What should an amateur do with a proof of an open problem? Assuming that somebody is not an employee of a university, just a math amateur, and makes a proper proof of some well known open math problem, what should he do with it? Publish on the internet for verification or send to some authorities? Is there a high risk that nobody will treat it seriously?
 A: There's a list out there that I'm sure someone else will be able to attribute that assigns crackpot points to a paper to determine whether or not it's worth reading ( I think John Baez? ).  Explicitly or not, most professionals decide whether or not to read a proof based on the ideas in that list.  
At any rate, regardless of what you've proved, and regardless of whether you're professional or amateur, your best bet to get your paper read is write a good summary introduction that introduces the main ideas and insights in your proof.  Your first couple pages should lay out the new ideas, the main technical tools used, etc  If your insight is obviously flawed, this will help an expert to quickly pick it out and point it out to you.  If it's an old idea known not to work ( but maybe this isn't known to you ), an expert will also be able to pick that up quickly and suggest you make sure you have addressed the old attempts.  If it's a genuinely new idea, though, then that's what's going to make the expert reader continue reading to the body of your proof.  
Agree with frogeyedpeas that arXiv is the best approach.  I think going directly to a journal would be very difficult.  You can try reaching out directly to some professional academics as well, but I strongly recommend you do not send it to the top people at top Universities.  Someone at a local college or University is a good bet.  If you've been able to formulate the new idea yourself without professional training, probably anyone at any University working in math will be able to follow it.  
Also, maybe goes without saying, but do write it up in LaTeX.  No one is going to read a Word document proof.  
A: If an amateur has created a proof of a problem the natural step is to publish it on either arXiv (or if you don't have access) viXra.
At that point you have documented you were the first to get the idea at a certain time. Now if you think it's material worthy of appearing in a paper you should attempt to reach out to relevant journals (which will have a standard method of submitting articles, etc.) and then submit your proof.
Additionally, you may want to reach out to relevant professors and researchers in that particular area to have them look over your stuff (this may take some time to coordinate so be patient!).
Again to be taken seriously in both areas is tricky. If the proof is nothing massively groundbreaking it shouldn't be a problem, but if you hypothetically did prove something huge like P=NP then chances are a good 99.9% of people will laugh you off (if you are an unestablished amateur) regardless whether you are correct or not.
This shouldn't be taken negatively, it has nothing to do with the mathematics community, it is human nature.
A: I'm no expert on this subject, but the impression that I have comes down to this: your work may be very worthwhile, but it's hard to be taken seriously if you cannot communicate your discoveries.
A particular difficulty in getting published in mathematics is that the subject is very technical -- to use a shaky metaphor, mathematicians speak their own language; and, within the field, each subject has its own dialect. If you're working without formal training or regular interaction with other mathematicians, it's naturally going to be difficult to gain the knowledge of terminology, notation, conventions, etc., that is needed to write research-level mathematics.
The problem is that lack of mastery of "the language" is often construed as a lack of understanding of concepts themselves. A professor of mine once mentioned that he receives many papers from amateurs, and in these there are often give-aways (like misuse of terminology and unintelligible symbolic manipulations) which indicate that reading the paper will simply be a waste of time. 
With this in mind, put effort into writing your results clearly and carefully. Strive to use the vernacular of the field in which you have made a discovery. Read/skim papers on subjects close to yours to get an idea how to do this. Post on MSE with terminology questions. If you have trustworthy friends with experience in mathematics, try to get them to read over your proof and give you their candid opinions (both about your exposition and about the validity of your arguments).
Probably far more helpful than my rambling, here is a webpage with extensive advice for amateurs, including suggestions for how to go about publishing. You might also find this MSE thread helpful.
A: If you have solved an open problem and can monetize it then this question is easy.  For example, consider factorization.
If you really developed an algorithm for efficiently factoring large numbers, you could write a paper in latex describing your algorithm and proving its correctness.  You could send that paper to me.  It is likely I will not understand your paper and nothing will happen.  You could send it to leading researchers in the field and they will probably think you are a crank and not read your paper and nothing will happen.
Or you could try some of the RSA challenges.  Collect a bunch of prizes and likely people will be coming to you asking you questions.
A: I think that there are three things you need to do.


*

*Type it in $\LaTeX$. No matter how good your results, or how clear your exposition, if it is written in Word using Equation Editor then noone will take it seriously and it will be lost to the world. Get the source-code of some good papers off of the arXiv to see how to do this well (find a paper, then go "Other formats $\rightarrow$ Download source").

*Establish precedent. I have never quite understood this one thought, but perhaps that is because I am a tad naive. However, I do not think that the best way to do this is to post it on the arXiv, because doing this lets the whole mathematical community see your work and if you have a simple error then you will have lost all credibility (note: you cannot delete papers from the arXiv).

*Send it to someone to proofread. However, this is difficult for all involved. Chances are, no professor will have the time to look at your work (they are busy people, who barely have time for their own research!). Thinking outside the box, I think the best thing you could do would be to cherry-pick a few PhD students and send it to them. They will be flattered, and will have more time to respond (or, at least, will not be getting dozens of similar e-mails every week!). If they think the work is worthy, then they will pass it up the tree. Anyway, if a professor does decide to look at your work then they will probably fob off the checking to a PhD student. For example, during my PhD there was a very insistent man who ended up frequenting the pavement outside my department, waiting for mathematicians to leave and then thrusting copies of his work into their slightly scared hands$^{\dagger}$. In the end, one professor took one for the team, and gave his first-year PhD student the challenge of finding the error.
$^{\dagger}$This is not a good way to make friends.
A: Hardy to Ramanujan: 'Let me put the matter plainly to you. You have in your possession now 3 [Ramanujan had only two] long letters of mine, in which I speak quite plainly about what you have proved or claim to be able to prove. I have shown your letters to Mr. Littlewood, Dr. Barnes, Mr. Berry, and other mathematicians. Surely it is obvious that, if I were to attempt to make any illegitimate use of your results, nothing would be easier for you than to expose me. You will, I am sure, excuse me stating the case with such bluntness. I should not do so if I were not genuinely anxious to see what can be done to give you a better chance of making the best of your obvious mathematical gifts.' Quoted - p181, The Man Who Knew Infinity, Robert Kanigel.
