Some unit-radius circles are drawn on the plane, and we color the center of each circle blue. We mark some points on the circumference of the circles in red such that exactly 2 red points lie on each circumference. What is the maximum number of blue points if there are a total of 25 colored points?
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Move a unit circle four times in the same direction by a distance of 0.0002.

![](https://cdn.mathpix.com/cropped/2024_05_02_9b69772ab09b921b6263g-1.jpg?height=228&width=336&top_left_y=222&top_left_x=884)

The centers of the five resulting unit circles lie on a segment of length 0.0008; therefore, any two of them have two common points, and no three of them have a common point. To prove the latter property, assume that some three circles have a common point $Q$. Then the centers of these three circles would lie on the circumference of a unit circle centered at $Q$, which is impossible since the centers of the five circles are collinear.

Now color the five centers red and the intersection points blue. Any two circles have two blue (intersection) points, which uniquely determine the two circles, so the number of blue points is $2 \cdot \binom{5}{2} = 20$. The conditions of the problem are thus satisfied by 20 unit circles whose centers are the blue points. Therefore, the number of blue points can be 20.

We will show that more than 20 is not possible. With more than 20 blue points, there could be at most 4 red points. However, only points that are the intersection points of the unit circles drawn around the red points can be blue, since otherwise they could not be at a unit distance from at least two red points. However, four circles can have at most $2 \cdot \binom{4}{2} = 12$ intersection points (any two circles can have at most two). Therefore, with at most 4 red points, there can be at most 12 blue points, which is clearly a contradiction.

With this, we are done: we have shown that there cannot be more than 20 blue points, and we have provided an example with 20.

Hegedüs Dániel (Budapesti Fazekas M. Gyak. Ált. Isk. és Gimn., 10th grade)

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