IMO Shortlist 1984 problem 8
Kvaliteta:
Avg: 0,0Težina:
Avg: 0,0 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
.
![O](/media/m/9/6/0/9601b72f603fa5d15addab9937462949.png)
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
![X](/media/m/9/2/8/92802f174fc4967315c2d8002c426164.png)
![C(X)](/media/m/a/6/c/a6caa58ca063e470c14eda7e7950d75d.png)
![O](/media/m/9/6/0/9601b72f603fa5d15addab9937462949.png)
![OX+{\angle AOX\over OX}](/media/m/5/a/9/5a9ed217f6d6bb805738956a11e9c824.png)
![\angle AOX](/media/m/e/7/d/e7d68c2031d149b78fab366a6cf6f375.png)
![[0,2\pi)](/media/m/2/3/f/23fd02ac89b9805356f69c51978fe402.png)
![X](/media/m/9/2/8/92802f174fc4967315c2d8002c426164.png)
![OA](/media/m/b/2/0/b206c115fb0e114a37cf644cba5338cb.png)
![C(X)](/media/m/a/6/c/a6caa58ca063e470c14eda7e7950d75d.png)
Izvor: Međunarodna matematička olimpijada, shortlist 1984