Germany Federal 1974 - 2022 (BWM) 96p

geometry problems from German Federal Mathematics Competitions

(Bundeswettbewerb Mathematik - BWM)

with aops links in the names

[this competition is independent to the German Math Olympiad,
and is short of a correspodence competition]

collected inside aops here

1974 -  2020

Round 1

1974 German Federal (BWM) Round 1 p1

Find the necessary and sufficient condition that a trapezoid can be formed out of a given four-bar linkage.

1975 German Federal (BWM) Round 1 p3

Describe all quadrilaterals with perpendicular diagonals which are both inscribed and circumscribed.

1977 German Federal (BWM) Round 1 p4

Show that the four perpendiculars dropped from the midpoints of the sides of a cyclic quadrilateral to the respective opposite sides are concurrent.

Note by Darij: A cyclic quadrilateral is a quadrilateral inscribed in a circle.

1978 German Federal (BWM) Round 1 p4

In a triangle $ABC$, the points $A_1, B_1, C_1$ are symmetric to $A, B,C$ with respect to $B,C, A$, respectively. Given the points $A_1, B_1,C_1$ reconstruct the triangle $ABC$.

1979 German Federal (BWM) Round 1 p2

The squares $OABC$ and $OA_1B_1C_1$ are situated in the same plane and are directly oriented. Prove that the lines $AA_1$ , $BB_1$, and $CC_1$ are concurrent.

1980 German Federal (BWM) Round 1 p2

In a triangle $ABC$, the bisectors of angles $A$ and $B$ meet the opposite sides of the triangle at points $D$ and $E$, respectively. A point $P$ is arbitrarily chosen on the line $DE$. Prove that the distance of $P$ from line $AB$ equals the sum or the difference of the distances of $P$ from lines $AC$ and $BC$.

1981 German Federal (BWM) Round 1 p2

Prove that if the sides $a, b, c$ of a non-equilateral triangle satisfy $a + b = 2c$, then the line passing through the incenter and centroid is parallel to one of the sides of the triangle.

1982 German Federal (BWM) Round 1 p2

In a convex quadrilateral $ABCD$ sides $AB$ and $DC$ are both divided into $m$ equal parts by points $A, S_1 , S_2 , \ldots , S_{m-1} ,B$ and $D,T_1, T_2, \ldots , T_{m-1},C,$ respectively (in this order). Similarly, sides $BC$ and $AD$ are divided into $n$ equal parts by points $B,U_1,U_2, \ldots, U_{n-1},C$ and $A,V_1,V_2, \ldots,V_{n-1}, D$. Prove that for $1 \leq i \leq m-1$ each of the segments $S_i T_i$ is divided by the segments $U_j V_j$ ($1\leq j \leq n-1$) into $n$ equal parts

1983 German Federal (BWM) Round 1 p1

The surface of a soccer ball is made up of black pentagons and white hexagons together. On the sides of each pentagon are nothing but hexagons, while on the sides of each border of hexagons alternately pentagons and hexagons. Determine from this information about the soccer ball , the number of its pentagons and its hexagons.

1983 German Federal (BWM) Round 1 p2

The radii of the circumcircle and the incircle of a right triangle are given. Cconstruct that triangle with compass and ruler, describe the construction and justify why it is correct.

1984 German Federal (BWM) Round 1 p2

Given is a regular $n$-gon with circumradius $1$. $L$ is the set of (different) lengths of all connecting segments of its endpoints. What is the sum of the squares of the elements of $L$?

1985 German Federal (BWM) Round 1 p2

Prove that in every triangle for each of its altitudes: If you project the foof of one altitude on the other two altitudes and on the other two sides of the triangle, those four projections lie on the same line.

1986 German Federal (BWM) Round 1 p3

The points $S$ lie on side $AB$, $T$ on side $BC$, and $U$ on side $CA$ of a triangle so that the following applies: $\overline{AS} : \overline{SB} = 1 : 2$, $\overline{BT} : \overline{TC} = 2 : 3$ and $\overline{CU} : \overline{UA} = 3 : 1$. Construct the triangle $ABC$ if only the points $S, T$ and $U$ are given.

1988 German Federal (BWM) Round 1 p2

Let $h_a$, $h_b$ and $h_c$ be the heights and $r$ the inradius of a triangle.

Prove that the triangle is equilateral if and only if $h_a + h_b + h_c = 9r$.

