If you put an absolute value around your polynomial, then this sort of max-min comes up in the construction of local height functions. In the terminology of Lang's *Fundamentals of Diophantine Geometry* (see especially Chapter 10, Theorem 3.5, page 261), the equation displayed at the top of page 262 is
$$
\lambda = \sup_j \inf_i \lambda_{ij}.
$$
Unsorting the definitions, one has essentially
$$
\lambda_{ij}(x_1,\ldots,x_n) = \log\bigl| f_{ij}(x_1,\ldots,x_n)\bigr|
$$
for a certain collection of polynomials (or maybe rational functions) $f_{ij}$. So aside from having taken logs, the function $\lambda$ is the IMO max-min with absolute values.

These Weil functions are associated to divisors and do satisfy an addition formula $\lambda_{D_1}+\lambda_{D_2}=\lambda_{D_1+D_2}$, which is related to the IMO question. The IMO question is less general, but more precise because the Weil function $\lambda_D$ associated to a divisor $D$ is only well-defined up to what Lang calls an $M_K$-bounded function. On the other hand, the proof of the addition formula probably comes down to doing the IMO problem (modulo those pesky absolute value signs).

Final Comment: In Lang's setup, the absolute value may be the usual one on $\mathbb R$ or $\mathbb C$, but it could also be a $p$-adic absolute value; and more generally, one really wants a collection of Weil height $\lambda_{D,v}$, one for each absolute value $v$, that have some sort of uniformity as one varies over $v$.

not askfor a solution to the problem, this would indeed be off-topic. It asks if a certain notion comes up in the (research) literature; as opposed to being only an artefact for some contest. This seems on-topic to me, especially since the name chosen in the contest might not be the name under which it actually comes up and it could be hard to find it. – quid Jun 28 at 15:56