IMO Shortlist 2012 problem A7


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3. studenoga 2013.
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We say that a function f:\mathbb{R}^k \rightarrow \mathbb{R} is a metapolynomial if, for some positive integer m and n, it can be represented in the form
f(x_1,\cdots , x_k )=\max_{i=1,\cdots , m} \min_{j=1,\cdots , n}P_{i,j}(x_1,\cdots , x_k),
where P_{i,j} are multivariate polynomials. Prove that the product of two metapolynomials is also a metapolynomial.
Izvor: Međunarodna matematička olimpijada, shortlist 2012