I would like to ask what is the correct preposition after the phrase "get stuck" when you want to figuratively express that you are bogged down. For example you are dealing with a logical task and you are not able to find out the correct solution. Is it "I have got stuck with/on/with…"?
Consider these possibilities:
I am stuck on the database problem.
This means you can't continue because you can't figure out how to solve the database problem.
You could also say
I am stuck at the database problem.
This means much the same thing, with the nuance that you are stuck at the point of the database problem among a set of tasks you are trying to address serially. You probably can't do anything else till you solve that one. If you use on it might suggest you could do other parts of the task while waiting for enlightenment on the database issue. But on and at are pretty much interchangeable here. I would expect to hear on more frequently.
I got stuck with the database problem.
This means someone left you or assigned you the database problem to solve (with overtones of it being an unpleasant or difficult task).
There are several possibilities:
You can get stuck with your mother in law for the weekend.
You can get stuck by the side of the road.
You can get stuck in the mud.
You can get stuck half way up/down a mountain.
You can get stuck climbing through a narrow window.
You can get stuck on a difficult problem - or a window ledge.
You can get stuck between a rock and a hard place.
You can get stuck for hours.
But in your example, you're probably stuck with or on...whichever suits.
If you are unable to find out a solution, you should use the preposition "on" after stuck, for example:
I have got stuck on this problem.
It's appropriate to use the preposition "with" instead of "on in this sentence. You use the "with" when you are unable to get rid of someyhing that you don't want. For example:
I don't like to wash dishes. I am stuck with this job.