Austria Junior Regional 2019-22 4p

 geometry problems from Austrian Junior Regional Competition with aops links in the names

collected inside aops here

2019 - 2021

2019 Austria  Junior Regional p2

A square $ABCD$ is given. Over the side $BC$ draw an equilateral triangle $BSD$ on the outside. The midpoint of the segment $AS$ is $N$ and the midpoint of the side $CD$ is $H$. Prove that $\angle NHC = 60^o$.

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(Karl Czakler)

2020 Austria  Junior Regional p3

Given is an isosceles trapezoid $ABCD$ with $AB \parallel CD$ and $AB> CD$. The The projection from $D$ on $AB$ is $E$. The midpoint of the diagonal $BD$ is $M$. Prove that $EM$ is parallel to $AC$.

(Karl Czakler)

2021 Austria  Junior Regional p2

A triangle $ABC$ with circumcenter $U$ is given, so that $\angle CBA = 60^o$ and $\angle CBU = 45^o$ apply. The straight lines $BU$ and $AC$ intersect at point $D$. Prove that $AD = DU$.

(Karl Czakler)

2022 Austria  Junior Regional p3

A semicircle is erected over the segment $AB$ with center $M$. Let $P$ be one point different from $A$ and $B$ on the semicircle and $Q$ the midpoint of the arc of the circle $AP$. The point of intersection of the straight line $BP$ with the parallel to $P Q$ through $M$ is $S$. Show that $PM = PS$ holds.

(Karl Czakler)