Austria Federal, part 2 1997 - 2021 37p

geometry problems from Austrian Federal Competition For Advanced Students. part 2 (final round)
with aops links in the names

collected inside aops here

1997 - 2021

final round, mislabeled as part 2 inside aops

1997 Austria Federal finals, p3
Let be given a triangle $ABC$. Points $P$ on side $AC$ and $Y$ on the production of $CB$ beyond $B$ are chosen so that $Y$ subtends equal angles with $AP$ and $PC$. Similarly, $Q$ on side $BC$ and $X$ on the production of $AC$ beyond $C$ are such that $X$ subtends equal angles with $BQ$ and $QC$. Lines $YP$ and $XB$ meet at $R$, $XQ$ and $YA$ meet at $S$, and $XB$ and $YA$ meet at $D$. Prove that $PQRS$ is a parallelogram if and only if $ACBD$ is a cyclic quadrilateral

1998 Austria Federal finals, p6
In a parallelogram $ABCD$ with the side ratio $AB : BC = 2 : \sqrt 3$ the normal through $D$ to $AC$ and the normal through $C$ to $AB$ intersects in the point $E$ on the line $AB$. What is the relationship between the lengths of the diagonals $AC$ and $BD$?

1999 Austria Federal finals , p2
Let $\epsilon$ be a plane and $k_1, k_2, k_3$ be spheres on the same side of $\epsilon$. The spheres $k_1, k_2, k_3$ touch the plane at points $T_1, T_2, T_3$, respectively, and $k_2$ touches $k_1$ at  $S_1$ and $k_3$ at $S_3$. Prove that the lines $S_1T_1$ and $S_3T_3$ intersect on the sphere $k_2$. Describe the locus of the intersection point.

2000 Austria Federal finals, p2
A trapezoid $ABCD$ with $AB \parallel CD$ is inscribed in a circle $k$. Points $P$ and $Q$ are chose on the arc $ADCB$ in the order $A-P -Q-B$. Lines $CP$ and $AQ$ meet at $X$, and lines $BP$ and $DQ$ meet at $Y$. Show that points $P,Q,X, Y$ lie on a circle.

2000 Austria Federal finals, p4
In a non-equilateral acute-angled triangle $ABC$ with $\angle C = 60^\circ$, $U$ is the circumcenter, $H$ the orthocenter and $D$ the intersection of $AH$ and $BC$. Prove that the Euler line $HU$ bisects the angle $BHD$.

2001 Austria Federal finals, p3
A triangle $ABC$ is inscribed in a circle with center $U$ and radius $r$. A tangent $c'$ to a larger circle $K(U, 2r)$ is drawn so that C lies between the lines $c = AB$ and $C'$. Lines $a'$ and $b'$ are analogously defined. The triangle formed by $a', b', c'$ is denoted $A'B'C'$. Prove that the three lines, joining the midpoints of pairs of parallel sides of the two triangles, have a common point.

2001 Austria Federal finals, p6
Let be given a semicircle with the diameter $AB$, and points $C,D$ on it such that $AC = CD$. The tangent at $C$ intersects the line $BD$ at $E$. The line $AE$ intersects the arc of the semicircle at $F$. Prove that $CF < FD$.

part 2 started in 2002, the previous final round is called since then as part 1

2002 Austria Federal part2 , p3
Let $ABCD$ and $AEFG$ be two similar cyclic quadrilaterals (with the vertices denoted counterclockwise). Their circumcircles intersect again at point $P$. Prove that $P$ lies on line $BE$.

2002 Austria Federal part2 , p6
Let $H$ be the orthocenter of an acute-angled triangle $ABC$. Show that the triangles $ABH,BCH$ and $CAH$ have the same perimeter if and only if the triangle $ABC$ is equilateral.


2003 Austria Federal part2 , p3

For every lattice point $(x, y)$ with $x, y$ non-negative integers, a square of side $\frac{0.9}{2^x5^y}$ with center at the point $(x, y)$ is constructed. Compute the area of the union of all these squares.

