Adjective clauses with prepositions

Question

Are all of the following sentences correct?

1. New York is a city to which I have traveled many times.
2. New York is a city which I have traveled to many times (not "many times to"?).
3. This is the problem for which there is no perfect solution.
4. This is the problem which there is no perfect solution for.

I used to think in adjective clauses with prepositions such as "which", "whom" we can put the preposition only before them.

All of the items are from the keys. The original task is to fill tha gaps (there might be several correct answers):

1. New York is a city _ which I have traveled _ many times. (to)
2. This is the problem _ which there is no perfect solution _. (for)

Answer 1

1. New York is a city to which I have traveled many times. (Formal, correct)
2. New York is a city which I have traveled to many times (Correct)
3. This is the problem for which there is no perfect solution. (Formal, correct)
4. This is the problem which there is no perfect solution for. (Sounds uncommon)

I would use "to" instead of "for":

• This is the problem to which there is no perfect solution.