Istmo Centroamericano 2017-19 3p

 geometry problems from Istmo Centroamericano MO from with aops links in the names

collected inside aops here

2017-19

it didn;t take place in 2020

2017 Istmo Centroamericano MO p1

Let $ABC$ be a triangle with $\angle ABC = 90^o$ and $AB> BC$. Let $D$ be a point on side $AB$ such that $BD = BC$. Let $E$ be the foot of the perpendicular from $D$ on $AC$, and $F$ the reflection of $B$ wrt $CD$. Show that $EC$ is the bisector of angle $\angle BEF$.

2018 Istmo Centroamericano MO p5

Let $ABC$ be an isosceles triangle with $CA = CB$. Let $D$ be the foot of the alttiude from $C$, and $\ell$ be the external angle bisector at $C$. Take a point $N$ on $\ell$ so that $AN> AC$ ,  on the same side as $A$ wrt $CD$. The bisector of the angle $NAC$ cuts $\ell$'at $F$. Show that $\angle NCD + \angle  BAF> 180^o.$

2019 Istmo Centroamericano MO p3

Let $ABC$ be an acute triangle, with $AB <AC$. Let $M$ be the midpoint of $AB$, $H$ the foot of the altitude from $A$, and $Q$ be point on side $AC$ such that $\angle  ABQ = \angle BCA$. Show that the circumcircles of the triangles  $ABQ$ and $BHM$ are tangent.

source: www.istmo-omcc.org/