Which libraries should I know to solve problems on C++?

Answer 1

It depends of what tipeof problems

http://en.cppreference.com/w/cpp/links/libs

Answer 2

That completely depends on the problems you need to solve.

Eg - For string related problems, <string>, <regex> and <sstream> are the headers you will probably need.

And most of the times, you don't need any, the language itself is sufficient.