How can an extremely mathematically talented young person be helped to fulfill his/her potential? Obviously, this question is not a research level mathematics question at all. But, I've just met an extremely mathematically talented $11$ years old student and I don't know how I can help him. For years I was working at a special school for young gifted and talented students. But, I had never met such talent at such a young age in my life. I talked to a mathematician friend of mine (who himself was an IMO gold medalist at a very young age), but he had no idea unfortunately. Basically, there is no one around with any idea. That is why I came to MO. Just in case that this question gets closed please e-mail me at asghari.amir at gmail if you have any idea.
 A: Hmm. I was that kid. I was/am very strong in math. I was in a very strong gifted program from gradeschool to high school. I went through various Academic Olympiads. I have a trophy where I was the second best HS kid in Chicago one year...
Why second? Because I didn't try. I didn't like trying. I flunked several classes in HS and college. I have severe anxiety at times, though getting better - a very supportive wife makes me wonder what I could have done if I dealt with anxiety sooner. My parents both had anxiety and depression. There were other home issues that made me more anxious and made "success" not that important. I read everything I could find, but the things I was supposed to read. No risk of judgement there.
My point being, all the books you could give me wouldn't have helped. My issues were mental, and some of safety. If you want this kid (or any kid for that matter) to reach their full potential, issues of mental state, safety, even food and shelter may be more important than any books you could give them. I'm not saying this person has my rollup of stuff, just to be aware that there is a lot more out there than access to contests. With the Internet, access to learning materials is 1000x easier than when I was that age, so your help will be even more needed on the mental side.
One big thing I forgot to add is - struggle, and how to deal with "failure". Though learning how to push through is hard for everyone, a smart kid has a lot of expectations on them. Help them deal with not getting it right the first time. The confusion of "this is normally so easy". Help them with patience to work it through, to not get frustrated or quit because something is hard for once. That's probably the lesson I most wish I learned earlier.
A: Give him some good books to read, and some olympiad (competition) problems to solve.
Of course the choice of books is individual. Much depends on his location/residence. In larger cities you can always find mathematicians who are willing to work with young students, "mathematical circles" etc. But sometimes in smaller cities these can be found too. Search in your (his/her) neighborhood. Sometimes, parents of such children are even willing to change their residence to the place where an
appropriate teacher is available. 
A: *

*He has to meet other "extremely mathematically talented young persons" on regular basis. Summer schools maybe? [He will learn more from kids of his age and this will help him to realize that he is actually talented.]

*Teach him how to read math books. [On the big scale that is the most important skill.
Make a reading course for him; i.e., make him to read a book, meet once in a week and discuss it.]
A: What matters here is not how to help him/her to fulfill his/her potential, but how to help him/her be happy in life. Being different is not easy to deal with, and often leads to loneliness or social difficulties. If I had such a kid, I'd probably prefer that he/she lived a normal life rather than spending his/her free time dealing with abstractions. Even though the mathematical world is amazingly beautiful, it's not the world we human beings live in. Automorphisms groups of L-functions and prime numbers won't give you love and happiness.
A: When I play basketball (recreationally of course) I prefer to play with those just slightly better than me because that will push me to the next level.  I generally avoid playing with someone who is totally out of my league because not only will I get my ass whooped, I would learn nothing and morale and drive would be depressed for a while - counterproductive to the goal of bettering myself.
You can apply the same strategy to the kid.
A: Wilmar H. Shiras wrote "In Hiding", a science fiction story about a young boy who was smart enough to make himself appear as a B-grade student, but led another life being incredibly creative, like a super Leonardo da Vinci.  I often wish I had done some of the things after the fashion of this young boy, with the self confidence that whatever I did would be of interest to me as well as beneficial to society, without having to worry about being "In Hiding".
I suggest not only not pushing, nor just asking this person to find their own way.  I recommend preparing yourself for such people by becoming an ideal mentor.  One aspect of this is to have a collection of resources on hand: your main gift as mentor is to hook them up quickly with something potentially helpful.  Inspirational stories (for me "In Hiding" was one, although it might not be appropriate for everyone), interesting ideas for projects, other people and fora to involve, occasional reminders that moderation is key (to paraphrase Henry Sanders, don't always believe that "Mathematics isn't everything; it's the ONLY thing!"), and constant modelling of core values (e.g. "It's OK to make mistakes, as long as you fix them", "writing thank you notes is good", "share the credit") are important to being a good mentor.  This young person will let you know if they need a coach rather than an advisor, in which case you can switch to training mode with the agreement of everyone involved.  Just don't start out in training mode, however much you may want to do so.
If you are a good mentor and friend, and this person does have the potential you believe he does, don't keep reminding them of their potential.  Do keep reminding them of the faith you have in their potential greatness, even if it should eventually lie outside of mathematics.  This is what I think you should do, as a good mentor and friend.
Gerhard "Treat Everybody As Potentially Great" Paseman, 2013.12.13
A: Why not suggesting him to play with math software? As a first idea I am thinking about wxMaxima and GeoGebra, both multi-platform and open source. There are more examples, some of them commercial software, you name it. The possibility to experiment with math, with an assistant that draws the graphs for you or performs the repetitive and boring tasks helps the user to buid intuition and get a playful feeling for basic concepts in math.
A: As Athanakor mentioned, don't push him too much, I think the best way is to provide him the right material, which means you have to choose and guess his interests.
I suggest testing: programming experience, history background in mathematics, and opening him to new fields. He has at least 7 years ahead, there are many things that he can do, the most important is that he know what exists, and what is interesting.


