## Task 1 - 070731

The sides of a hexagon, in which no side is parallel to another, are extended beyond the vertices.

What is the maximum number of new intersection points that can thus be created?
The solution boils down to the question: "How many intersection points can 6 lines have at most, if none of them is parallel to another?" At the end, the number of vertices of the hexagon is to be subtracted.

Each line can have at most 5 intersection points with the remaining 5 lines. With 6 lines, this results in at most $6 \cdot \frac{5}{2}=15$ intersection points, since each intersection point in this case belongs to exactly 2 lines. Since the vertices of the hexagon represent 6 of these intersection points, at most 9 new intersection points can thus arise.