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0,1 in nxn board, ways that sum of products of all numbers in k-th row is even

Source: KJMO 2010 p2

May 3, 2019
combinatoricsSumProductgridboardsquare grid

Problem Statement

Let there be a n×nn\times n board. Write down 00 or 11 in all n2n^2 squares. For 1≤k≤n1 \le k \le n, let AkA_k be the product of all numbers in the kkth row. How many ways are there to write down the numbers so that A1+A2+...+AnA_1 + A_2 + ... + A_n is even?
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