Problem:
There's $n$ red and black circles(that means, the number of the circles is $n$) on a plane, where $n$ is even. Their radius don't need to be equal, and they cannot completely overlap the other circles(they can intesect). Paint the tangent point of a red circle and a black circle blue. Prove that the maximum of blue points is $\dfrac{n^2}{4}$.
My attempt: $\dfrac{n^2}{4}$ means to hold $n/2$ red circles and $n/2$ black circles, and each circle is tangent with all other circles with the other color. However no matter how I construct, the answer always seems to be $\Theta (n)$, and I don't know how to proceed further.
Also please notice that internally tangent are also supported.
Any hint or advice is appreciated.
UPD: Sorry for my mistake. The circles can intersect with each other.