[ Plane, divided by lines ]

How many maximum parts can 5 segments divide a plane into?

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Each subsequent segment intersects with p already drawn segments in no more than p points.

## Solution

Let $\mathrm{n}$ segments have already been drawn, and they divide the plane into K(n) parts. Draw the ( $\mathrm{n}+1$ )-th segment. It intersects with the n already drawn segments in no more than n points, which can divide it into no more than (n-1) segments. Each of these segments can divide an existing part of the plane into two. Thus, drawing the $(\mathrm{n}+1)$-th segment adds no more than (n-1) new parts to the plane, i.e., K(n+1) does not exceed K(n)+n-1 (for $\mathrm{n}>1$). From this, it is easy to see (since K(2)=1) that K(5) does not exceed $1+(1+2+3)=7$. An example with seven parts of the plane can be constructed as follows. Draw 5 lines in general position and take 5 segments on these lines, containing all pairwise intersection points of these lines.

## Answer

7.00