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nice set of lines if every lines intersects odd no of other lines of set

Source: Dutch IMO TST 2018 day 3 p1

August 30, 2019
combinatorial geometrycombinatoricslines

Problem Statement

A set of lines in the plan is called nice i f every line in the set intersects an odd number of other lines in the set. Determine the smallest integer k≥0k \ge 0 having the following property: for each 20182018 distinct lines ℓ1,ℓ2,...,ℓ2018\ell_1, \ell_2, ..., \ell_{2018} in the plane, there exist lines ℓ2018+1,ℓ2018+2,...,ℓ2018+k\ell_{2018+1},\ell_{2018+2}, . . . , \ell_{2018+k} such that the lines ℓ1,ℓ2,...,ℓ2018+k\ell_1, \ell_2, ..., \ell_{2018+k} are distinct and form a nice set.
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