1984 IMO Problems/Problem 3
Problem
Given points and
in the plane. Every point in the plane is colored with one of a finite number of colors. Given a point
in the plane, the circle
has center
and radius
, where
is measured in radians in the range
. Prove that we can find a point
, not on
, such that its color appears on the circumference of the circle
.
Solution
Let be a sequence of positive reals such that
. For each
, let
be the circle centered at
with radius
.
Because of , we can find points
such that for all
we have
. We now forget about all the other points, and work only with the matrix
.
Suppose we use colors. There must be one,
, which appears infinitely many times on the first row of
, in, say, points
. Then
cannot appear on the lines
,
. Next, there is a color
which appears infinitely often among the points
,
. But then
cannot appear on the lines
for such
. Repeating this procedure, we reach a stage where we have a row of
(infinitely many actually) on which none of our
colors
can appear. This is a contradiction.
This solution was posted and copyrighted by grobber. The original thread for this problem can be found here: [1]
See Also
1979 IMO (Problems) • Resources | ||
Preceded by Problem 2 |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 4 |
All IMO Problems and Solutions |