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Ukraine geometry
Problem
In the plane six lines are given such that no three of them are coincident. Can it happen that they have exactly:
a) ;
b) different intersection points?
a) ;
b) different intersection points?
Solution
a) It is possible. It is enough to take lines that have intersection points among themselves, and also parallel lines, each of which intersects each of the initial lines.
b) It is not possible. The first line intersects the others in at most points, the second adds at most intersection points, and so on. In total, we have at most different intersection points.
b) It is not possible. The first line intersects the others in at most points, the second adds at most intersection points, and so on. In total, we have at most different intersection points.
Final answer
a) Yes; b) No
Techniques
Constructions and lociColoring schemes, extremal arguments