IMO 1986 LL IRE46

Origin: IRE

Problem

We wish to construct a matrix with 19 rows and 86 columns, with entries xij \in{0, 1, 2} (1 \leqi \leq19, 1 \leqj \leq86), such that: (i) in each column there are exactly k terms equal to 0; (ii) for any distinct j, k \in{1, . . . , 86} there is i \in{1, . . . , 19} with xij + xik = 3. For what values of k is this possible?