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0s and 1s in square board, maximum number of 1s

Source: Croatia MO 2001 4th Grade P4

April 22, 2021
combinatoricsgame

Problem Statement

Suppose that zeros and ones are written in the cells of an n×nn\times n board, in such a way that the four cells in the intersection of any two rows and any two columns contain at least one zero. Prove that the number of ones does not exceed n2(1+4n−3)\frac n2\left(1+\sqrt{4n-3}\right).
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