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Statements by mathematicians about contest math
Statements by distinguished mathematicians who participated in contest math while in school
Terry Tao, pre-teen IMO participant and Fields Medalist
Terence Tao was a child prodigy who represented Australia at the International Mathematical Olympiad at the ages of 11, 12, and 13. He joined college and finished his Ph.D. at a young age and is now a professor at the University of California, Los Angeles. His research has earned him a Fields Medal, the highest honor for mathematical research.
Tao has written an advice article for high school students considering contest math here. Relevant passage:
But mathematical competitions are very different activities from mathematical learning or mathematical research; don’t expect the problems you get in, say, graduate study, to have the same cut-and-dried, neat flavour that an Olympiad problem does.(While individual steps in the solution might be able to be finished off quickly by someone with Olympiad training, the majority of the solution is likely to require instead the much more patient and lengthy process of reading the literature, applying known techniques, trying model problems or special cases, looking for counterexamples, and so forth.)
Also, the “classical” type of mathematics you learn while doing Olympiad problems (e.g. Euclidean geometry, elementary number theory, etc.) can seem dramatically different from the “modern” mathematics you learn in undergraduate and graduate school, though if you dig a little deeper you will see that the classical is still hidden within the foundation of the modern. For instance, classical theorems in Euclidean geometry provide excellent examples to inform modern algebraic or differential geometry, while classical number theory similarly informs modern algebra and number theory, and so forth. So be prepared for a significant change in mathematical perspective when one studies the modern aspects of the subject. (One exception to this is perhaps the field of combinatorics, which still has large areas which closely resemble its classical roots, though this is changing also.)In summary: enjoy these competitions, but don’t neglect the more “boring” aspects of your mathematical education, as those turn out to be ultimately more useful.
See also an interview with Tao here.