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IMO Long List 1986 matrix problem-Find values of k

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August 29, 2010
linear algebramatrixlinear algebra unsolved

Problem Statement

We wish to construct a matrix with 1919 rows and 8686 columns, with entries xij∈{0,1,2} (1≤i≤19,1≤j≤86)x_{ij} \in \{0, 1, 2\} \ (1 \leq i \leq 19, 1 \leq j \leq 86), such that:
(i) in each column there are exactly kk terms equal to 00;
(ii) for any distinct j,k∈{1,...,86}j, k \in \{1, . . . , 86\} there is i∈{1,...,19}i \in \{1, . . . , 19\} with xij+xik=3.x_{ij} + x_{ik} = 3.
For what values of kk is this possible?
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