2003 Austria Federal part2 , p6
Let $ABC$ be an acute-angled triangle. The circle $k$ with diameter $AB$ intersects $AC$ and $BC$ again at $P$ and $Q$, respectively. The tangents to $k$ at $A$ and $Q$ meet at $R$, and the tangents at $B$ and $P$ meet at $S$. Show that $C$ lies on the line $RS$.

2004 Austria Federal part2 , p3
A trapezoid $ABCD$ with perpendicular diagonals $AC$ and $BD$ is inscribed in a circle $k$. Let $k_a$ and $k_c$ respectively be the circles with diameters $AB$ and $CD$. Compute the area of the region which is inside the circle $k$, but outside the circles $k_a$ and $k_c$.

2004 Austria Federal part2 , p6
Over the sides of an equilateral triangle with area $1$ are triangles with the opposite angle $60^{\circ}$ to each side drawn outside of the triangle. The new corners are $P$, $Q$ and $R$. (and the new triangles $APB$, $BQC$ and $ARC$)
1) What is the highest possible area of the triangle $PQR$?
2) What is the highest possible area of the triangle whose vertexes are the midpoints of the inscribed circles of the triangles $APB$, $BQC$ and $ARC$?

2005 Austria Federal part2 , p3
Triangle $DEF$ is acute. Circle $c_1$ is drawn with $DF$ as its diameter and circle $c_2$ is drawn with $DE$ as its diameter. Points $Y$ and $Z$ are on $DF$ and $DE$ respectively so that $EY$ and $FZ$ are altitudes of triangle $DEF$ . $EY$ intersects $c_1$ at $P$, and $FZ$ intersects $c_2$ at $Q$. $EY$ extended intersects $c_1$ at $R$, and $FZ$ extended intersects $c_2$ at $S$. Prove that $P$, $Q$, $R$, and $S$ are concyclic points.

2005 Austria Federal part2 , p6
Let $Q$ be a point inside a cube. Prove that there are infinitely many lines $l$ so that $AQ=BQ$ where $A$ and $B$ are the two points of intersection of $l$ and the surface of the cube.

2006 Austria Federal part2 , p3
The triangle $ABC$ is given. On the extension of the side $AB$ we construct the point $R$ with $BR = BC$, where $AR > BR$ and on the extension of the side $AC$ we construct the point $S$ with $CS = CB$, where $AS > CS$. Let $A_1$ be the point of intersection of the diagonals of the quadrilateral $BRSC$.
Analogous we construct the point $T$ on the extension of the side $BC$, where $CT = CA$ and $BT > CT$ and on the extension of the side $BA$ we construct the point $U$ with $AU = AC$, where $BU > AU$. Let $B_1$ be the point of intersection of the diagonals of the quadrilateral $CTUA$.
Likewise we construct the point $V$ on the extension of the side $CA$, where $AV = AB$ and $CV > AV$ and on the extension of the side $CB$ we construct the point $W$ with $BW = BA$ and $CW > BW$. Let $C_1$ be the point of intersection of the diagonals of the quadrilateral $AVWB$.
Show that the area of the hexagon $AC_1BA_1CB_1$ is equal to the sum of the areas of the triangles $ABC$ and $A_1B_1C_1$.

2007 Austria Federal part2 , p3
Determine all rhombuses $ABCD$ with the given length $2a$ of ist sides by giving the angle $\alpha = \angle BAD$, such that there exists a circle which cuts each side of the rhombus in a chord of length $a$.

2007 Austria Federal part2 , p6
The triangle $ABC$ with the circumcircle $k(U,r)$ is given. On the extension of the radii $UA$ a point $P$ is chosen. The reflection of the line $PB$ on the line $BA$ is called $g$. Likewise the reflection of the line $PC$ on the line $CA$ is called $h$. The intersection of $g$ and $h$ is called $Q$. Find the geometric location of all possible intersections $Q$, while $P$ passes through the extension of the radii $UA$.

