Is it morally right and pedagogically right to google answers to homework? This is a soft question that I have been struggling with lately.

My professor sets tough questions for homework (around 10 per week).

The difficulty is such that if I attempt the questions entirely on my own, I usually get stuck for over 2 hours per question, with no guarantee of succeeding.

Some of the questions are in fact theorems proved by famous mathematicians like Gauss, or results from research papers.

As much as I dislike to search for answers on the internet, I am often forced to by time constraints if I even expect to complete the homework in time for submission. (I am taking 2 other modules and writing an undergraduate thesis too).

My school does not have explicit rules against googling for homework, so I guess it is not a legal issue.

However, it often goes against my conscience, and I wonder if this practice is counterproductive for my mathematical development.

Any suggestions and experience dealing with this?
Let me explain why I, and almost all faculty members I know, do not want students searching for homework problems online. 


It destroys our ability to calibrate the course difficulty. Twenty hours of homework a week is very high for a math course; higher than I would expect from any course that was not promoted as a "boot camp" style course. Either you are falling behind the rest of the class, or other people are turning in much scantier work than you are, or everyone is googling the problems. The first two situations are obvious, and your professor should be adjusting to it. The last situation is invisible. We had an analysis course at MI last year pedagogically ruined because everyone kept solving the homework problems, so the professor kept increasing his pace, until an in class test revealed that no one was actually doing the homework themselves.
It forces us to use more obscure, and often not as good, problems. There are some fields where there are computations every student should do -- and, as a result, they are written up in books and online sources everywhere. It hurts my ability to design good problem sets if I can't put this fundamental problems on the problem set. Even in fields where there are not such key problems, there are often only so many ways to set up an example so that it is doable in a reasonable amount of time. If I can't use the examples which are already online, then I need to pick larger and stranger values for my parameters, which makes the problem set harder.
I do not believe that students will learn as much from reading a solution as finding it themselves; this is probably uncontroversial. Moreover, I think that hearing a solution from a classmate with whom you have been discussing the problem together is better than hearing it from a classmate who solved it separately; hearing it from a classmate is better than hearing it from a faculty member; and hearing it from a faculty member is better than reading it in a textbook or here on math.SE. I think that the more interactive and the less polished the presentation, the more you have to engage your own understanding to process and take in the answer. This is why I almost never leave full answers to questions that look like homework here; I think it is harmful.


Let me quote the policy I will have for the combinatorial representation theory course I will be teaching this Fall:


  Homework Policy: You are welcome to consult each other provided (1)
  you list all people and sources who aided you, or whom you aided and
  (2) you write-up the solutions independently, in your own language. If
  you seek help from other mathematicians/math students, you should be
  seeking general advice, not specific solutions, and must disclose this
  help. I am, of course, glad to provide help!
  
  I don't intend for you to need to consult books and papers outside
  your notes. If you do consult such, you should be looking for
  better/other understanding of the definitions and concepts, not
  solutions to the problems.
  
  You MAY NOT post homework problems to internet fora seeking solutions.
  Although I know of cases where such fora are valuable, and I
  participate in some, I feel that they have a major tendency to be too
  explicit in their help. You may post questions asking for clarification and 
  alternate perspectives on concepts and results we have covered.


You should ask your professor for his or her policy, but I think that this is on the permissive side of what most math professors would write if they thought about a policy.
Write up your homework solutions in tex and cite your sources.  Soon the professor will want you to help with research.