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Note: by "elementary" I mean "without using more advanced theory and tools".

Students are sometimes required or encouraged to solve very difficult problems using limited number of tools and machinery. It is not uncommon in competition-style exams that problems in algebraic/analytic number theory are solved using elementary number theory (i.e. without analysis or abstract algebra). Various other exams in the world also have a limit on the number of tools that can be used. For example, sometimes, people are required to prove something similar to mean value theorem for a specific given function (with an explicit expression) solely by very complicated algebraic computations, without Calculus, because the rigorous $\epsilon$-$\delta$ definition of derivatives and limits are not yet taught.

What are the reasons why we wish to tackle hard problems with elementary methods? And is it beneficial or not?

If we are building a theory from axioms, when we write proofs, we must only use things that are already proven. But in the situation I describe above, we are not building up a theory; instead, we are applying some theory to solve a problem, so in this situation, what are the reasons why we sometimes limit our range of tools? What are some pros/cons of this?

This might be, to some extent, a matter of taste, but it is still interesting to know reasons for this.

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    $\begingroup$ This sort of feels like another question where there's an unstated assumption that the goals of IMO competitions should be the same as the goals of advanced math education. Who cares if someone wants to run a competition like IMO? How's it hurting you? Should we outlaw marathon races because the practical way to travel 26 miles is by bicycle (or car)? Note also that this question is amazingly similar to your previous question: matheducators.stackexchange.com/questions/16858/… $\endgroup$ – guest Oct 22 at 14:23
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    $\begingroup$ In the case of students, this can be like weight-training for football players, and for researchers, see my comment that begins with as I began to realize how one can sometimes. $\endgroup$ – Dave L Renfro Oct 22 at 16:46
  • $\begingroup$ I just don't think using too low-level tools is beneficial to solving a problem because it's like reinventing the wheel or doing much strenuous effort than necessary. eg there's nothing wrong with using all of the course of Algebra to prove FTA with Galois theory. There's an easier topological proof too. FTA cannot be proven with just high school algebra - it can but it takes much more effort, handwaving and sacrificing some rigor? $\endgroup$ – Lenny Oct 22 at 17:59
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    $\begingroup$ Regarding low level tools, it's well known that you can get all sorts of information from hard analysis proofs in analysis (e.g. explicit $\epsilon$-$\delta$ error estimates and the like that typically don't require highly sophisticated ideas) that you can't get from soft analysis proofs in analysis (e.g. Baire category existence results), and vice-versa. See here and here for these terms. $\endgroup$ – Dave L Renfro Oct 22 at 20:23
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    $\begingroup$ Perhaps similar to this. In the Boston Marathon, you are required to travel on foot,you are not allowed to use bicycles or motorcycles. $\endgroup$ – Gerald Edgar Nov 16 at 11:36
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Sharing the impressions of a person who earned 2 IMO bronze medals in his youth, but whose dreams of a successful research career were never truly fulfilled :-)

  1. Mathematics is, indeed, not only about problem solving, but it isn't only about building theory either. Individual mathematicians may place themselves near one of the end points of this "spectrum", but the entire line segment is somewhat continuously populated.
  2. The end points feed (and need) each other. The need to solve problems drives theory. Think of all the theory Newton and Leibniz developed to solve a few physics problems. Think of all the theory developed by people trying to solve Fermat's Last. On the other hand, a new theory usually spawns new problems (calculus again).
  3. In that process, whenever a new kind of a problem emerges, the first attacks on it are chiefly taking the existing tools to their limits. This is what, IMHO, learning to look for elementary solutions prepares the students for. If those attacks fail then we may need some new theory.
  4. On the other hand, the theories we teach to the students also open up new ways of thinking. I'm not sure how to best phrase this, so I poke randomly with: A) theory allows us to put a problem into a new framework, B) conceptualizing a problem helps. Somewhat unsatisfactory word-dropping here, sorry.
  5. I somewhat equate using elementary tools only to getting some actual dirt on my hands. This is necessary. Students need to see a lot of that. Too many kids coming out of high school think that math is all about learning which theory/tool applies to which problem. This is fine for an engineer, may be even a physicist, not sure about a teacher. If we let them go through an entire undergrad program without unlearning that, I think of that as a failure. I mean, when facing a new problem the first reaction should not be to search for an entirely new method.
  6. Giving the students problems that require taking their (necessarily elementary) tools to the limit is also essential for motivating the gifted ones (and sieving them out of the rest of the pack).
  7. Consider an analogy with sports. The professional athletes don't come out of a vacuum. They competed in college, they competed in high school, they competed in junior high, probably earlier. We need to do the same.
  8. I do concede that the type of problems/exercises you seem to be concerned about are not for everybody. A student can become a very competent teacher or engineer without ever needing to strain themselves with elementary tools only (though the engineer may still benefit from developing a similar mindset).
  9. If your concern is that some of the contest problems have become a craft as opposed to an art that is something I can associate with. Such problems are not the ones the contestants will remember though :-)
  10. I cannot reach a conclusion here. My brain is not strong enough. May be it is just that the elementary tools have their own appeal? Selberg got a Fields medal and a position at IAS (Princeton) for an elementary proof of the prime number theorem (among other things).
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