## Task 1 - 130511

a) Determine the maximum number of intersection points that 6 different circles of the same size can have with each other! Here, intersection points are understood to be the points where two circles intersect.

b) Draw an example where 6 circles have the maximum number of intersection points determined in a), and where the circle centers all lie on the same straight line! Choose a radius of $r=3 \text{~cm}$ and number the intersection points.
a) Two circles can intersect at most at 2 points. The maximum number of intersection points of the six circles occurs when each circle intersects each of the other five circles twice. The apparent number of intersection points $6 \cdot 5 = 60$ must be divided by 2, as each intersection point is counted twice in this product. Therefore, there are at most 30 intersection points.

b) The centers of the six circles must be less than 2 times the radius $r = 3 \text{ cm}$ apart from each other, as only then can they intersect pairwise at two points each.

A possible example is:

![](https://cdn.mathpix.com/cropped/2024_06_06_b85a894aa8dbf722a7b3g-0061.jpg?height=643&width=1307&top_left_y=1135&top_left_x=383)