CR-MO 2019

Let $x, y, z \in R$ , find all triples $(x, y, z)$ that satisfy the following system of equations: $2x^2 - 3xy + 2y^2 = 1$ $y^2 - 3yz + 4z^2 = 2$ $z^2 + 3zx - x^2 = 3$

Let $H$ be the orthocenter and $O$ the circumcenter of the acute triangle $\vartriangle ABC$ . The circle with center $H$ and radius $HA$ intersects the lines $AC$ and $AB$ at points $P$ and $Q$ , respectively. Let point $O$ be the orthocenter of triangle $\vartriangle APQ$ , determine the measure of $\angle BAC$ .