Alexey86 posted:

Should I put "a" before "problem/question" in sentences like,

We solved (a) problem after (a) problem.
I answered (a) question after (a) question.

Or, are these examples of so called bare noun coordination?

Hi, Alexey86,

You can say:

1. We solved problem after problem:
2. I answered question after question.

or

3. We solved one problem after another.
4. I answered one question after another.

or use, if you know the number of problems or use "all," an adverbial like "in a row":

- We solved all the problems in a row.
- I answered the five questions in a row.

You cannot use "a," but I wouldn't call it a case of "bare noun coordination" (as in "I pronounce you man and wife") for the simple reason that, unlike "and," "after" is not a coordinating conjunction. 

Alexey86 posted:

You cannot use "a" (sorry, I don't know how to make a quote)

Is it because such constructions function like adverbs, i.e. they describe the mode of action unlike sentences like, "I bought a kettle after a frying pan"?

Those structures do look like adverbial expressions such as "day after day" or "step by step," but considering they come after a transitive verb at least the first noun works as a direct object (a typical nominal function). It's hard to say if the prepositional phrase "after + Noun" forms part of the direct object or is an adverbial, but the truth is that most idiomatic phrases are hard to parse.

However, in your sentences you need a direct object. You cannot say:

- Problem after problem, I solved.
- Question after question, I answered.

I agree that those structures can be adverbials (possibly of manner) if there is some other word or phrase functioning as object:

- I solved everything, problem after problem.
- I answered everything, question after question.

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