Austria Regional 2001-22 22p

geometry problems from Austrian Regional Competition For Advanced Students
with aops links in the names

collected inside aops here

2001 - 2022

2001 Austria Regional p3
In a convex pentagon $ABCDE$, the area of the triangles $ABC, ABD, ACD$ and $ADE$ are equal and have the value $F$. What is the area of the triangle $BCE$ ?

2002 Austria Regional p3
In the convex $ABCDEF$ (has all interior angles less than $180^o$) with the perimeter $s$ the triangles $ACE$ and $BDF$ have perimeters $u$ and $v$ respectively.
a) Show the inequalities $\frac{1}{2} \le   \frac{s}{u+v}\le 1$
b) Check whether $1$ is replaced by a smaller number or $1/2$ by a larger number can the inequality remains valid for all convex hexagons.

2003 Austria Regional p3
Given are two parallel lines $g$ and $h$ and a point $P$, that lies outside of the corridor bounded by $g$ and $h$. Construct three lines $g_1$, $g_2$ and $g_3$ through the point $P$. These lines intersect $g$ in $A_1,A_2, A_3$ and $h$ in $B_1, B_2, B_3$ respectively. Let $C_1$ be the intersection of the lines $A_1B_2$ and $A_2B_1$, $C_2$ be the intersection of the lines $A_1B_3$ and $A_3B_1$ and let $C_3$ be the intersection of the lines $A_2B_3$ and $A_3B_2$. Show that there exists exactly one line $n$, that contains the points $C_1,C_2,C_3$ and that $n$ is parallel to $g$ and $h$.

2004 Austria Regional p3
Given is a convex quadrilateral $ABCD$ with $\angle ADC=\angle BCD>90^{\circ}$.
Let $E$ be the point of intersection of the line $AC$ with the parallel line to $AD$ through $B$ and $F$ be the point of intersection of the line $BD$ with the parallel line to $BC$ through $A$. Show that $EF$ is parallel to $CD$

2005 Austria Regional p2
Construct the semicircle $h$ with the diameter $AB$ and the midpoint $M$. Now construct the semicircle $k$ with the diameter $MB$ on the same side as $h$. Let $X$ and $Y$ be points on $k$, such that the arc $BX$ is $\frac{3}{2}$ times the arc $BY$. The line $MY$ intersects the line $BX$ in $D$ and the semicircle $h$ in $C$. Show that $Y$ ist he midpoint of  $CD$.

2006 Austria Regional p3
In a non isosceles triangle $ABC$ let $w$ be the angle bisector of the exterior angle at $C$. Let $D$ be the point of intersection of $w$ with the extension of $AB$. Let $k_A$ be the circumcircle of the triangle $ADC$ and analogy $k_B$ the circumcircle of the triangle $BDC$. Let $t_A$ be the tangent line to $k_A$ in A and $t_B$ the tangent line to $k_B$ in B. Let $P$ be the point of intersection of $t_A$ and $t_B$. Given are the points $A$ and $B$. Determine the set of points $P=P(C )$ over all points $C$, so that $ABC$ is a non isosceles, acute-angled triangle.

2007 Austria Regional p4
Let $M$ be the intersection of the diagonals of a convex quadrilateral $ABCD$. Determine all such quadrilaterals for which there exists a line $g$ that passes through $M$ and intersects the side $AB$ in $P$ and the side $CD$ in $Q$, such that the four triangles $APM$, $BPM$, $CQM$, $DQM$ are similar.

2008 Austria Regional p3
Given is an acute angled triangle $ABC$. Determine all points $P$ inside the triangle with
$1\leq\frac{\angle APB}{\angle ACB},\frac{\angle BPC}{\angle BAC},\frac{\angle CPA}{\angle CBA}\leq2$

2009 Austria Regional p3

Let $D$, $E$, $F$ be the feet of the altitudes wrt sides $BC$, $CA$, $AB$ of acute-angled triangle $\triangle ABC$, respectively. In triangle $\triangle CFB$, let $P$ be the foot of the altitude wrt side $BC$. Define $Q$ and $R$ wrt triangles $\triangle ADC$ and $\triangle BEA$ analogously. Prove that lines $AP$, $BQ$, $CR$ don't intersect in one common point.

2010 Austria Regional p3
Let $\triangle ABC$ be a triangle and let $D$ be a point on side $\overline{BC}$. Let $U$ and $V$ be the circumcenters of triangles $\triangle ABD$ and $\triangle ADC$, respectively. Show, that $\triangle ABC$ and $\triangle AUV$ are similar.

2011 Austria Regional p3
Let $k$ be a circle centered at $M$ and let $t$ be a tangentline to $k$ through some point $T\in k$. Let $P$ be a point on $t$ and let $g\neq t$ be a line through $P$ intersecting $k$ at $U$ and $V$. Let $S$ be the point on $k$ bisecting the arc $UV$ not containing $T$ and let $Q$ be the the image of $P$ under a reflection over $ST$. Prove that $Q$, $T$, $U$ and $V$ are vertices of a trapezoid.

