How to mentor an exceptional high school student? I have a unique and, quite truthfully, humbling opportunity. The parents of an exceptionally talented high school freshman have reached out to me and asked if I might be able to help.
This kid is seriously good; he came to our state high school math contest and blew away the competition. He is active in a local Math Circle and is in extremely capable hands, and his parents inform me that he is extremely active in the Art of Problem Solving and the Worldwide Online Olympiad Training program. 
I think my role, insofar as I could help, would be to introduce him to advanced topics and/or research. I am thinking about suggesting to him that he read Apostol's intro book on analytic number theory, or Stillwell's Naive Lie Theory; he might also enjoy some serious combinatorics and/or learning about the partition function. (Or, ... ?)
Does anyone have any suggestions for helping such a student?
Thank you! -Frank
 A: You have a sweet problem on your hands. I had a similar student a few years ago. (He twice won the Clay Institute  Prize  for  the most original solution to an Olympiad Problem. He just completed his PhD at Stanford in Computer Science.)  
Keep stimulating  his intellect with varied problems. The book of Polya and Szego on problems in analysis is a good source.  Old Putnam problems  are also  good stimuli.  Encourage him to participate at competitions. Once he reaches the national level of the Olympiad he will meet other students  of his caliber  and  he will have a nice  "gang" of his own.   Good  books  on the history of math or biographies of famous mathematicians also help.  Books that made a difference  in your education  could help him too.   
My math teacher gave me this advise  (that goes back to Abel?): learn from the masters, not their disciples, and looking back I realized that he   was   strictly adhering to it when he gave me  things to read.
Very likely  he is more sophisticated in his usage of computers than    you are  and you can learn a thing or two from him in that respect.
A: Tell him about his other opportunities (although perhaps being on AoPS he is already aware of them). Summer programs like 


*

*Ross

*PROMYS

*the Canada/USA Mathcamp

*HCSSiM
and others come highly recommended, and several of their alumni are here on MO, even. (Full disclosure: I went to PROMYS myself.) In a few years I would also recommend that the student apply to the Research Science Institute. (Full disclosure: I also went to RSI.) 
When I was in a similar position (or so I flatter myself!) I would also have appreciated being introduced to the math blogosphere earlier (which I didn't discover until freshman year of university). Tim Gowers (minus the recent Elsevier stuff which is a little less relevant), Terence Tao, and John Baez are all great places to start. 
A: I don't suppose anyone can make meaningful specific suggestions as to what the student might read, since it's probably best for you to suggest a book in a topic you love and are excited about, so that you can pass on your enthusiasm and insight most effectively.
If you think the student could begin a meaningful research problem with you in the timeframe you have to work with him (and if you think you have the time to give), I would suggest seeking a problem which (a) is worth your time research- and career-wise to think about (and thus publishable if finished), (b) has an experimental/exploration aspect, perhaps involving difficult but doable computations either by hand or on computer, and (c) has a wide spectrum for success, ranging from producing new conjectures based on computational evidence to theorems relating to existing literature.
I would then choose a focused literature curriculum for the student based on preparation for the particular problem, rather than geared towards breadth/culture.  No matter how focused you try to be, you will find he has a nearly insurmountable amount to learn, in order to even get started.  That will be a lot of work for you both, but very useful to the student.  I would plan to work through problems and examples in person with the student when you are first getting started, and taper off in some systematic way, so that they have a chance to make progress on their own, but after having seen you be confused and having been confused with you, so they know how that works, and that it is okay.
If you are successful, and the student gets even a glimpse at real research and discovery, it will sustain him through a lot of dull lectures in his future education.  And he will have some insight into the process of turning intuition and computations into definitions and theorems that most of us get only in grad school.   Of course, item (a) is already not so easy for oneself or a graduate student, and (a)-(c) will be very challenging indeed.  But it's a guideline.  I think (a) is important for your morale and his, while (b) is a logistical matter - gives you a place to start. (c) makes sure you have a place to end.
Note that if a high school student does any research, this can be submitted to various competitions like Siemens, Intel, ISEF, state and regional science fairs, etc.  The experience of preparing for these is very worthwhile, in my experience.  If nothing else, it provides deadlines, which force the student to write things down and explain himself.
In addition to the excellent suggestions of Qiaochu, I'd also note that some REU's under special circumstances will take high school students (basically only if they are prepared far better than the average undergrad applicant, as I understand it).
A: I want to make a few quick points. 
First, I think that at the high school stage it is really crucial for a motivated and talented student to have a healthy balance between learning something meaningful (i.e. not get stuck in difficult toy problems leading nowhere) and actually attempting problems where he or she can make some visible progress (i.e. not just reading advanced textbooks but seeing that new things learned can be applied to prove something new and reasonably exciting). 
Second, and this is based on the way I was brought up as a mathematician, I tend to find a problem-based approach to learning more exciting, though of course there are people who learn in a different way. One great instance of a book which is based on this approach, and is free of two potential threats I mentioned above is "Abel's Theorem in Problems and Solutions" by Alekseev (there is both an "official" English translation, and some translation freely available online, I am not in the position to compare them at the moment). This actually, I believe, is an outstanding choice of a book for a talented high school student to work through.
Third, and this is something much more subtle, however exciting it is to do this mentoring job, it is important to remember with what level of responsibility it comes (which I am sure you know!). By the very nature of maths it is very easy to get depressed because of not having any visible outcome of what you are doing, even though on some level progress is being made. It is very important to detect such situations and do something about them.....
A: And don't forget to play ball with him every once in a while.
A: Ask him what his set of values/goals are.  Have him pick a few people he admires or might admire, and ask him what he thinks it will take to get him there (what characteristics of those that he would like to adopt).  If possible, encourage him to introduce himself to those people and ask what they did to get to where they got (and where he thinks he wants to be).  Then tell him what you can do to help him on his way, and do it.
If this is to be a long term association, have him revisit and perhaps rework his values/goals.  Encourage him to explore other aspects of life occasionally, or relate his studies/activities to things other than mathematical.  Test his ability to communicate/teach others.
Gerhard "Ask Me About System Design" Paseman, 2012.02.27
