2005 Canadian MO Problems/Problem 3
Let be a set of points in the interior of a circle.
- Show that there are three distinct points and three distinct points on the circle such that is (strictly) closer to than any other point in , is closer to than any other point in and is closer to than any other point in .
- Show that for no value of can four such points in (and corresponding points on the circle) be guaranteed.
(a) Let be the convex hull of , and choose any three vertices of .
Now, draw a line through such that all points of lie on one side of . (This is possible because is convex and contains .) Define and similarly.
For , let be the line perpendicular to passing through , and let hit the circle at point on the side of opposite . Now, each is closer to than any other point in . Since vertices are also in , we are done.
(b) Describe the circle as the curve in polar coordinates. We will construct a set with an arbitrary number of elements satisfying the desired property. Let . It is easy to see that for any point on the circle, is at most away from one of the . Therefore, if we also include points on any of the line segments ( is the center), where then no point can be chosen on the circle which is closer to one of the then the .
Thus is as desired.
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