# Does there exist an expression?

For which positive integers does there exist expression

where each is a disk of radius such that each point belongs to either the boundary of some or to precisely interiors of the sets , , ?

(Euler Competition , 2017)

Solution [official]

For none . Assume , on the contrary , that such expression exists for some value of . Let us first note that the set of disks is countable hence the Lebesgue measure of the union of all the circles is .

Further, consider any disk and let us show that the number of the other disks it intersects is finite. Indeed, each disk that intersects would have to be a subset of the disk or radius co-centric with . All such disks together would cover an area of at most so that each point is covered by at most disks. Hence the total area of disks that intersect is at most and therefore there are not more than of them.

Now, let us introduce the function whose value is the number of disks that belongs to. Note that

• for almost all
• is constant on every connected component of .

What remains to be shown is that this is impossible. Since the disk intersects only with finitely many other disks , let us consider the circular arc between two such intersection points such that the arc in the figure above. But then obviously the value for points on one side of the arc and for points on the other side of the arc differ by . For the collections of disks that cover the two regions is different – is only on one side of the circular arc.