1959–1966 IMO Longlist Problems/Czechoslovakia 1
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?
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.