Problem 6. (8 points) In the plane, there is a non-closed, non-self-intersecting broken line consisting of 31 segments (adjacent segments do not lie on the same straight line). For each segment, the line defined by it is constructed. It is possible for some of the 31 constructed lines to coincide. What is the minimum number of different lines that can be obtained?

Answer. 9.
Solution. Evaluation. Besides the ends, the broken line has 30 vertices, each of which is the intersection point of two lines. If the lines are no more than 8, then the intersection points are no more 

![](https://cdn.mathpix.com/cropped/2024_06_03_376d2fa4074cfcd15764g-2.jpg?height=700&width=722&top_left_y=1569&top_left_x=1027)
than 7.8/2 = $28<30$, a contradiction. An example is shown in the drawing.