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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. – Eric Naslund Sep 27 '12 at 6:29
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. – Qiaochu Yuan Sep 27 '12 at 8:17

When I am working on my research, or on MO/math.SE questions, I often find myself thinking in a way which reminds me of the feel of solving Olympiad problems. If I then solve the problem, I try to find a very special case of my problem which is still challenging, and can be stated and solved using only material on the Olympiad curriculum. I then e-mail Kiran Kedlaya and say "Hey Kiran, do you think this would be a good Olympiad problem?" If he thinks so, he proposes it to the USAMO committe.

I wrote Problem 2 on the 2010 USAMO in this way; it is Theorem 3.2 of this paper specialized to the case that $W_0$ is the group $S_n$. The fact that the "total number of moves" referred to in the theorem is at most $\binom{n}{3}$ is computed in Section 5.2.

I send Kiran about 1-2 problems a year; I don't think any of the others have appeared yet.

UPDATE Problem B4 of the 2014 Putnam was mine. Let $F(x,y,z) = \sum F_{ijk} x^i y^j z^k$ be a homogenous polynomial of degree $n$ with positive real coefficients. We say that $F$ is hyperbolic with respect to the positive orthant if, for all $(u,v,w) \in \mathbb{R}_{> 0}^3$, the polynomial $f(t) = F(tu,tv,tw)$ has $n$ negative real roots.

This paper show that there are constants $V_1$ and $V_2$ (dependent on $n$) so that,

(1) if $F$ is hyperbolic with respect to the positive orthant, then $F_{i(j+1)(k+1)} F_{(i+1)j(k+1)} > V_1 F_{i(j+1)(k+1)} F_{(i+2)jk}$ and the same for all permutations of the indices

(2) if $F_{i(j+1)(k+1)} F_{(i+1)j(k+1)} > V_2 F_{i(j+1)(k+1)} F_{(i+2)jk}$ and the same for all permutations of the indices, then $F$ is hyperbolic with respect to the positive orthant.

The proof is nonconstructive; I also (Theorem 20) give an explicit value of $V_1$. I was thinking about whether I could give a concrete value for $V_2$. The problem was too hard, so I thought instead about homogenous polynomials in two variables, which is the same as inhomogenous polynomials in one variable. At this point, I was basically looking for a converse to Newton's inequality: I wanted a constant $C$ so that, if $a_k^2 > C a_{k-1} a_{k+1}$, then all the roots of $\sum_{k=0}^n a_k z^k$ are real. The result in one variable wasn't worth publishing, but I figured I could make a nice problem by choosing a particular polynomial and asking people to prove the roots were real.

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Thanks, these stories are both very interesting! I've also mentioned them on the AoPS forums (…; for USAMO 2010 #2 and for Putnam 2014 B4), since the problems probably get the most traffic over there. I wish more problem authors wrote these stories---I think it would lead to a healthier contest atmosphere. (FWIW, some of my own stories are at…) – Victor Wang Jan 18 '15 at 3:56
This is brilliant. It's exactly how Cambridge examiners used to write the Tripos in the old days (1800's). The problem flowed directly from their research. If I'm not mistaken, one question in the old Tripos asked to prove Stokes' Theorem (or Green's, I don't know multivariable calculus so I can't remember which one it was). Of course, this was proposed by Stoke/Green himself. This is why the old Tripos examination was so difficult; if I remember well, the highest score anyone got was ~half marks. – Ark Mar 4 '15 at 21:08

If you ask me, I suppose the problems don't get invented - rather, I think they arise as word-problem forms of well-known theorems/results etc.

That is to say, they are not made for the competitions as such - they existed before. For example, regarding the grasshopper problem, it may be that it was known that points could be chosen in any order without having to choose any point $p$ such that $p\in M$, and they just added the jumps and the grasshopper.

That's my 2 cents.

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Very few Olympiad problems are word problems. – Christopher A. Wong Sep 27 '12 at 7:36
I'm not sure why this answer was downvoted, considering David Speyer described doing exactly this. Of course "well known" is debatable. – Antonio Vargas Sep 27 '12 at 14:21

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