Iranian Mathematical Olympiad

Point $O$ is the center of the circumcircle $\omega$ of the acute-angled triangle $ABC$. A circle centered at $O$ tangent to side $BC$ of the triangle is drawn. Let $X$ and $Y$ be the intersection points of tangents from $A$ to this circle with side $BC$ in such a way that points $X$ and $B$ are on one side of the line $AO$. A line parallel to side $AC$ is drawn from $X$ to intersect the tangent at point $B$ to the circle $\omega$ at $T$. Similarly, a line parallel to side $AB$ is drawn from $Y$ to intersect the tangent at point $C$ to the circle $\omega$ at $S$. Prove that line $TS$ is tangent to $\omega$.