*

*Programming: gives you a nice entry point in logic manipulation. If he delves enough he will encounter very abstract Computer Science problems. Programming gives you a nice way to interact with your objects, as well as practical skills, which are always useful. 

*History: I understand he is very gifted, but let's not underestimate the complexity of "ancient maths". I'm thinking about demonstrating Euclide theorems, maybe the Chinese Remainder Theorem as well, which gives him natural entry points to number theory and algebra.

*New fields: physics, biology, sociology, computer science. He may be bright in maths but he is still under development and may have different interests (I might receive death threats stating that on a maths forum).

A: I agree with other suggestions regarding books and correspondence programs. I would be cautious though regarding Olympiads. 
Although they are good for introducing kids to their peers interested in the same things and present interesting challenges, IMO (in my opinion) IMOs (international math olympiads) are counter-productive in the long run for the following reason:
Success usually comes as a syzygy of talent and tenacity. It is important at early age to learn how to stay with a problem for a long time, try different approaches, keep it in the back of your mind when doing something else. Olympiads, on the other hand, train participants to attack a problem quickly, focus on it for perhaps 20 min, and then switch completely to the next unrelated one. Good for focus, bad for tenacity. 
The difference between the mode of mathematical thinking promoted by Olympiads and the one required for successful research is akin 100 m dash versus 10000 m run, which require completely different stamina.
I am not trying to say here that Olympiads are necessarily bad. Of course there's something very useful there, in particular exposure to other kids interested in Mathematics. However, they are only good in moderation, and the kid should be completely clear that Olympiads are not representative at all of how one should study and research Mathematics. A socializing even, a side show, but not the main feature.
EDIT: to clarify, I am not advocating against Olympiads as one of the tools for learning Mathematics, I am advocating against winning Olympiads becoming a major goal for learning Mathematics. As discussed in the comments, training to win Olympiads promotes focus span different from the one needed for research.
A: It depends on how talented. 


*

*If it is the case of someone who still has to discover what 'pros' call maths, then yes, books, Khan Academy, Olympiads, Gelfand Correspondence program (Russian or American version), programming...whatever it takes to keep him/her entertained and aware that there is more to it. 

