Question 5
Circles $\omega$ and $\Omega$ meet at $A,B$. Let $M$ be the midpoint of the arc $AB$ of $\omega$ lying inside $\Omega$. A chord $MP$ of $\omega$ intersects $\Omega$ at $Q$ inside $\omega$. Let $\ell_P$ be the tangent to $\omega$ at $P$ and $\ell_Q$ the tangent to $\Omega$ at $Q$. Prove that the circumcircle of the triangle formed by $\ell_P,\ell_Q,AB$ is tangent to $\Omega$.
Proposed by Ilya Bogdanov, Russia, and Medeubek Kungozhin, Kazakhstan.
Newly drawn configuration:
Proposed by Ilya Bogdanov, Russia, and Medeubek Kungozhin, Kazakhstan.
Newly drawn configuration: