Given points
![O](/media/m/9/6/0/9601b72f603fa5d15addab9937462949.png)
and
![A](/media/m/5/a/e/5ae81275ee67d638485e903bdc0e9cde.png)
in the plane. Every point in the plane is colored with one of a finite number of colors. Given a point
![X](/media/m/9/2/8/92802f174fc4967315c2d8002c426164.png)
in the plane, the circle
![C(X)](/media/m/a/6/c/a6caa58ca063e470c14eda7e7950d75d.png)
has center
![O](/media/m/9/6/0/9601b72f603fa5d15addab9937462949.png)
and radius
![OX+{\angle AOX\over OX}](/media/m/5/a/9/5a9ed217f6d6bb805738956a11e9c824.png)
, where
![\angle AOX](/media/m/e/7/d/e7d68c2031d149b78fab366a6cf6f375.png)
is measured in radians in the range
![[0,2\pi)](/media/m/2/3/f/23fd02ac89b9805356f69c51978fe402.png)
. Prove that we can find a point
![X](/media/m/9/2/8/92802f174fc4967315c2d8002c426164.png)
, not on
![OA](/media/m/b/2/0/b206c115fb0e114a37cf644cba5338cb.png)
, such that its color appears on the circumference of the circle
![C(X)](/media/m/a/6/c/a6caa58ca063e470c14eda7e7950d75d.png)
.
%V0
Given points $O$ and $A$ in the plane. Every point in the plane is colored with one of a finite number of colors. Given a point $X$ in the plane, the circle $C(X)$ has center $O$ and radius $OX+{\angle AOX\over OX}$, where $\angle AOX$ is measured in radians in the range $[0,2\pi)$. Prove that we can find a point $X$, not on $OA$, such that its color appears on the circumference of the circle $C(X)$.