2005 Canadian MO Problems/Problem 3
Problem
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.
Solution
(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.
See also
2005 Canadian MO (Problems) | ||
Preceded by Problem 2 |
1 • 2 • 3 • 4 • 5 | Followed by Problem 4 |