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I was going through an "article" on the "Buffalo Way", where the author said that one should NEVER use the Buffalo Way for proving inequalities in actual real-time contests as it is "highly inelengant". What is the reason behind this notion ? In Mathematics, there are a whole lot of ways to attempt a given question. If the BW provides a proof for some inequality, then why it is given the downvote ?

Michael Rozenberg
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    It's a matter of elegance. In principle there is nothing wrong with brute force methods or methods that require many different cases, but if the same result can be proven conceptually in an organized and streamlined fashion, people prefer the latter. – Mathematician 42 Jan 30 '17 at 12:58
  • @Mathematician42 Can you provide an example to support your comment ? –  Jan 30 '17 at 13:03
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    Compare it to the following extreme: Show that all numbers between $0$ and $1000$ whose sum of digits are divisible by $3$ are divisble by $3$. This can be proven easily using the basics of modular arithmetics. But you could also prove it by listing all such numbers and check for each of them whether it holds. Obviously the latter is not a very smart proof. The Buffalo Way is looked down upon in much the same fashion. Especially in contests where there probably are shorter arguments. – Mathematician 42 Jan 30 '17 at 13:04
  • I don't think it's quite as bad as my example above. The Buffalo Way does give you a more or less algorithmic approach to certain problems and might transform them into easier problems. So I don't believe it is that bad of a method, albeit tedious. In contests people want to evaluate your mathematical creativity as well, and having a fixed tedious solution is certainly not creative. I guess they more like solutions you came up with yourself. – Mathematician 42 Jan 30 '17 at 13:12
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    An extra reason: Olympiad questions are written very cleverly by very clever people. When they write an inequality question (or any other kind of question, really), they have in mind certain methods which are "elegant enough" for their tastes. Before they put it on the actual Olympiad, however, they will check to see if brute force methods are a viable approach; if a brute force method is found to work within a reasonable period of time, then they will probably adjust the question so that the brute force method takes longer - probably so long that it becomes unviable. – Will R Jan 30 '17 at 13:44

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I think we can not delete BW from all methods, with there help we can prove polynomial inequalities.

This method is very useful. See here: http://www.artofproblemsolving.com/community/c6h522084

There are polynomial inequalities, which we can prove by nice way,

but to find this way during a competition is very hard.

By the way, BW can give a quick proof sometimes.

There are polynomial inequalities for which there is a proof by BW only.

See here: Olympiad Inequality $\sum\limits_{cyc} \frac{x^4}{8x^3+5y^3} \geqslant \frac{x+y+z}{13}$

By the way, there are polynomial inequalities for which BW does not help.

See here: Inequality $\sum\limits_{cyc}\frac{a^3}{13a^2+5b^2}\geq\frac{a+b+c}{18}$

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Michael Rozenberg
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