Albanians Cup in Mathematics 2020-21 2p

  geometry problems from Albanians Cup in Mathematics, with aops links in the names

2020-21

2020 Albanians Cup in Mathematics p4

Let $ABC$ be a triangle with its orthocenter $H$. Circle with diameter $AH$ intersect again the circumcircle of traingle of $ABC$ at point $P$. Line which pass through $H$ and parallel with $AP$ intersect lines $AC, AB$ at points $E, F$, respectively. Let $X$ be the intersection point of the lines $BE$ and $CF$. Prove that line $PH$ bisect the segment $AX$.

Viktor Ahmeti, Kosovo

2021 Albanians Cup in Mathematics p2

Angle bisector at $A$, altitude from $B$ to $CA$ and altitude of $C$ to $AB$ on a scalene triangle $ABC$ forms a triangle $\triangle$. Let $P$ and $Q$ points on lines $AB$ and $AC$, respectively, such that the midpoint of segment $PQ$ is the orthocenter of the triangle $\triangle$. Prove that the points $B, C, P$ and $Q$ lie on a circle.