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
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.
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.
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