55rd Ukrainian National Mathematical Olympiad - Third Round (Second Tour)

Circles $w_1$ and $w_2$ with centers $O_1$ and $O_2$ respectively intersect in points $A$ and $B$. A straight line $O_1{O}_2$ intersects $w_1$ in a point $Q$, that is not inside $w_2$, and $w_2$ in a point $X$, that is inside $w_1$. Around the triangle $O_{1}AX$ a circle $w_3$ is circumscribed and intersects $w_1$ for the second time in a point $T$. A straight line $QT$ intersects $w_3$ in a point $K$, and a straight line $QB$ intersects $w_2$ a second time in a point $H$. Prove that



Fig. 24

a) points $T$, $X$, $B$ are collinear; b) points $K$, $X$, $H$ are collinear.