TITLE: A circle with $500$ points in its interior
QUESTION [9 upvotes]: Given any $1000$ points in the plane, show that there is a circle which contains exactly $500$ of the points in its interior, and none on its circumference.

How do I approach this problem? I feel it has something to do with the pigeonhole principle but I am not able to find the right argument for it.

REPLY [4 votes]: Here is another approach.  First, you can draw a line $L$ that splits the plane in two halves, each containing $500$ points.  To do this, choose a direction such that no line with this direction contains more that one of the $1000$ points.  Then draw a line with this direction which does not contain any of the points.  If one the half planes contains more than $500$ points, you can move the line towards it until it passes exactly one point; repeat until the points are equally distributed in both side of the line.  Call this line $L$.
Next, you can choose a circle containing all $500$ points on one side of the line.  To see this, draw a line segment $S$ perpendicular to $L$, and look at circles tangent to $L$ with center on $S$.  Any point of the half-plane belongs to one of these circles; simply choose one that is big enough to contain all $500$ by putting its center far enough from $L$.