2012 Austria Regional p4
In a triangle $ABC$, let $H_a$, $H_b$ and $H_c$ denote the base points of the altitudes on the sides $BC$, $CA$ and $AB$, respectively. Determine for which triangles $ABC$ two of the lengths $H_aH_b$, $H_bH_c$ and $H_aH_c$ are equal.

2013 Austria Regional p4

We call a pentagon distinguished if either all side lengths or all angles are equal. We call it very distinguished if in addition two of the other parts are equal. i.e. $5$ sides and $2$ angles or $2$ sides and $5$ angles.Show that every very distinguished pentagon has an axis of symmetry.

2014 Austria Regional p4

For a point $P$ in the interior of a triangle $ABC$ let $D$ be the intersection of $AP$ with $BC$, let $E$ be the intersection of $BP$ with $AC$ and let $F$ be the intersection of $CP$ with $AB$.Furthermore let $Q$ and $R$ be the intersections of the parallel to $AB$ through $P$ with the sides $AC$ and $BC$, respectively. Likewise, let $S$ and $T$ be the intersections of the

parallel to $BC$ through $P$ with the sides $AB$ and $AC$, respectively.In a given triangle $ABC$, determine all points $P$ for which the triangles $PRD$, $PEQ$and $PTE$ have the same area.

2015 Austria Regional p4
Let $ABC$ be an isosceles triangle with $AC = BC$ and $\angle ACB < 60^\circ$. We denote the incenter and circumcenter by $I$ and $O$, respectively. The circumcircle of triangle $BIO$ intersects the leg $BC$ also at point $D \ne B$.
(a) Prove that the lines $AC$ and $DI$ are parallel.
(b) Prove that the lines $OD$ and $IB$ are mutually perpendicular.

Walther Janous

2016 Austria Regional p4
Let $ABC$ be a triangle with $AC > AB$ and circumcenter $O$. The tangents to the circumcircle at $A$ and $B$ intersect at $T$. The perpendicular bisector of the side $BC$ intersects side $AC$ at $S$.
(a) Prove that the points $A$, $B$, $O$, $S$, and $T$ lie on a common circle.
(b) Prove that the line $ST$ is parallel to the side $BC$.

Karl Czakler

2017 Austria Regional p2
Let $ABCD$ be a cyclic quadrilateral with perpendicular diagonals and circumcenter $O$. Let $g$ be the line obtained by reflection of the diagonal $AC$ along the angle bisector of $\angle BAD$. Prove that the point $O$ lies on the line $g$.

 Theresia Eisenkölbl

2018 Austria Regional p2
Let $k$ be a circle with radius $r$ and $AB$ a chord of $k$ such that $AB > r$. Furthermore, let $S$ be the point on the chord $AB$ satisfying $AS = r$. The perpendicular bisector of $BS$ intersects $k$ in the points $C$ and $D$. The line through $D$ and $S$ intersects $k$ for a second time in point $E$. Show that the triangle $CSE$ is equilateral.

Stefan Leopoldseder

2019 Austria Regional p2

The convex pentagon $ABCDE$ is cyclic and $AB = BD$. Let point  $P$ be the intersection of the diagonals $AC$ and $BE$. Let the straight lines $BC$ and $DE$ intersect at point $Q$. Prove that the straight line $PQ$ is parallel to the diagonal $AD$.

Gottfried Perz

2020 Austria Regional p1

Let a triangle $ABC$ be given with $AB <AC$. Let the inscribed center of the triangle be $I$. The perpendicular bisector of side $BC$ intersects the angle bisector of $BAC$ at point $S$ and the angle bisector of $CBA$ at point $T$. Prove that the points $C, I, S$ and $T$ lie on a circle.

Karl Czakler

2021 Austria Regional p2

Let $ABC$ be an isosceles triangle with $AC = BC$ and circumcircle $k$. The point $D$ lies on the shorter arc of $k$ over the chord $BC$ and is different from $B$ and $C$. Let $E$ denote the intersection of $CD$ and $AB$. Prove that the line through $B$ and $C$ is a tangent of the circumcircle of the triangle $BDE$.

Karl Czakler

2022 Austria Regional p2

Let $ABC$ denote a triangle with $AC\ne BC$. Let $I$ and $U$ denote the incenter and circumcenter of the triangle $ABC$, respectively. The incircle touches $BC$ and $AC$ in the points $D$ and E, respectively. The circumcircles of the triangles $ABC$ and $CDE$ intersect in the two points $C$ and $P$. Prove that the common point $S$ of the lines $CU$ and $P I$ lies on the circumcircle of the triangle $ABC$.

Karl Czakler