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Hello,

I have looked around quite a bit on Math Overflow and haven't found a similar question, so I'm posting it here. I understand that this isn't the typical forum for such a question but it's as good as I can find.

Basically, I'm a rising 9th grader and this past year I finished a standard single variable calculus course. I've also had a fair amount of exposure to discrete mathematics and number theory, through through my own 'exploration' and not olympiads, like many other studenst of my age

At the moment I am quite certain that I will want to be doing research in math in some flavor or another. My dilemma at the moment is one torn between spending time channelling into the depths on undergraduate math or working on problem solving skills in preparation for a competition like the Putnam. I have found many a site on the web saying that a top Putnam score is now a rite of passage in this day and age and few low-scorers end up, for some reason unknown to me, doing research at a top level.

I love math for its elegance and ubiquity and the prospect of working through hundreds of mechanical olympiad problems for a single-day competition seems rather dull. When I contrast that to getting an understanding of some of the basic underpinnings of math through books like Axler and Artin, I find myself seriously doubting the merit in that approach.

Does anybody have any advice for someone in my position?

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closed as off topic by Felipe Voloch, MTS, Noah Snyder, Igor Rivin, Ryan Budney May 29 '12 at 1:13

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"...few low-scorers end up, for some reason unknown to me, doing research at a top level." This is false. –  Tyler Lawson May 29 '12 at 0:59
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This is probably not the right forum for this question. I suggest that you try to find a mentor - maybe a professor at a nearby university - that can help you determine what the right path is for you. Personally, I'd say that you should follow your interests. If it seems more interesting to work on undergrad math stuff like linear or abstract algebra, then do that. For what it's worth, lots of people become researchers, even at the top levels, without getting high scores on the Putnam, or even writing the Putnam for that matter. Nobody becomes a researcher without learning linear algebra. –  MTS May 29 '12 at 1:02
    
Not even in high school and you already are pretty sure you know what you want to do. I know this isn't uncommon for extremely advanced students, but I was half-way through college and didn't really know what I wanted to do. I find it hard to imagine anyone disagreeing with this, but do not waste a lot of time preparing for competitions. These have essentially nothing to do with research mathematics. A lot of the best grad students at my university never competed in a single one of these. –  Matt May 29 '12 at 1:17
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If you don't find competition math fun, you really shouldn't feel any pressure to do it. There are certainly excellent mathematicians who enjoyed competition math and did well at it, and there are plenty of mathematicians who didn't enjoy it and never took the Putnam. Just keep learning math you're interested in. Two things that were very useful to me at your age were talking to people in the math department at a local college or university (in my case, Franklin and Marshall) and going to a math summer program (in my case, the Ross program). –  Noah Snyder May 29 '12 at 1:20
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This is irrelevant, but on the subject of being quite sure that you want to do research in math: "A friend of mine who is a quite successful doctor complains constantly about her job. When people applying to medical school ask her for advice, she wants to shake them and yell 'Don’t do it!' (But she never does.) How did she get into this fix? In high school she already wanted to be a doctor. And she is so ambitious and determined that she overcame every obstacle along the way—including, unfortunately, not liking it. Now she has a life chosen for her by a high-school kid." - Paul Graham –  Qiaochu Yuan May 29 '12 at 1:26

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