Pair of Circles Intersect If $S$ is a collection of circles s.t. for each point $c$ on the x-axis there is a circle in $S$ passing through the point $(c,0)$ and at the same time has the x-axis as a tangent to the circle at $(c,0)$: I want to show that $S$ has a pair of circles that intersect. 
Attempt:
Well I want to use countability in my proof. I've sketched it out and I can see this geometrically makes sense. The fact two circles intersect which the distance between the centre of the two circles is less than the sum of the two radii. $c$ is a real so I want to use the properties of the reals being uncountable. I haven't quite yet been able to tie these thoughts altogether, any help will be appreciated. 
 A: Let $\mathcal{C}$ be the collection of circles satisfying the required conditions.  For $x\in\mathbb{R}$, write $\omega(x)$ for a circle in $\mathcal{C}$ tangent to the $X$-axis at the point $(x,0)$.  There are uncountably many pairs of circles in $\mathcal{C}$ that intersect, but are not tangent to one another. 
Suppose contrary that there are only countably many pairs which intersect.  Define $f:\mathbb{R}\to\mathbb{Q}^2$ as follows.  For $x\in\mathbb{R}$, let $f(x)$ be the coordinates of a rational point within the interior of $\omega(x)$.  Let $T\subseteq \mathbb{R}$ be the set of all $x\in\mathbb{R}$ such that $\omega(x)$ has an intersection with some other circles in $\mathcal{C}$.  Then, $T$ is countable, whence $\mathbb{R}\setminus T$ is uncountable.  Furthermore, $f|_{\mathbb{R}\setminus T}:(\mathbb{R}\setminus T)\to\mathbb{Q}^2$ is an injective function from an uncountable set into a countable set.  This is a contradiction.  Hence, $T$ must be uncountable, and our proof is now complete.
P.S.: Let $n\in\mathbb{N}$.  If, in $\mathbb{R}^{n+1}$, a collection of $n$-spheres satisfies the condition that, for any $x\in\mathbb{R}^n$, there exists an $n$-sphere in the collection that meets the hyperplane $\left\{(t,0)\,|\,t\in\mathbb{R}^n\right\}$ at the point $(t,0)$.  Then, there are uncountably many pairs of $n$-spheres in this collection that intersect, but are not tangent to one another.