2008 Austria Federal part2 , p3
We are given a line $g$ with four successive points $P$, $Q$, $R$, $S$, reading from left to right. Describe a straightedge and compass construction yielding a square $ABCD$ such that $P$ lies on the line $AD$, $Q$ on the line $BC$, $R$ on the line $AB$ and $S$ on the line $CD$.

2008 Austria Federal part2 , p6
We are given a square $ABCD$. Let $P$ be a point not equal to a corner of the square or to its center $M$. For any such $P$, we let $E$ denote the common point of the lines $PD$ and $AC$, if such a point exists. Furthermore, we let $F$ denote the common point of the lines $PC$ and $BD$, if such a point exists. All such points $P$, for which $E$ and $F$ exist are called acceptable points. Determine the set of all acceptable points, for which the line $EF$ is parallel to $AD$.

2009 Austria Federal part2 , p3
Let $P$ be the point in the interior of $\vartriangle ABC$. Let $D$ be the intersection of the lines $AP$ and $BC$ and let $A'$ be the point such that $\overrightarrow{AD} = \overrightarrow{DA'}$. The points $B'$ and $C'$ are defined in the similar way. Determine all points $P$ for which the triangles $A'BC, AB'C$, and $ABC'$ are congruent to $\vartriangle ABC$.

2009 Austria Federal part2 , p6
The quadrilateral $PQRS$ whose vertices are the midpoints of the sides $AB, BC, CD, DA$, respectively of a quadrilateral $ABCD$ is called the midpoint quadrilateral of $ABCD$. Determine all circumscribed quadrilaterals whose mid-point quadrilaterals are squares.

2010 Austria Federal part2 , p3

On a circular billiard table a ball rebounds from the rails as if the rail was the tangent to the circle at the point of impact. A regular hexagon with its vertices on the circle is drawn on a circular billiard table. A (point-shaped) ball is placed somewhere on the circumference of the hexagon, but not on one of its edges. Describe a periodical track of this ball with exactly four points at the rails.

With how many different directions of impact can the ball be brought onto such a track?

2010 Austria Federal part2 , p6

A diagonal of a convex hexagon is called long if it decomposes the hexagon into two quadrangles.

Each pair of long diagonals decomposes the hexagon into two triangles and two quadrangles.

Given is a hexagon with the property, that for each decomposition by two long diagonals the resulting triangles are both isosceles with the side of the hexagon as base.

Show that the hexagon has a circumcircle.

2011 Austria Federal part2 , p3
We are given a non-isosceles triangle $ABC$ with incenter $I$. Show that the circumcircle $k$ of the triangle $AIB$ does not touch the lines $CA$ and $CB$.
Let $P$ be the second point of intersection of $k$ with $CA$ and let $Q$ be the second point of intersection of $k$ with $CB$. Show that the four points $A$, $B$, $P$ and $Q$ (not necessarily in this order) are the vertices of a trapezoid.

2011 Austria Federal part2 , p6
Two circles $k_1$ and $k_2$ with radii $r_1$ and $r_2$ touch each outside at point $Q$. The other endpoints of the diameters through $Q$ are $P$ on $k_1$ and $R$ on $k_2$. We choose two points $A$ and $B$, one on each of the arcs $PQ$ of $k_1$. ($PBQA$ is a convex quadrangle.) Further, let $C$ be the second point of intersection of the line $AQ$ with $k_2$ and let $D$ be the second point of intersection of the line $BQ$ with $k_2$. The lines $PB$ and $RC$ intersect in $U$ and the lines $PA$ and $RD$ intersect in $V$ . Show that there is a point $Z$ that lies on all of these lines $UV$

2012 Austria Federal part2 , p3
We call an isosceles trapezoid $PQRS$ interesting, if it is inscribed in the unit square $ABCD$ in such a way, that on every side of the square lies exactly one vertex of the trapezoid and that the lines connecting the midpoints of two adjacent sides of the trapezoid are parallel to the sides of the square. Find all interesting isosceles trapezoids and their areas.

2012 Austria Federal part2 , p6
Given an equilateral triangle $ABC$ with sidelength 2, we consider all equilateral triangles $PQR$ with sidelength 1 such that $P$ lies on the side $AB$,$Q$ lies on the side $AC$, and $R$ lies in the inside or on the perimeter of $ABC$. Find the locus of the centroids of all such triangles $PQR$.

