Question 8
Let $\omega$ be a circle, and let $A$ and $B$ be points on a line not intersecting $\omega$. Given a point $X_0$ on $\omega$, define a sequences $X_0, X_1, X_2, \ldots$ and $Y_0, Y_1, Y_2, \ldots$ as follows: $Y_n$ is the second intersection of the line $AX_n$ with $\omega$ and $X_{n+1}$ is the second intersection of the line $BY_n$ with $\omega$. Prove that if for some integer $k$, it happens that $X_k = X_0$ for some choice of $X_0$ then $X_k = X_0$ for any choice of $X_0$.