, 2012

The statement trivially holds for $n = 2$ and $n = 3$. We will show that it does not hold for any $n \ge 4$.

First suppose that $n$ is even. Setup a coordinate, and place a point at $(0,0)$. Place $n/2 - 1$ points on the negative $x$-axis and $n/2 - 1$ points on the positive $x$-axis at $(-1/2^i, 0)$ and $(2^i, 0)$ for $i = 1, 2, \dots, n-1$. Place the last point on the positive $y$-axis, so that the perpendicular bisector of the segment joining that point and any of the previously placed point divides the plane into two parts, one of which contains only the last point. (It is clear that we can do this, and that this construction serves as a counterexample to the given statement.)

If $n$ is odd, we simply add another point on the negative $y$-axis, so that perpendicular bisector of the segment joining that point and any of the previously placed point of the $x$-axis divides the plane into two parts, one of which contains only the last point. Also, the distance from the $x$-axis of the point on the positive $y$-axis and the point on the negative $y$-axis should be distinct.