2013 Austria Federal part2 , p3
A square and an equilateral triangle are inscribed in a same circle. The seven vertices form a convex heptagon $S$ inscribed in the circle ($S$ might be a hexagon if two vertices coincide). For which positions of the triangle relative to the square does $S$ have the largest and smallest area, respectively?

2013 Austria Federal part2 , p6
Consider a regular octahedron $ABCDEF$ with lower vertex $E$, upper vertex $F$, middle cross-section $ABCD$, midpoint $M$ and circumscribed sphere $k$. Further, let $X$ be an arbitrary point inside the face $ABF$. Let the line $EX$ intersect $k$ in $E$ and $Z$, and the plane $ABCD$ in $Y$. Show that $\sphericalangle{EMZ}=\sphericalangle{EYF}$.

2014 Austria Federal part2 , p6
Let $U$ be the center of the circumcircle of the acute-angled triangle $ABC$. Let  $M_A, M_B$ and $M_C$ be the circumcenters of triangles $UBC,  UAC$ and  $UAB$ respecrively. For which triangles $ABC$ is the triangle $M_AM_BM_C$ similar to the starting triangle (with a suitable order of the vertices)?

2015 Austria Federal part2 , p2
We are given a triangle $ABC$. Let $M$ be the mid-point of its side $AB$. Let $P$ be an interior point of the triangle. We let $Q$ denote the point symmetric to $P$ with respect to $M$. Furthermore, let $D$ and $E$ be the common points of $AP$ and $BP$ with sides $BC$ and $AC$, respectively. Prove that points $A$, $B$, $D$, and $E$ lie on a common circle if and only if $\angle ACP = \angle QCB$ holds.

Karl Czakler

2015 Austria Federal part2 , p5
Let I be the incenter of triangle $ABC$ and let $k$ be a circle through the points $A$ and $B$. The circle intersects
* the line $AI$ in points $A$ and $P$
* the line $BI$ in points $B$ and $Q$
* the line $AC$ in points $A$ and $R$
* the line $BC$ in points $B$ and $S$
with none of the points $A,B,P,Q,R$ and $S$ coinciding and such that $R$ and $S$ are interior points of the line segments $AC$ and $BC$, respectively.
Prove that the lines $PS$, $QR$, and $CI$ meet in a single point.

Stephan Wagner

2016 Austria Federal part2 , p2
Let $ABC$ be a triangle. Its incircle meets the sides $BC, CA$ and $AB$ in the points $D, E$ and $F$, respectively. Let $P$ denote the intersection point of $ED$ and the line perpendicular to $EF$ and passing through $F$, and similarly let $Q$ denote the intersection point of $EF$ and the line perpendicular to $ED$ and passing through $D$.
Prove that $B$ is the mid-point of the segment $PQ$.

 Karl Czakler

2017 Austria Federal part2 , p5
Let $ABC$ be an acute triangle. Let $H$ denote its orthocenter and $D, E$ and $F$ the feet of its altitudes from $A, B$ and $C$, respectively. Let the common point of $DF$ and the altitude through $B$ be $P$. The line perpendicular to $BC$ through $P$ intersects $AB$ in $Q$. Furthermore, $EQ$ intersects the altitude through $A$ in $N$. Prove that $N$ is the midpoint of $AH$.

Karl Czakler

2018 Austria Federal part2 , p2
Let $A, B, C$ and $D$ be four different points lying on a common circle in this order. Assume that the line segment $AB$ is the (only) longest side of the inscribed quadrilateral $ABCD$. Prove that the inequality $AB + BD > AC + CD$ holds.

