TH-TMO 2023 Problem 5

Let $\ell$ be a line in the plane and let $90^\circ<\theta<180^\circ$ . Consider any distinct points $P,Q,R$ that satisfy the following:

(i) $P$ lies on $\ell$ and $PQ$ is perpendicular to $\ell$ (ii) $R$ lies on the same side of $\ell$ as $Q$ , and $R$ doesn't lie on $\ell$ (iii) for any points $A,B$ on $\ell$ , if $\angle ARB=\theta$ then $\angle AQB \geq \theta$ .

Find the minimum value of $\angle PQR$ .