Olympiad problems

Cono Sur Shortlist - geometry, 2003.G1

Let $O$ be the circumcenter of the isosceles triangle $ABC$ ($AB = AC$). Let $P$ be a point of the segment $AO$ and $Q$ the symmetric of $P$ with respect to the midpoint of $AB$. If $OQ$ cuts $AB$ at $K$ and the circle that passes through $A, K$ and $O$ cuts $AC$ in $L$, show that $\angle ALP = \angle CLO$.

2022 EGMO, 1

Tags: angle chase , angle chasing , collinear , incenter , geometry

Let $ABC$ be an acute-angled triangle in which $BC<AB$ and $BC<CA$. Let point $P$ lie on segment $AB$ and point $Q$ lie on segment $AC$ such that $P \neq B$, $Q \neq C$ and $BQ = BC = CP$. Let $T$ be the circumcenter of triangle $APQ$, $H$ the orthocenter of triangle $ABC$, and $S$ the point of intersection of the lines $BQ$ and $CP$. Prove that $T$, $H$, and $S$ are collinear.

2007 Sharygin Geometry Olympiad, 9

Tags: geometry , angle chasing , convex quadrilateral , angle chase

Suppose two convex quadrangles are such that the sides of each of them lie on the perpendicular bisectors of the sides of the other one. Determine their angles,

2017 IOM, 1

Tags: geometry , parallelogram , angle chase

Let $ABCD$ be a parallelogram in which angle at $B$ is obtuse and $AD>AB$. Points $K$ and $L$ on $AC$ such that $\angle ADL=\angle KBA$(the points $A, K, C, L$ are all different, with $K$ between $A$ and $L$). The line $BK$ intersects the circumcircle $\omega$ of $ABC$ at points $B$ and $E$, and the line $EL$ intersects $\omega$ at points $E$ and $F$. Prove that $BF||AC$.

2017 Baltic Way, 14

Tags: perpendicular lines , angle chase , geometry

Let $P$ be a point inside the acute angle $\angle BAC$. Suppose that $\angle ABP = \angle ACP = 90^{\circ}$. The points $D$ and $E$ are on the segments $BA$ and $CA$, respectively, such that $BD = BP$ and $CP = CE$. The points $F$ and $G$ are on the segments $AC$ and $AB$, respectively, such that $DF$ is perpendicular to $AB$ and $EG$ is perpendicular to $AC$. Show that $PF = PG$.