1988 German Federal (BWM) Round 1 p3

Consider an octagon with equal angles and with rational sides. Prove that it has a center of symmetry.

1989 German Federal (BWM) Round 1 p2

A trapezoid has area $2\, m^2$ and the sum of its diagonals is $4\,m$. Determine the height of this trapezoid.

1990 German Federal (BWM) Round 1 p4

Suppose that every two opposite edges of a tetrahedron are orthogonal. Show that the midpoints of the six edges lie on a sphere.

1992 German Federal (BWM) Round 1 p3

Given is a triangle $ABC$ with side lengths $a, b,c$. Three spheres touch each other in pairs and also touch the plane of the triangle at points $A,B$ and $C$, respectively. Determine the radii of these spheres.

1993 German Federal (BWM) Round 1 p4

Given is a triangle $ABC$ with side lengths $a, b, c$ ($a = \overline{BC}$, $b = \overline{CA}$, $c = \overline{AB}$) and area $F$. The side $AB$ is extended beyond $A$ by a and beyond $B$ by $b$. Correspondingly, $BC$ is extended beyond $B$ and $C$ by $b$ and $c$, respectively. Eventually $CA$ is extended beyond $C$ and $A$ by $c$ and $a$, respectively. Connecting the outer endpoints of the extensions , a hexagon if formed with area $G$. Prove that $\frac{G}{F}>13$.

1994 German Federal (BWM) Round 1 p3

Given a triangle $A_1 A_2 A_3$ and a point $P$ inside. Let $B_i$ be a point on the side opposite to $A_i$ for $i=1,2,3$, and let $C_i$ and $D_i$ be the midpoints of $A_i B_i$ and $P B_i$, respectively. Prove that the triangles $C_1 C_2 C_3$ and $D_1 D_2 D_3$ have equal area.

1995 German Federal (BWM) Round 1 p2
A line $g$ and a point $A$ outside $g$ are given in a plane. A point $P$ moves along $g$. Find the locus of the third vertices of equilateral triangles whose two vertices are $A$ and $P$.

1996 German Federal (BWM) Round 1 p3
Four lines are given in a plane so that any three of them determine a triangle. One of these lines is parallel to a median in the triangle determined by the other three lines. Prove that each of the other three lines also has this property.

1997 German Federal (BWM) Round 1 p3
A square $S_a$ is inscribed in an acute-angled triangle $ABC$ with two vertices on side $BC$ and one on each of sides $AC$ and $AB$. Squares $S_b$ and $S_c$ are analogously inscribed in the triangle. For which triangles are the squares $S_a,S_b$, and $S_c$ congruent?

1998 German Federal (BWM) Round 1 p3
Let $F$ be the midpoint of side $BC$ or triangle $ABC$. Construct isosceles right triangles $ABD$ and $ACE$ externally on sides $AB$ and $AC$ with the right angles at $D$ and $E$ respectively. Show that $DEF$ is an isosceles right triangle.

1999 German Federal (BWM) Round 1 p3
In the plane are given a segment $AC$ and a point $B$ on the segment. Let us draw the positively oriented isosceles triangles $ABS_1, BCS_2$, and $CAS_3$ with the angles at $S_1,S_2,S_3$ equal to $120^o$. Prove that the triangle $S_1S_2S_3$ is equilateral.

2000 German Federal (BWM) Round 1 p3
A convex quadrilateral $ABCD$ is inscribed in a semicircle with diameter $AB$. The diagonals $AC,BD$ intersect at $S$, and $T$ is the projection of $S$ on $AB$. Show that $ST$ bisects angle $CTD$.

2001 German Federal (BWM) Round 1 p3
Let $ABC$ be a triangle. Points $A',B',C'$ are on the sides $BC, CA, AB,$ respectively such that we have $\overline{A'B'} = \overline{B'C'} = \overline{C'A'}$ and $\overline{AB'} = \overline{BC'} = \overline{CA'}$. Prove that triangle $ABC$ is equilateral.

2002 German Federal (BWM) Round 1 p3
The circumference of a circle is divided into eight arcs by a convex quadrilateral $ABCD$ with four arcs lying inside the quadrilateral and the remaining four lying outside it. The lengths of the arcs lying inside the quadrilateral are denoted by $p,q,r,s$ in counter-clockwise direction. Suppose $p+r = q+s$. Prove that $ABCD$ is cyclic.

