I already posted this on math.stackexchange, but I'm also posting it here because I think that it might get more and better answers here! Hope this is okay.
We all know that problems from, for example, the IMO and the Putnam competition can sometimes have lovely connections to "deeper parts of mathematics". I would want to see such problems here which you like, and, that you would all add the connection it has.
Some examples:
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Stanislav Smirnov mentions the following from the 27th IMO:
"To each vertex of a regular pentagon an integer is assigned in such a way that the sum of all five numbers is positive. If three consecutive vertices are assigned the numbers $x,y,z$ respectively, and $y <0$, then the following operation is allowed: the numbers $x,y,z$ are replaced by $x+y$, $-y$, $z+y$, respectively. Such an operation is performed repeatedly as long as at least one of the five numbers is negative. Determine whether this procedure necessarily comes to an end after a finite number of steps".He mentions that a version of this problem is used to prove the Kottwitz-Rapoport conjecture in algebra(!). Further, a version of this has appeared in at least a dozen research papers.
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(Taken from Gerry Myerson in this thread https://math.stackexchange.com/questions/33109/contest-problems-with-connections-to-deeper-mathematics)
On the 1971 Putnam, there was a question, show that if $n^c$ is an integer for $n=2,3,4$,… then $c$ is an integer.
If you try to improve on this by proving that if $2^c$, $3^c$, and $5^c$ are integers then $c$ is an integer, you find that the proof depends on a very deep result called The Six Exponentials Theorem.
And if you try to improve further by showing that if $2^c$ and $3^c$ are integers then $c$ is an integer, well, that's generally believed to be true, but it hadn't been proved in 1971, and I think it's still unproved.
The most interesting part would be to see solutions to these problems using both elementary methods, and also with the more abstract "deeper methods".
Best Answer
[Found another of these Putnam problems; it seems that the protocol here is to post separate big-list examples separately rather than add them to one big-answer.]
2002 Problem B-6. Let $p$ be a prime number. Prove that the determinant of the matrix $$ \left(\begin{array}{lll}x&y&z\cr x^p &y^p&z^p\cr x^{p^2}&y^{p^2}&z^{p^2}\end{array}\right) $$ is congruent modulo $p$ to a product of polynomials of the form $ax+by+cz$ where $a,b,c$ are integers.
This actually works for the analogous $n\times n$ determinant for each $n$, and also over any finite field $k$ rather than just the prime field $\mathbf{Z} / p \mathbf{Z}$. The case $n=2$ is basically Fermat's little theorem; the general case is Moore's $q$-analogue of the Vandermonde determinant, used by Dickson to find the subring of $k[x_1,\ldots,x_n]$ invariant under all $k$-linear transformations of the $x_i$: they're the polynomials in $n$ fundamental invariants of degrees $q^n - q^i$ ($i=0,1,2,\ldots,n-1$), with the invariant of degree $q^n-1$ being the $q-1$ power of the Moore determinant. Moreover, replacing that power with the Moore determinant itself yields the invariants for $SL_n(k)$ instead of $GL_n(k)$. This century-old theorem has found applications ranging from algebraic topology to Diophantine and algebraic geometry.
References:
E.H.Moore: A two-fold generalization of Fermat's theorem, Bull. AMS 2 #7 (1896), 189-199.
L.E.Dickson: A fundamental system of invariants of the general modular linear group with a solution of the form problem. Trans. AMS 12 (1911), 75-98