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Here is a problem I've been trying to solve for some time now. Maybe you could help me.

We have two sets

[TEX]\mathcal {Q}[/TEX] is a set of those circles in the plane such that for any [TEX]x \in \mathbb{R}[/TEX] there exists a circle [TEX]O \in \mathcal {Q}[/TEX] which intersects [TEX]x[/TEX] axis in [TEX](x,0)[/TEX].

[TEX]\mathcal {T}[/TEX] is a set of those circles in the plane such that for any [TEX]x \in \mathbb{R}[/TEX] there exists a circle [TEX]O \in \mathcal {T}[/TEX] which is tangent to [TEX]x[/TEX] axis in [TEX](x,0)[/TEX].

We need to show that in each of these sets there exist at least two different circles whose intersection isn't empty.

It seems obvious that [TEX]card (Q) \ge card (\mathbb{R})[/TEX]. Maybe we could somehow identify each circle with a different rational number?