2003 German Federal (BWM) Round 1 p3
Let $ABCD$ be a parallelogram. Let $M$ be a point on the side $AB$ and $N$ be a point on the side $BC$ such that the segments $AM$ and $CN$ have equal lengths and are non-zero. The lines $AN$ and $CM$ meet at $Q$. Prove that the line $DQ$ is the bisector of the angle $\measuredangle ADC$.

Alternative formulation.
Let $ABCD$ be a parallelogram. Let $M$ and $N$ be points on the sides $AB$ and $BC$, respectively, such that $AM=CN\neq 0$. The lines $AN$ and $CM$ intersect at a point $Q$. Prove that the point $Q$ lies on the bisector of the angle $\measuredangle ADC$.

2004 German Federal (BWM) Round 1 p2
Consider a triangle whose sidelengths $a$, $b$, $c$ are integers, and which has the property that one of its altitudes equals the sum of the two others. Then, prove that $a^2+b^2+c^2$ is a perfect square.

2004 German Federal (BWM) Round 1 p4
A cube is decomposed in a finite number of rectangular parallelepipeds such that the volume of the cube's circum sphere volume equals the sum of the volumes of all parallelepipeds' circum spheres. Prove that all these parallelepipeds are cubes.

2005 German Federal (BWM) Round 1 p3
Let $ABC$ be a triangle with sides $a$, $b$, $c$ and (corresponding) angles $A$, $B$, $C$.
Prove that if $3A + 2B = 180^{\circ}$, then $a^2+bc=c^2$.

2006 German Federal (BWM) Round 1 p3
Let $a,b,c$ be the sidelengths of a triangle such that $a^2+b^2 > 5c^2$ holds. Prove that $c$ is the shortest side of the triangle.

2007 German Federal (BWM) Round 1 p3
In triangle $ABC$ points $E$ and $F$ lie on sides $AC$ and $BC$ such that segments $AE$ and $BF$ have equal length, and circles formed by $A,C,F$ and by $B,C,E,$ respectively, intersect at point $C$ and another point $D.$ Prove that that the line $CD$ bisects $\angle ACB.$

2008 German Federal (BWM) Round 1 p3
Prove: In an acute triangle $ABC$ angle bisector $w_{\alpha},$ median $s_b$ and the altitude $h_c$ intersect in one point if $w_{\alpha},$ side $BC$ and the circle around foot of the altitude $h_c$ have vertex $A$ as a common point.

2009 German Federal (BWM) Round 1 p3
Let $P$ be a point inside the triangle $ABC$ and $P_a, P_b ,P_c$ be the symmetric points wrt the midpoints of the sides $BC, CA,AB$ respectively. Prove that that the lines $AP_a, BP_b$ and $CP_c$ are concurrent.

2010 German Federal (BWM) Round 1 p3
On the sides of a triangle $XYZ$ to the outside construct similar triangles $YDZ, EXZ ,YXF$ with circumcenters $K, L ,M$ respectively. Here are  $\angle ZDY = \angle ZXE = \angle FXY$ and  $\angle YZD = \angle EZX =  \angle YFX$. Show that the triangle $KLM$ is similar to the triangles 

2011 German Federal (BWM) Round 1 p3

The diagonals of a convex pentagon divide each of its interior angles into three equal parts.
Does it follow that the pentagon is regular?

2012 German Federal (BWM) Round 1 p3

An equilateral triangle $DCE$ is placed outside a square $ABCD$. The center of this triangle is denoted as $M$ and the intersection of the straight line $AC$ and $BE$ with $S$. Prove that the triangle $CMS$ is isosceles.

2013 German Federal (BWM) Round 1 p3
In the interior of the square $ABCD$, the point $P$ lies in such a way that $\angle DCP = \angle CAP=25^{\circ}$. Find all possible values of $\angle PBA$.

2014 German Federal (BWM) Round 1 p3
A regular hexagon with side length $1$ is given. Using a ruler construct points in such a way that among the given and constructed points there are two such points that the distance between them is $\sqrt7$.

Notes: ''Using a ruler construct points $\ldots$'' means: Newly constructed points arise only as the intersection of straight lines connecting two points that are given or already constructed. In particular, no length can be measured by the ruler.

2015 German Federal (BWM) Round 1 p3

Let $M$ be the midpoint of segment $[AB]$ in triangle $\triangle ABC$. Let $X$ and $Y$ be points such that $\angle{BAX}=\angle{ACM}$ and $\angle{BYA}=\angle{MCB}$. Both points, $X$ and $Y$, are on the same side as $C$ with respect to line $AB$. Show that the rays $[AX$ and $[BY$ intersect on line $CM$.

