SELECTION TESTS FOR THE 2019 BMO AND IMO

The internal bisectrix $AI$ and the perpendicular bisectrix $MN$ of the side $BC$ cross at the midpoint $K$ of the arc $BEC$. It is a fact that the circle $BIC$ is centred at $K$.

Let now the line $MP$ cross the circle $BIC$ again at $Q$, to infer that the arcs $BL$ and $CQ$ of this circle have equal angular spans, so $L$ and $Q$ are reflexions of one another in the perpendicular bisectrix $KMN$ of the chord $BC$. Project $Q$ orthogonally to $Q^{\prime }$ on $BC$ and refer to standard notation in the triangle $ABC$: $a$, $b$, $c$ denote the lengths of the sides $BC$, $CA$, $AB$, respectively, $s=(a+b+c)/2$ denotes its semiperimeter, $r$ its inradius, and $S$ its area. With reference to standard formulae, write $C{Q}^{\prime }=BD=s-b$ and $$Q{Q}^{\prime }=DL=\frac{DB\cdot DC}{DI}=\frac{(s-b)(s-c)}{r}=\frac{s}{s-a},$$ to infer that $Q$ is the $A$-excentre of the triangle $ABC$, so it lies on the line $AIK$. Finally, write $\angle PQA=\angle PQI=\angle PLI=\angle ELI=\angle EAI=\angle DAQ$, to conclude that the lines $AD$ and $MP$ are indeed parallel.