Niech rozwiążę ten problem sounds a bit like you are thinking aloud ("Let's see if I can solve this"). Adding ja creates a stronger contrast with other people, making it sound more like "Let me solve this problem (perhaps because someone else couldn't)."
How does this differ from saying Pozwól mi rozwiązać ten problem?
Pozwól mi... literally means "Allow me to..." and is a direct command asking a specific person for permission. Niech ja rozwiążę... is an indirect command expressing a strong declaration of intent. You aren't necessarily asking for a specific person's permission; you are asserting that you should be the one to do it.
How would the meaning change if we used the imperfective verb rozwiązywać instead?