2016 German Federal (BWM) Round 1 p2
A triangle $ABC$ with area $1$ is given. Anja and Bernd are playing the following game: Anja chooses a point $X$ on side $BC$. Then Bernd chooses a point $Y$ on side $CA$ und at last Anja chooses a point $Z$ on side $AB$. Also, $X,Y$ and $Z$ cannot be a vertex of triangle $ABC$. Anja wants to maximize the area of triangle $XYZ$ and Bernd wants to minimize that area.
What is the area of triangle $XYZ$ at the end of the game, if both play optimally?

2016 German Federal (BWM) Round 1 p3
Let $A,B,C$ and $D$ be points on a circle in this order. The chords $AC$ and $BD$ intersect in point $P$. The perpendicular to $AC$ through C and the perpendicular to $BD$ through $D$ intersect in point $Q$. Prove that the lines $AB$ and $PQ$ are perpendicular

2017 German Federal (BWM) Round 1 p3
Let $M$ be the incenter of the tangential quadrilateral $A_1A_2A_3A_4$. Let line $g_1$ through $A_1$ be perpendicular to $A_1M$; define $g_2,g_3$ and $g_4$ similarly. The lines $g_1,g_2,g_3$ and $g_4$ define another quadrilateral $B_1B_2B_3B_4$ having $B_1$ be the intersection of $g_1$ and $g_2$; similarly $B_2,B_3$ and $B_4$ are intersections of $g_2$ and $g_3$, $g_3$ and $g_4$, resp. $g_4$ and $g_1$.
Prove that the diagonals of quadrilateral $B_1B_2B_3B_4$ intersect in point $M$.





















2018 German Federal (BWM) Round 1 p3
Let $H$ be the orthocenter of the acute triangle $ABC$. Let $H_a$ be the foot of the perpendicular from $A$ to $BC$ and let the line through $H$ parallel to $BC$ intersect the circle with diameter $AH_a$ in the points $P_a$ and $Q_a$. Similarly, we define the points $P_b, Q_b$ and $P_c,Q_c$.
Show that the six points $P_a,Q_a,P_b,Q_b,P_c,Q_c$ lie on a common circle.

2019 German Federal (BWM) Round 1 p3
Let $ABCD$ be a square. Choose points $E$ on $BC$ and $F$ on $CD$ so that $\angle EAF=45^\circ$ and so that neither $E$ nor $F$ is a vertex of the square. The lines $AE$ and $AF$ intersect the circumcircle of the square in the points $G$ and $H$ distinct from $A$, respectively. Show that the lines $EF$ and $GH$ are parallel.

2020 German Federal (BWM) Round 1 p3

Let $AB$ be the diameter of a circle $k$ and let $E$ be a point in the interior of $k$. The line $AE$ intersects $k$ a second time in $C \ne A$ and the line $BE$ intersects $k$ a second time in $D \ne B$. Show that the value of $AC \cdot AE+BD\cdot BE$ is independent of the choice of $E$.

2021 German Federal (BWM) Round 1 p1

A cube with side length $10$ is divided into two cuboids with integral side lengths by a straight cut. Afterwards, one of these two cuboids is divided into two cuboids with integral side lengths by another straight cut. What is the smallest possible volume of the largest of the three cuboids?

2021 German Federal (BWM) Round 1 p2

Consider a triangle $ABC$ with $\angle ACB=120^\circ$. Let $A’, B’, C’$ be the points of intersection of the angular bisector through $A$, $B$ and $C$ with the opposite side, respectively.

Determine $\angle A’C’B’$.

2022 German Federal (BWM) Round 1 p3

A circle $k$ touches a larger circle $K$ from inside in a point $P$. Let $Q$ be point on $k$ different from $P$. The line tangent to $k$ at $Q$ intersects $K$ in $A$ and $B$.

Show that the line $PQ$ bisects $\angle APB$.

Round 2

1974 German Federal (BWM) Round 2 p3

A circle $K_1$ of radius $r_1 = 1\slash 2$ is inscribed in a semi-circle $H$ with diameter $AB$ and radius $1.$ A sequence of different circles $K_2, K_3, \ldots$ with radii $r_2, r_3, \ldots$ respectively are drawn so that for each $n\geq 1$, the circle $K_{n+1}$ is tangent to $H$, $K_n$ and $AB.$ Prove that $a_n = 1\slash r_n$ is an integer for each $n$, and that it is a perfect square for $n$ even and twice a perfect square for $n$ odd.

