Problem 869

【Analysis】First, prove that each vertex on the convex hull of these $n$ points lies on a balance line.

Assume $R$ is a vertex on the convex hull of the known $2 n$ points. Without loss of generality, let $R$ be a red point. Thus, there exists a line $l$ such that all points (excluding $R$) are on the same side of the line $l$. If the line $l$ is rotated clockwise around point $R$, it will pass through all the blue points in a certain order.

Let the blue points that the line $l$ passes through in sequence when rotated clockwise around point $R$ be $B_{1}, B_{2}, \cdots, B_{n}$. When $l$ rotates to point $B_{i}$, the number of blue points and red points that $l$ has passed through are denoted as $b_{i}$ and $r_{i}$, respectively. Then,
$$\begin{array}{l} b_{i}=i-1(i=1,2, \cdots, n), \\ 0 \leqslant r_{1} \leqslant r_{2} \leqslant \cdots \leqslant r_{n} \leqslant n-1 . \\ \text { Define } d_{i}=r_{i}-b_{i}(i=1,2, \cdots, n) . \\ \text { Hence } d_{1}=r_{1} \geqslant 0, d_{n}=r_{n}-b_{n}=r_{n}-(n-1) \leqslant 0, \end{array}$$

Thus, the sequence $d_{1}, d_{2}, \cdots, d_{n}$ starts with non-negative integers and ends with non-positive integers. As $i$ increases, $r_{i}$ does not decrease, and $b_{i}$ increases by 1, indicating that the difference between consecutive terms in the sequence $d_{1}, d_{2}, \cdots, d_{n}$ does not exceed 1.
In fact, for $i=1,2, \cdots, n-1$, we have
$d_{i}-d_{i+1}=r_{i}-r_{i+1}+b_{i+1}-b_{i} \leqslant 0+1=1$.
Therefore, there must exist some point $i \in\{1,2, \cdots, n\}$ such that $d_{i}=0$.
Thus, $r_{i}=b_{i}$.
Hence, $R B_{i}$ is a balance line.
Since the convex hull of $2 n(n \geqslant 2)$ points has at least three vertices, and each vertex of the convex hull lies on a balance line, there are at least two different balance lines.