IMO Shortlist 2010 problem N6


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June 23, 2013
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The rows and columns of a 2^n \times 2^n table are numbered from 0 to 2^{n}-1. The cells of the table have been coloured with the following property being satisfied: for each 0 \leq i,j \leq 2^n - 1, the j-th cell in the i-th row and the (i+j)-th cell in the j-th row have the same colour. (The indices of the cells in a row are considered modulo 2^n.) Prove that the maximal possible number of colours is 2^n.

Proposed by Hossein Dabirian, Sepehr Ghazi-nezami, Iran
Source: Međunarodna matematička olimpijada, shortlist 2010