1977 German Federal (BWM) Round 2 p2

On a plane are given three non-collinear points $A, B, C$. We are given a disk of diameter different from that of the circle passing through $A, B, C$ large enough to cover all three points. Construct the fourth vertex of the parallelogram $ABCD$ using only this disk (The disk is to be used as a circular ruler, for constructing a circle passing through two given points).

1978 German Federal (BWM) Round 2 p1

Let $a, b, c$ be sides of a triangle. Prove that

$$\frac{1}{3} \leq \frac{a^2 +b^2 +c^2 }{(a+b+c)^2 } < \frac{1}{2}$$ and show that $\frac{1}{2}$ cannot be replaced with a smaller number.

1979 German Federal (BWM) Round 2 p2

A circle $k$ with center $M$ and radius $r$ is given. Find the locus of the incenters of all obtuse-angled triangles inscribed in $k$.

1980 German Federal (BWM) Round 2 p3

In a triangle $ABC$, points $P, Q$ and $R$ distinct from the vertices of the triangle are chosen on sides $AB, BC$ and $CA$, respectively. The circumcircles of the triangles $APR$, $BPQ$, and $CQR$ are drawn. Prove that the centers of these circles are the vertices of a triangle similar to triangle $ABC$.

1981 German Federal (BWM) Round 2 p2

A bijective mapping from a plane to itself maps every circle to a circle.

Prove that it maps every line to a line.

1982 German Federal (BWM) Round 2 p2

Decide whether every triangle $ABC$ in space can be orthogonally projected onto a plane such that the projection is an equilateral triangle $A'B'C'$.

1983 German Federal (BWM) Round 2 p2

The figure shows a triangular pool table with sides $a$, $b$ and $c$. Located at point $S$ on $c$ a sphere - which can be assumed as a point. After kick-off, as indicated in the figure, it runs through as a result of reflections to $a, b, a, b$ and $c$ (in $S$) always the same track. The reflection occurs according to law of reflection. Characterize entilrely all triangles $ABC$, which allow such an orbit, and determine the locus of $S$.

1984 German Federal (BWM) Round 2 p4

A sphere is touched by all the four sides of a (space) quadrilateral. Prove that all the four touching points are in the same plane.

1985 German Federal (BWM) Round 2 p2

The insphere of any tetrahedron has radius $r$. The four tangential planes parallel to the side faces of the tetrahedron cut from the tetrahedron four smaller tetrahedrons whose in-sphere radii are $r_1, r_2, r_3$ and $r_4$. Prove that $$r_1 + r_2 + r_3 + r_4 = 2r$$

1986 German Federal (BWM) Round 2 p2

A triangle has sides $a, b,c$, radius of the incircle $r$ and radii of the excircles $r_a, r_b, r_c$: Prove that:

a) The triangle is right-angled if and only if: $r + r_a + r_b + r_c = a + b + c$.

b) The triangle is right-angled if and only if: $r^2 + r^2_a + r^2_b + r^2_c = a^2 + b^2 + c^2$.

1988 German Federal (BWM) Round 2 p3

Prove that all acute-angled triangles with the equal altitudes $h_c$ and the equal angles $\gamma$ have orthic triangles with same perimeters.

1989 German Federal (BWM) Round 2 p3

Over each side of a cyclic quadrilateral erect a rectangle whose height is equal to the length of the opposite side. Prove that the centers of these rectangles form another rectangle.

1992 German Federal (BWM) Round 2 p3

Provided a convex equilateral pentagon. On every side of the pentagon We construct equilateral triangles which run through the interior of the pentagon. Prove that at least one of the triangles does not protrude the pentagon's boundary.

1993 German Federal (BWM) Round 2 p3

In the triangle $ABC$, let $A'$ be the intersection of the perpendicular bisector of $AB$ and the angle bisector of $\angle BAC$ and define $B', C'$ analogously. Prove that

a) The triangle $ABC$ is equilateral if and only if $A' =B'.$

b) If $A', B'$ and $C'$ are distinct, we have $\angle B' A' C' = 90^{\circ} - \frac{1}{2} \angle BAC.$

1994 German Federal (BWM) Round 2 p3

Let $A$ and $B$ be two spheres of different radii, both inscribed in a cone $K$. There are $m$ other, congruent spheres arranged in a ring such that each of them touches $A, B, K$ and two of the other spheres. Prove that this is possible for at most three values of $m.$

1995 German Federal (BWM) Round 2 p3
Each diagonal of a convex pentagon is parallel to one side of the pentagon. Prove that the ratio of the length of a diagonal to that of its corresponding side is the same for all five diagonals, and compute this ratio.

