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The problems that appear in difficult math competitions such as the IMO or the Putnam exam are usually very difficult and require some ingenuity to solve. They also usually don't look like they can be solved by simply knowing more advanced theory and the such.

How do people typically come up with these problems? Do they arise naturally from advanced mathematics (the somewhat infamous 'grasshopper problem' from the 2009 IMO comes to mind - to my not exactly knowledgeable mind this problem looks like it popped out of basically nowhere)? What is the perspective that mathematicians take when seemingly "inventing" these problems with no theoretical motivation to them whatsoever?

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    Sometimes people work backwards. For example, here is an integral question I came up with while reading MSE: Prove that $$\int_{-\infty}^{\infty}\int_{0}^{\infty}\frac{\log\left(a^{2}+1\right)}{\left(1+x^{2}\right)\left(1+x^{a}\right)\left(1+a^{2}\right)}dxda=\pi^{2}\log2.$$ Although, I think that many questions arise naturally at some point from research, or from thinking about other mathematical problems.2012-09-27
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    It is not clear to me that mathematicians take a single perspective when writing down Olympiad problems. Probably different mathematicians have many different ways of writing down Olympiad problems.2012-09-27
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    A single mathematician is not obliged to use a single way. I construct them as special cases from general theorems, I take something that comes up in research or in a talk, I construct them from the solution techniques I have in mind, I doodle around in geogebra, CylindricalDecomposition or with graphs on paper, and so on. Most often, my perspective is that I am bored during exam surveillance, but have to stay too much focused on the students to do something serious.2017-03-03
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    PS: The grasshopper problem looks to me exactly like something doodled during a talk or an exam.2017-03-03

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