50*. What is the greatest number of circles of radius 1 that can be placed on a plane so that they all intersect a certain fixed unit circle $S$ and no one of them contains the center of $S$ or the center of another circle inside it?
50. Fig. 134 shows that 18 circles satisfying the condition of the problem can be placed (six circles in Fig. 134 have centers at the vertices of a regular hexagon inscribed in $S$, the other 12 - at the vertices of squares constructed on the sides of the inscribed hexagon outside it; compare with Fig. 130). The fact that more than 18 circles cannot be placed follows from the result of problem 48: otherwise, the centers of all circles (including the center $O$ of circle $S$) would form a system of 20 or more points enclosed within a circle of radius 2 centered at one of these points $O$ and such that the distance between any two of these points is not less than 1, while according to the theorem of problem 48, such a system of points cannot exist.

![](https://cdn.mathpix.com/cropped/2024_05_21_962b75a91b7185035ab9g-165.jpg?height=587&width=590&top_left_y=510&top_left_x=180)

Fig. 134.