MathDB

Determine the least integer

n

such that for any set of

n

lines in the 2D plane, there exists either a subset of

1001

lines that are all parallel, or a subset of

1001

lines that are pairwise nonparallel.
Proposed by Samuel Wang
Solution.

\boxed{1000001}

Since being parallel is a transitive property, we note that in order for this to not exist, there must exist at most

1001

groups of lines, all pairwise intersecting, with each group containing at most

1001

lines. Thus,

n = 1000^2 + 1 = \boxed{1000001}

3C