Question 3
Let $ABC$ be an equilateral triangle. Let $P\in AC$ and $Q\in AB$ be such that both triangles $ABP,ACQ$ are acute. Let $R,S$ be their respective orthocentres, and $T=BP\cap CQ$. Find all possible values of $\angle CBP$ and $\angle BCQ$ such that $TRS$ is equilateral.
Editorial convention: the intersection $T$ is unique and triangle $TRS$ is nondegenerate. The simultaneous choice $P=C,Q=B$ makes $BP=CQ=BC$ and is therefore excluded.
Editorial convention: the intersection $T$ is unique and triangle $TRS$ is nondegenerate. The simultaneous choice $P=C,Q=B$ makes $BP=CQ=BC$ and is therefore excluded.