Karl Czakler

2018 Austria Federal part2 , p4
Let $ABC$ be a triangle and $P$ a point inside the triangle such that the centers $M_B$ and $M_A$ of the circumcircles $k_B$ and $k_A$ of triangles $ACP$ and $BCP$, respectively, lie outside the triangle $ABC$. In addition, we assume that the three points $A, P$ and  $M_A$ are collinear as well as the three points $B, P$ and $M_B$. The line through $P$ parallel to side $AB$ intersects circles $k_A$ and $k_B$ in points $D$ and $E$, respectively, where $D, E \ne P$. Show that $DE = AC + BC$.

Walther Janous

2019 Austria Federal part2 , p2

A (convex) trapezoid $ABCD$ is good, if it is inscribed in a circle, sides $AB$ and $CD$ are the bases and $CD$ is shorter than $AB$. For a good trapezoid $ABCD$ the following terms are defined:

$\bullet$ The parallel to $AD$ passing through $B$ intersects the extension of side $CD$ at point $S$.

$\bullet$ The two tangents passing through $S$ on the circumircle of the trapezoid touch the circle at $E$ and $F$, where $E$ lies on the same side of the straight line $CD$ as $A$.

Give the simplest possible equivalent condition (expressed in side lengths and / or angles of the trapezoid) so that with a good trapezoid $ABCD$ the two angles $\angle BSE$ and $\angle FSC$ have the same measure.

Walther Janous

2019 Austria Federal part2 , p5

Let $ABC$ be an acute-angled triangle. Let $D$ and $E$ be the feet of the altitudes on the sides $BC$ or $AC$. Points $F$ and $G$ are located on the lines $AD$ and $BE$ in such a way that$\frac{AF}{FD}=\frac{BG}{GE}$. The line passing through $C$ and $F$ intersects $BE$ at point $H$, and the line passing through $C$ and $G$ intersects $AD$ at point $I$. Prove that points $F, G, H$ and $I$ lie on a circle.

Walther Janous

2020 Austria Federal part2 , p1

Let $ABCD$ be a convex cyclic quadrilateral with the diagonal intersection $S$. Let further be $P$ the circumcenter of the triangle $ABS$ and $Q$ the circumcenter of the triangle $BCS$. The parallel to $AD$ through $P$ and the parallel to $CD$ through $Q$ intersect at point $R$. Prove that $R$ is on $BD$.

Karl Czakler

2020 Austria Federal part2 , p5

Let $h$ be a semicircle with diameter $AB$. Let $P$ be an arbitrary point inside the diameter $AB$. The perpendicular through $P$ on $AB$ intersects $h$ at point $C$. The line $PC$ divides the semicircular area into two parts. A circle will be inscribed in each of them that touches $AB, PC$ and $h$. The points of contact of the two circles with $AB$ are denoted by $D$ and $E$, where $D$ lies between $A$ and $P$. Prove that the size of the angle $DCE$ does not depend on the choice of $P$.

Walther Janous

2021 Austria Federal part2 , p5

Let $ABCD$ be a convex cyclic quadrilateral with diagonals $AC$ and $BD$. Each of the four vertixes are reflected across the diagonal on which the do not lie.

(a) Investigate when the four points thus obtained lie on a straight line and give as simple an equivalent condition as possible to the cyclic quadrilateral $ABCD$ for it.

(b) Show that in all other cases the four points thus obtained lie on one circle.

Theresia Eisenkölbl

2022 Austria Federal part2 , p2

Let $ABC$ be an acute-angled, non-isosceles triangle with orthocenter $H$, $M$ midpoint of side $AB$ and $w$ bisector of angle $\angle ACB$. Let $S$ be the point of intersection of the perpendicular bisector of side $AB$ with $w$ and $F$ the foot of the perpendicular from $H$ on $w$. Prove that the segments $MS$ and $MF$ are equal.

Karl Czakler

2022 Austria Federal part2 , p5

Let $ABC$ be an isosceles triangle with base $AB$. We choose a point $P$ inside the triangle on altitude through $C$. The circle with diameter $CP$ intersects the straight line through $B$ and $P$ again at the point $D_P$ and the Straight through $A$ and $C$ one more time at point $E_P$. Prove that there is a point $F$ such that for any choice of $P$ the points $D_P , E_P$ and $F$ lie on a straight line.

Walther Janous