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I've been solving a lot of math contest inequality problems last few days and sometimes when I solve the problem I can easily ''see'' the idea behind it's creation (for an example, one clever substitution and you get something equivalent to some well-known inequality).

But sometimes, even after solving the problem, I don't have the faintest idea how someone could even conjecture it. Usually, I'm left with the feeling I took the long way to the solution, and that there should be a shorter, more elegant one.

I'm sure there are people on math.se who contribute to math competitions and I would really like to see their process behind creating an inequality problem (on some example contest problem). I hope this question isn't too soft.

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    $\begingroup$ In the case of contest math, it is often the other way round: You start with a well-known inequality and make a clever transformation away from it. So the problem inequalities are not conjectured and then proved, they are hand-crafted for the contest right away. (I know what I am talking about: Long ago, during my own training for the IMO, in one of the training lectures we were shown some very nice examples of how to create intriguing inequality problems from simple standard iniequalities) $\endgroup$ – Hagen von Eitzen Aug 18 '14 at 10:31
  • $\begingroup$ What you ask for might be related to the concept of majorization. Hardy, Littlewood and Polya had already developed the theory to a very advanced level. If you are willing to pursue the study, you should probably check the book "Inequalities: Theory of Majorization and Its Applications". But again my suggestion could be useless, if it is as @HagenvonEitzen has suggested, that it is only the contest you focus on. $\endgroup$ – Troy Woo Aug 18 '14 at 10:37
  • $\begingroup$ @HagenvonEitzen How come you participated IMO and out of the business now? You clearly love math a lot. I'm just curious ;-). $\endgroup$ – Troy Woo Aug 18 '14 at 10:39
  • $\begingroup$ @HagenvonEitzen Yes, I'm aware of that. But I would like to see a concrete example. $\endgroup$ – ante.ceperic Aug 20 '14 at 23:25
  • $\begingroup$ Khabibullin's conjecture on integral inequalities. en.wikipedia.org/wiki/… $\endgroup$ – user153012 Aug 20 '14 at 23:47
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As some of the comments have mentioned, contest math problems are often contrived. As an avid competitor, had having written some contest problems myself, problem writers tend to draw upon topics which happen to have been in their heads recently.

For instance, an inequality fan might for instance take the Cauchy-Schwarz inequality, substitute some variables, and then "reverse-simplify", or make the problem appear more complicated using some substitution. The intended solution is to undo that substitution, simplify, and then see the equivalence of the original problem to Cauchy-Schwarz.

Sorry I can't provide any examples here (copyrights), but if you look at the solutions to the example problems in the first chapter of Problems From the Book (Andreescu and Dospinescu), you'll see what I'm talking about.

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