Question 3
Let $ABC$ be an acute scalene triangle with no angle equal to $60^\circ$.
Let $\omega$ be the circumcircle of $ABC$.
Let $\Delta_B$ be the equilateral triangle with three vertices on $\omega$,
one of which is $B$.
Let $\ell_B$ be the line through the two vertices of $\Delta_B$ other than $B$.
Let $\Delta_C$ and $\ell_C$ be defined analogously.
Let $Y$ be the intersection of $AC$ and $\ell_B$,
and let $Z$ be the intersection of $AB$ and $\ell_C$.
Let $N$ be the midpoint of minor arc $BC$ on $\omega$.
Let $\mathcal{R}$ be the triangle formed by $\ell_B$, $\ell_C$, and the tangent to $\omega$ through $N$.
Prove that the circumcircle of $AYZ$ and the incircle of $\mathcal{R}$ are tangent.
Proposed by Rogelio Guerrero.
Manual essay; 1 editorial point, not official contest scoring. Automatic answer fields are intentionally empty.
Let $\omega$ be the circumcircle of $ABC$.
Let $\Delta_B$ be the equilateral triangle with three vertices on $\omega$,
one of which is $B$.
Let $\ell_B$ be the line through the two vertices of $\Delta_B$ other than $B$.
Let $\Delta_C$ and $\ell_C$ be defined analogously.
Let $Y$ be the intersection of $AC$ and $\ell_B$,
and let $Z$ be the intersection of $AB$ and $\ell_C$.
Let $N$ be the midpoint of minor arc $BC$ on $\omega$.
Let $\mathcal{R}$ be the triangle formed by $\ell_B$, $\ell_C$, and the tangent to $\omega$ through $N$.
Prove that the circumcircle of $AYZ$ and the incircle of $\mathcal{R}$ are tangent.
Proposed by Rogelio Guerrero.
Manual essay; 1 editorial point, not official contest scoring. Automatic answer fields are intentionally empty.