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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

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)
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)
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)
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)

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