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6n colored points in a line

Source: Brazilian Math Olympiad 2008, Problem 2

October 28, 2008

combinatorial geometrycombinatorics unsolvedcombinatorics

Problem Statement

Let

S

be a set of

6n

points in a line. Choose randomly

4n

of these points and paint them blue; the other

2n

points are painted green. Prove that there exists a line segment that contains exactly

3n

points from

S

2n

of them blue and

n

of them green.