Question 63
$AB$ is a diameter of circle $O$. $X$ is a point on $AB$ such that $AX = 3 BX$. Distinct circles $\omega_1$ and $\omega_2$ are tangent to $O$ at $T_1$ and $T_2$ and to $AB$ at $X$. The lines $T_1 X$ and $T_2 X$ intersect $O$ again at $S_1$ and $S_2$. What is the ratio $\frac{T_1 T_2}{S_1 S_2}$?