I'm looking for a news site for Mathematics which particularly covers recently solved mathematical problems together with the unsolved ones. Is there a good site MO users can suggest me or is my only bet just to google for them?

As a counterpoint to my somewhat flippant previous answer (which only really applies if one is a specialist in the field), if you are looking at a field in which you are not as much a specialist in, I suggest reading the articles from the Bulletin of the AMS. The articles are designed to be fairly uptodate and expository in nature, and often gives the state of the art in their reviews. Of course, a similar caveat as that to Helge's answer applies: the "news" maybe several months out of date. But considering the glacial paces at which a lot of mathematical refereeing takes place, I think it is quite okay. In the spirit of this answer, you may also find Which journals publish expository work? to be useful. 


Another suggestion Annals: to appear. Also other top journals. If a big problem gets solved, its solution probably gets submitted to a journal of this type, so its to appear lists are what you are looking for. Of course, you only learn about the solution of the problem a few years late (refereeing takes time), but you can be almost certain that the solution is actually correct. 


The Wikipedia page List of unsolved problems in mathematics has a specific (and long) sublist for recently solved problems. 


arXiv.org Any paper worth reading should include some background material and a description of general progress in its introduction section. This is especially true of papers that actually solve a problem, rather than chipping away at some small technicality. 


There is also a list about group theory open problems here : http://www.grouptheory.info/ 

