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 $C_i, C_j \in \mathcal F, C_i \not = C_j$

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 $C \subset \{1,2,\ldots,n\}$ for every $C \in \mathcal F$ and $|C_i \cap C_j| < 2 $ for all $C_i, C_j \in \mathcal F, C_i \not = C_j$

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.

If we change 10 to 3 the problem become trivial as we can choose $n$ red points on the plane such that there are no 4 point lie on the same circle, the set of blue point will be the set of center of circumscribed circle crossing every three red points.

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 $C_i, C_j \in \mathcal F, C_i \not = C_j$

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 $C_i, C_j \in \mathcal F, C_i \not = C_j$