Question 9
Circles $\Omega_1$ and $\Omega_2$ with different radii intersect at two points, denote one of them by $P$. A variable line $\ell$ passing through $P$ intersects the arc of $\Omega_1$ which is outside of $\Omega_2$ at $X_1$, and the arc of $\Omega_2$ which is outside of $\Omega_1$ at $X_2$. Let $R$ be the point on segment $X_1X_2$ such that $X_1P = RX_2$. The tangent to $\Omega_1$ through $X_1$ meets the tangent to $\Omega_2$ through $X_2$ at $T$. Prove that line $RT$ is tangent to a fixed circle, independent of the choice of $\ell$. (Josef Tkadlec)