Question 12
We have two different points in the plane, $A$ and $O$. For each point $X$ of the plane other than $O$, we denoted by $\alpha(X)$ the angle measurement between $OA$ and $OX$, in radians, and counted counterclockwise from $OA$ $(0 \leq \alpha(X)\lt 2\pi)$. Let $C(X)$ the center circumference $O$ and radius of length $OX + \frac{\alpha(X)}{OX}$. We have a finite number of colors and color each of the dots on the plane with them. Prove that there is a point $Y$ such that $\alpha(Y)\gt 0$ and such that its color appears on the circumference of $C(Y)$.