*If it is someone who has already opened and understood undergraduate (or graduate) level textbooks, I would say do nothing. This is someone who does not need to be pushed, or who has already been pushed hard. No need to make her/him a curiosity. Since that person has presumably a long life in front of him/her, and will surely be noticed soon enough, he/she should enjoys the free time to think independently of new things we have not thought about. There are nice introductions to current mathematical challenges available on the internet (e.g. Terry Tao's blog) if he/she is interested, but roaming freely might be best at first.
A: Do anything but more mathematics, make sure that he doesn't get trapped in a parallel universe where finding an improved method for prime factorization is more important than everything else.
Truly great minds are those who master more than one field, those who can reason with consideration for a plethora of different fields have the potential to do something truly revolutionary. He better find out himself what to study beyond maths, it doesn't have to be a traditional or anything apparently useful, just make sure that it is not more maths.
But above all else make sure that he has a life beyond being a prodigy. People get weird if they ain't allowed to be normal at least some of the time.
A: Books. Also, introduction to programming. It is more experimental, and helps a lot with learning new things. For example, trigonometry, functions, maps from the plane to the plane, Mobius maps and such things can be experienced with programming in a natural way. 
Wikipedia is great for a brief introduction to a lot of nice mathematics, then there is also Khan academy. In this young age, it is more important to experience the different types of mathematics, and the "fun" mathematics, (problem solving), rather than diving deep down into technical definitions.
A: Everybody seems have opinions and comments to help him/her to be of great success. I doubt whether this is feasible, My point is that try what you can and leave other things alone. 
there are too many variables that you can't deal with. if there do exist such methods that define the process to help him fulfill his potential, then basically everyone on the earth is fulfilling his potential.
Besides, put what you think is the correct thing upon others is always not suggested by me.
Try what you can, and leave other things alone.
A: The advice so far sounds good to me. However, I just want to point out that "extremely mathematical talented" is an extremely ambiguous description, and how one should proceed depends on where the kid lies within the spectrum.
Let me describe one extreme example. I am currently working with a preteen who fits the description of "extremely mathematical talented", and my instinct was to proceed as others here have advised. However, a colleague who was much more experienced than me in mentoring such kids advised me, after he had met the kid, to introduce him to Herstein's book on abstract algebra. I was dubious about doing this to such a young kid, but tried it anyway. There was no real harm, since we could abandon it if the kid showed no aptitude or interest. We went through the whole book, except linear algebra, in a semester. After that we started working through baby Rudin, which he initially didn't like but got the hang of fairly quickly. This spring, again on the advice of my colleague and against my instincts, he is sitting in on two graduate analysis courses, real analysis and topology. Although I had to provide some help at the start with the topology course, he is now able to follow the course rather easily and is in fact one of the better students in the class. I could go on, but I hope this is enough to get the idea.
A: He is a kid.
Maybe brilliant but just a kid.I believe that the best you can do is to make sure that he is good at almost all lessons in school and in mathematics you should give him as much as he wants.
If he wants to learn more,give him more. This way you will not push him or  make him anxious  to be the best among all.
Feed him only when he is hungry.
Consider Gauss.He was the best among all, but he did not have anyone to push him.(as far as i know)
Let him push you.If he does,he might be so gifted and he finds interesting mathematics indeed.
If he does not, let him live his life. 
A: Beware of the one trick pony. Young kids have flexible minds; he may today have an extraordinary mathematical ability, but it is people who can synthesize across multiple fields / disciplines who will really be able to change the world for the better.
I would encourage him to expand his horizons - not just computers, physics, chemistry, biology; but also - and more so - "non-hard" subjects. Learn poetry, psychology, history, languages ... most of all, ENCOURAGE CURIOSITY, and ALLOW MISTAKES. A little competition doesn't hurt - but if you're having fun (winning is fun of course) you will put in the hours needed.
The world does not need 15 year olds with PhDs in mathematics; it needs 15 year olds who are on track to make a truly meaningful contribution (be it in mathematics, or elsewhere) because they can see a little farther than others - and in different directions.
Help this 11 year old become that 15 year old.
A: Although a bit early at this point, I highly suggest that you encourage him to pursue math research, with the support of MIT's PRIMES-USA program. While I personally have not participated in this program, a number of my mathematically-gifted peers have achieved great success with it (e.g., Jonathan Tidor and Rohil Prasad).
A: It seems by your description that the kid is already taking undergraduate math classes (number theory), so Gelfand's school would be a step down. If you have some classes in Iran (I think, this is where you are) for math-olympiad type problems, that might be also good. I think you can simply encourage him to take core  undergraduate and then graduate math classes your university offers. (Real and Complex Analysis, Combinatorics, Topology, Abstract Algebra, etc.) I know, it was done in the case of this user at University of Oregon so you may want to ask him directly about the experience. In our department we are currently doing this with a 10 year old, who is currently taking upper division real analysis. Lastly, the best math book I can recommend to a kid is Courant and Robbins "What is Mathematics", I do not know if it is translated to Persian though. 
Addendum: Two more excellent books at this level, although primarily geometry-oriented: "Geometry and imagination" (by Hilbert and Cohn-Vossen) and 2-volume "Geometry" by Berger. The main reason is that by going straight to university-level classes, the kid might skip much of the geometry which would be a real shame. 
