Question 135
There are distinct points $O$, $A$, $B$, $K_1$, $\ldots$, $K_n$, $L_1$, $\ldots$, $L_n$ on a plane such that no three points are collinear. The open line segments $K_1L_1$, $\ldots$, $K_nL_n$ are coloured red, other points on the plane are left uncoloured. An *allowed path* from point $O$ to point $X$ is a polygonal chain with first and last vertices at points $O$ and $X$, containing no red points. For example, for $n = 1$, and $K_1 = (-1; 0)$, $L_1 = (1; 0)$, $O = (0; -1)$, and $X = (0; 1)$, $OK_1X$ and $OL_1X$ are examples of allowed paths from $O$ to $X$; there are no shorter allowed paths. Find the least positive integer $n$ such that it is possible that the first vertex that is not $O$ on any shortest possible allowed path from $O$ to $A$ is closer to $B$ than to $A$, and the first vertex that is not $O$ on any shortest possible allowed path from $O$ to $B$ is closer to $A$ than to $B$.