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.