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Find the greatest possible m of the m x n table

Source: 2012 European Girls’ Mathematical Olympiad P2

April 13, 2012
combinatoricsEGMOEGMO 2012Combinatorial Number Theorynumber theory

Problem Statement

Let nn be a positive integer. Find the greatest possible integer mm, in terms of nn, with the following property: a table with mm rows and nn columns can be filled with real numbers in such a manner that for any two different rows [a1,a2,…,an]\left[ {{a_1},{a_2},\ldots,{a_n}}\right] and [b1,b2,…,bn]\left[ {{b_1},{b_2},\ldots,{b_n}} \right] the following holds: max⁡(∣a1−b1∣,∣a2−b2∣,...,∣an−bn∣)=1\max\left( {\left| {{a_1} - {b_1}} \right|,\left| {{a_2} - {b_2}} \right|,...,\left| {{a_n} - {b_n}} \right|} \right) = 1 Poland (Tomasz Kobos)
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