A.K.

In the plane, there is an open, non-self-intersecting broken line with 31 segments (adjacent segments do not lie on the same straight line). Through each segment, a line containing this segment was drawn. As a result, 31 lines were obtained, some of which may have coincided. What is the smallest number of different lines that could have resulted?

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Evaluation. Except for the ends, the broken line has 30 vertices, and each is the intersection of two lines. If there are no more than eight lines, then there are no more than $7 \cdot 8: 2=28$ intersection points - a contradiction.

Example - in the figure.

![](https://cdn.mathpix.com/cropped/2024_05_06_9a42dc41a6897b3a1545g-11.jpg?height=720&width=749&top_left_y=0&top_left_x=669)

## Answer

9 lines.