Arab Math Olympiad 2018, 2020 2p

geometry problems from Arab Math Olympiad, an international contest for Arab countries, that started in 2018 with aops links

collected inside aops here

2018, 2020

2018  Arab Math Olympiad p2

Let $ABC$ be an acute triangle of circumcenter $O$. The line $(AO)$ intersects $(BC)$ at $D$. The parallel line through $D$ to $(AB)$ intersects $(BO)$ at $S$. $(AS)$ and $(BC)$ intersect at $T$. Show that if $O,D,S$ and $T$ lie on the same circle, then $ABC$ is an isosceles triangle.

2020  Arab Math Olympiad p3

Let $ABC$ be an oblique triangle and $H$ be the foot of the altitude passing through the vertex $A$. We denote by $I, J, K$ the respective midpoints of the segments $AB,AC$ and $IJ$. Show that the circle $c_1$ passing through the point $K$ and tangent to line $AB$ at $I$, and the circle $c_2$ passing through the point $K$ and tangent to line $AC$ at $J$, intersect at second point $K'$ , and that $H,K$ and $K'$ are collinear.