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I am relatively new to the Olympiad style maths problems and I find the word "bashing" being thrown around a lot in the community and I haven't really understood it. I do know about "co-ordinate bashing" using Barycentric co-ordinates and complex numbers but I've also seen people use the term "bashing" in Number Theory, Combinatorial and Algebraic questions and don't know why.

My Question- What does the term "bashing" generally mean in Mathematics(in the broader sense)?

Also, if anyone could provide me some other powerful problem solving tools like Bashing which are useful for Maths Olympiads I would be highly obliged. By "Problem Solving Tools" I refer to concepts like Inversive, Projective and Homothety in geometry;LTE and Chinese Remainder Theorem in Number Theory; Vieta Root Jumping in Algebra. I only know a few of these and would like to see a comprehensive list of all(or most) of such important techniques. Even a link would suffice.

Thanks in advance.(First question on Stack Exchange).

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    $\begingroup$ Clarification of the term "bashing", and a request for additional problem-solving tools, should probably be separate questions. $\endgroup$ – Blue Mar 4 at 8:12
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"Bashing" is a term for brute force methods, applied with very little cleverness. These are looked down on in contest mathematics, both because they aren't "pretty" and because they tend to take more time and computational effort than is practical in a live contest.

How do you tell if a solution is "bashing"? That's an entirely subjective judgment. The more you like it, the less likely you are to call it bashing.

What's the advantage of methods that might get called bashing? Reliability. Often, that method is something you know will work if you put enough time and effort into it. For example, consider the two top-rated answers to this recently active inequality problem; one is a short and sweet application of a classical inequality to eliminate the square roots, while the other is a very long slog of multivariable calculus and numerical root-finding, finding all twenty critical points of a function in order to find its minimum. The latter is certainly fairly bashy - I might not have gone through with it if I had realized how much work it was from the start - but it's also a complete solution, where the former isn't. After the simplification, that attempt stalls out with no clear next step.

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  • $\begingroup$ This isn't exclusively used in mathematics either. If you're driving a nail into a piece of wood with anything except for a hammer or naildriver, you're bashing it in. It's a common term for brute-forcing anything with whatever tools you got on hand, instead of doing it the right way. Except for those cases where bashing is the only right way, of-course. $\endgroup$ – Mast Mar 4 at 16:09
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I think in general the term bashing is used to describe an approach which is about applying a small set of manipulations over and over until you get an answer. These approaches usually don't require much insight.

One other trait of bashing which I can think of is that it is often very computational. This will usually go hand in hand with bashing's non-insightfulness.

The example you gave of using coordinate geometry to solve geo problems is a prime example of bashing as it is just a bunch of messy algebra after you translate the terms of the problem to coordinate geometry.

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In inequalities examples of bashing they are:

  1. Buffalo Way. See here: https://artofproblemsolving.com/community/c6h522084

  2. uvw . See here: https://artofproblemsolving.com/community/c6h278791

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