2015 IMO Problems/Problem 1
Problem
We say that a finite set in the plane is balanced
if, for any two different points
,
in
, there is
a point
in
such that
. We say that
is center-free if for any three points
,
,
in
, there is no point
in
such that
.
- Show that for all integers
, there exists a balanced set consisting of
points.
- Determine all integers
, such that there exists a balanced centre-free set consisting of
points.
Solution
Part (a): We explicitly construct the sets . For
odd
,
can be taken to be the vertices of
regular polygons
with
sides: given any two vertices
and
, one of the two open half-spaces into
which
divides
contains an odd number of
of vertices of
. The
vertex encountered while moving from
to
along the circumcircle of
is therefore equidistant from
and
.
If is even, choose
to be the largest
integer such that
Hence
. Consider a circle
with centre
, and let
be distinct points placed counterclockwise
(say) on
such that
(for
). Hence for any line
, there is a line
such that
(using the facts that
, and
odd). Thus
,
and
form an equilateral triangle. In other words, for
arbitrary
, there exists
equidistant to
and
. Also given {\em any}
such that
,
is equidistant to
and
. Hence
the
points
form a balanced set.
Part (b): Note that if is odd, the set
of
vertices of a regular polygon
of
sides forms a balanced set
(as above) and a centre-free set (trivially, since the centre of the
circumscribing circle of
does not belong to
).
For even, we prove that a balanced, centre free set consisting of
sides does not exist. Assume that
is center-free. Pick an
arbitrary
, and let
be the number of
distinct non-ordered pairs of points
(
) to
which
is equidistant. Any
two such pairs are disjoint (for, if there were two such pairs
and
with
distinct, then
would be
equidistant to
,
, and
, violating the centre-free
property). Hence
(we use the fact that
is even here), which means
. Hence there are at least
non-ordered pairs
such that no point in
is equidistant to
and
.
See Also
2015 IMO (Problems) • Resources | ||
Preceded by Problem 0 |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 2 |
All IMO Problems and Solutions |