#16
Suppose a circle is given and 2 billion points inside of it. Is it possible to draw a straight line
such that there would be exactly 1 billion points on each side of it?
solution:
Denote the set of points we have by X. Let’s draw a straight line through each pair of points. A
vast amount of straight lines is obtained, but a finite amount. Each straight line intersects the
circle in two points. In such way we obtain a finite set of points on the circle, denote this set M.
Let’s choose a point on the circle which is not in M, and denote this point A. The important fact
about point A is that no straight line passing through it passes through more than one point in X
(otherwise A would be in M). Now, draw through point A a tangent line to the circle, and begin
to turn it in a certain direction (around the point A). Just turn it until exactly one billion of points
is left on each side of it.