"A/an" or "any"

[subhajit123]

Are the following sentences grammatically correct?
1- Do you know a woman who can solve this problem?
2- Do you know any woman who can solve this problem?

Dear teachers, I am not a native speaker. I doubt if my second sentence is also grammatically correct because I think "any" is used with plural countable nouns. So the correct sentence might be "Do you know any women who can solve this problem?". I see many people use "any" followed by singular nouns with this kind of sentence structure. can someone tell me what the rule of using "any" is? And does the first sentence mean the same as the second?

Any suggestions would be helpful.

 

[GoesStation]

Number 1 is correct and natural. Number 2 is possible in a specific situation.

"I'm looking for a woman who can solve this. Can Ms. Merkel solve it?"
"No."
"Can Ms. May solve it?"
"No."
"Can Ms. Palin solve it?"
"Definitely not!"
"Do you know any woman who can solve this problem?"