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I recently came up with a seemingly very easy problem I couldn't solve beyond the $ n =2 $ case.
Consider a set consisting of (the union of) $n+1$ circles (which might have different centers or radii). Show that if it is included (in terms of points) in a same kind of set with only $n$ circles, then at least two circles in the original set have same the center and radius. This seems really straightforward - I tried the pigeonhole principle to no avail.

Context: This "lemma" formalizes an idea I had to solve the following elementary problem (high-school level)

Show that the image of the set of nonzero complex numbers $z$ s.t. $\left|z+\frac{1}{z}\right|=2$ is the union of two circles, which shall be precised.

See Exercise 353 of https://www.louislegrand.fr/wp-content/uploads/2022/01/EXOS-TERMINALE3-3-AVECDESSIN.pdf (in French). As I couldn't find any solution for the initial problem on MSE (there's an unsatisfying/uneducational proof elsewhere where the author just drops the circles equations and checks the calculations), I might link to a typed version of mine when I have time.

Relevant MSE posts:

François Mortier
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The intersection of any two different circles is a finite set of points, so a union of finitely many circles can only contain finitely many points of another circle that isn't one of the union members. Therefore, if a union of $n$ circles contains a whole circle $S$, then $S$ must be one of the $n$ original circles. The pigeonhole argument follows.

Karl
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  • Got it, thanks ! I didn't think about the basic property "$S \subset U \Rightarrow S \cap U = S$ but it all hinges on that. – François Mortier Jan 05 '24 at 18:41
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    @exploreandwonder Hmm, I wouldn't say it hinges on that. That property holds for any sets $S$ and $U$, but the "intersection of two different circles is a finite set" is important here. Notice that your theorem isn't true if you replace circles with line segments. – Karl Jan 05 '24 at 18:50
  • You're right, I just meant that it's the part I missed, whereas I had thought about the "intersection of two different circles is a finite set" property. – François Mortier Jan 05 '24 at 18:53