Question 3
In a plane two different points $O$ and $A$ are given. For each point $X\ne O$ of the plane denote by $\alpha(X)$ the angle $AOX$ measured in radians $(0\le\alpha(X)\lt 2\pi)$ and by $C(X)$ the circle with center $O$ and radius $OX+\frac{\alpha(X)}{OX}$. Suppose each point of the plane is colored by one of a finite number of colors. Show that there exists a point $X$ with $\alpha(X)\gt 0$ such that its color appears somewhere on the circle $C(X)$.