Problem 1075

Prove that taking the convex hull $\Omega$ of these $2n$ points, we prove: each vertex on $\Omega$ lies on a balanced line.

If the above proposition is proven, then by the condition, $\Omega$ has at least 3 distinct vertices, and each balanced line passes through at most two vertices of $\Omega$, so there are at least two balanced lines.

Now we prove (1) holds. For any vertex $A$ of $\Omega$, assume $A$ is a red point, draw a line $l$ through $A$ such that $\Omega$ is on one side of $l$. Then, rotate $l$ counterclockwise around $A$, each time passing through a blue point, to get $n$ lines $A B_{1}, A B_{2}, \cdots, A B_{n}$. The number of blue points on the left side of the line $A B_{i}$ is $b_{i}=i-1$, and the number of red points on this side is denoted as $r_{i}$, then $0 \leqslant r_{1} \leqslant r_{2} \leqslant \cdots \leqslant r_{n} \leqslant n-1$.
We need to prove: there exists $i \in\{1,2, \cdots, n\}$, such that $b_{i}=r_{i}$ (thus, (1) holds).
In fact, let $d_{i}=r_{i}-b_{i}, i=1,2, \cdots, n$, then $d_{1}=r_{1} \geqslant 0, d_{n}=r_{n}-(n-1) \leqslant 0$. Therefore, the sequence $d_{1}, \cdots, d_{n}$ starts from a number not less than zero and ends with a number not greater than zero. Furthermore, there is
$$d_{i}-d_{i+1}=\left(r_{i}-r_{i+1}\right)+\left(b_{i+1}-b_{i}\right) \leqslant 0+1=1,$$

So, from $d_{1}$ to $d_{n}$, it decreases by at most 1 each time. Therefore, (2) holds.
The proposition is proven.