1959–1966 IMO Longlist Problems/Czechoslovakia 1
Problem
Given points in the plane such that no three of the points are collinear. Does there exist a circle passing through (at least) 3 of the given points and not containing any other of the
points in its interior?
Solution
The answer is yes.
Since any finite set of at least three coplanar points is contained by a convex hull with vertices in the set of points, we can select adjacent points and
on this convex hull. Clearly all of the other
points will lie on the same side of the line
. Of these other points, we select the point
such that the angle
is maximized. Then
satisfy the conditions of the problem, because if there were some point
inside the circle, since it would be on the same side of line
as
, the angle
would be greater than the angle
, which is a contradiction. Q.E.D.