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Part of 2017 Benelux

Problems(1)

Benelux n-square

Source: Benelux Mathematical Olympiad 2017, Problem 4

5/6/2017
A Benelux n-square (with n≥2n\geq 2) is an n×nn\times n grid consisting of n2n^2 cells, each of them containing a positive integer, satisfying the following conditions: ∙\bullet the n2n^2 positive integers are pairwise distinct. ∙\bullet if for each row and each column we compute the greatest common divisor of the nn numbers in that row/column, then we obtain 2n2n different outcomes.
(a) Prove that, in each Benelux n-square (with n≥2n \geq 2), there exists a cell containing a number which is at least 2n2.2n^2. (b) Call a Benelux n-square minimal if all n2n^2 numbers in the cells are at most 2n2.2n^2. Determine all n≥2n\geq 2 for which there exists a minimal Benelux n-square.
number theorynumber theory proposedBeneluxOlympiadmath olympiadalgebracombinatorics