There are some blue points and red points on the plane such that in the **boundary** of every unit circle centered at one blue point there are **exactly** 10 red point. Can the number of blue points strictly greater than the number of red points ?

This is a problem I found a very long time ago with any solution the only hint I found is this problem will use probabilistic method somewhere

I figured out that if we change exactly 10 to at least 10, the answer is yes, it is because we can chessboard coloring a large grid and scale it down by a factor of $1/ \sqrt{N}$ where $N$ satisfy that the solution $x^2+y^2 = N$ has at least 10 positive integer solution.

Translating the problem into a set-intersecting problem will not work here, there exist a large number $n$ and a family of set $\mathcal F$ with $|\mathcal F| = O(n^2)$ such that $|C| = 10$ and $C \subset \{1,2,...n\}$ for every $C \in \mathcal F$ and $|C_i \cap C_j| < 2 $ for all $C_i, C_j \in \mathcal F, C_i \not = C_j$