How inequalities are made

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.

Solution 1:

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.