6. Variant 1.

In the figure, an example is given of how 3 rays divide the plane into 3 parts. Into what maximum number of parts can 11 rays divide the plane?

![](https://cdn.mathpix.com/cropped/2024_05_06_dc91a791fc316ca400e2g-06.jpg?height=731&width=902&top_left_y=677&top_left_x=634)
Answer: 56.

## Solution.

If the $(n+1)$-th ray is drawn so that it intersects all $n$ previous rays, then $n$ intersection points on it divide it into $n+1$ segments. The segment closest to the vertex does not add new parts to the partition, while each of the other $n$ segments ( $n-1$ segments and 1 ray) divides some existing part of the partition into two new parts, i.e., $n$ new parts will be added. (Accordingly, if the ray intersects fewer rays, then fewer new parts will be added to the partition).

1 ray - 1 part.

2 rays - $1+1$.

3 rays - $1+1+2$.

...

$n$ rays - $1+1+2+\ldots+(n-1)=1+n(n-1)/2=(n^2-n+2)/2$.

For $n=11$, the answer is $(121-11+2)/2=56$.