# Task 5. (12 points)

On a plane, there are four different circles. We will call an intersection point a point where at least two circles intersect. Find the maximum possible number of intersection points of four circles.

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# Solution

Any two circles can intersect at no more than two points. From four circles, we can choose 6 different pairs of circles. Therefore, the number of intersection points cannot exceed 12.

The figure below shows a case where there are exactly 12 intersection points.

![](https://cdn.mathpix.com/cropped/2024_05_06_0748dce8a8149f40dec4g-3.jpg?height=591&width=599&top_left_y=887&top_left_x=777)

Answer: 12.