1996 German Federal (BWM) Round 2 p3
Let $ABC$ be a triangle, and erect three rectangles $ABB_1A_2$, $BCC_1B_2$, $CAA_1C_2$ externally on its sides $AB$, $BC$, $CA$, respectively. Prove that the perpendicular bisectors of the segments $A_1A_2$, $B_1B_2$, $C_1C_2$ are concurrent.

1997 German Federal (BWM) Round 2 p3
A semicircle with diameter $AB = 2r$ is divided into two sectors by an arbitrary radius. To each of the sectors a circle is inscribed. These two circles touch A$B$ at $S$ and $T$. Show that $ST \ge 2r(\sqrt{2}-1)$.

1998 German Federal (BWM) Round 2 p3
A triangle $ABC$ satisfies $BC = AC +\frac12 AB$. Point $P$ on side $AB$ is taken so that $AP = 3PB$. Prove that $\angle PAC = 2\angle CPA$.

1999 German Federal (BWM) Round 2 p3
Let $P$ be a point inside a convex quadrilateral $ABCD$. Points $K,L,M,N$ are given on the sides $AB,BC,CD,DA$ respectively such that $PKBL$ and $PMDN$ are parallelograms. Let $S,S_1$, and $S_2$ be the areas of $ABCD, PNAK$, and $PLCM$. Prove that $\sqrt{S}\ge  \sqrt{S_1} +\sqrt{S_2}$.

2000 German Federal (BWM) Round 2 p3
For each vertex of a given tetrahedron, a sphere passing through that vertex and the midpoints of the edges outgoing from this vertex is constructed. Prove that these four spheres pass through a single point.

2001 German Federal (BWM) Round 2 p3
Let $ABC$ an acute triangle with circumcircle center $O.$ The line $(BO)$ intersects the circumcircle again in $D,$ and the extension of the altitude from $A$ intersects the circle in $E.$ Prove that the quadrilateral $BECD$ and the triangle $ABC$ have the same area.

2002 German Federal (BWM) Round 2 p4
In an acute-angled triangle $ABC$, we consider the feet $H_a$ and $H_b$ of the altitudes from $A$ and $B$, and the intersections $W_a$ and $W_b$ of the angle bisectors from $A$ and $B$ with the opposite sides $BC$ and $CA$ respectively. Show that the centre of the incircle $I$ of triangle $ABC$ lies on the segment $H_aH_b$ if and only if the centre of the circumcircle $O$ of triangle $ABC$ lies on the segment $W_aW_b$.

2003 German Federal (BWM) Round 2 p3
Consider a cyclic quadrilateral $ABCD$, and let $S$ be the intersection of $AC$ and $BD$. Let $E$ and $F$ the orthogonal projections of $S$ on $AB$ and $CD$ respectively. Prove that the perpendicular bisector of segment $EF$ meets the segments $AD$ and $BC$ at their midpoints.

2004 German Federal (BWM) Round 2 p3
Given two circles $k_1$ and $k_2$ which intersect at two different points $A$ and $B$. The tangent to the circle $k_2$ at the point $A$ meets the circle $k_1$ again at the point $C_1$. The tangent to the circle $k_1$ at the point $A$ meets the circle $k_2$ again at the point $C_2$. Finally, let the line $C_1C_2$ meet the circle $k_1$ in a point $D$ different from $C_1$ and $B$. Prove that the line $BD$ bisects the chord $AC_2$.

2005 German Federal (BWM) Round 2 p3
Two circles $k_1$ and $k_2$ intersect at two points $A$ and $B$. Some line through the point $B$ meets the circle $k_1$ at a point $C$ (apart from $B$), and the circle $k_2$ at a point $E$ (apart from $B$). Another line through the point $B$ meets the circle $k_1$ at a point $D$ (apart from $B$), and the circle $k_2$ at a point $F$ (apart from $B$). Assume that the point $B$ lies between the points $C$ and $E$ and between the points $D$ and $F$. Finally, let $M$ and $N$ be the midpoints of the segments $CE$ and $DF$. Prove that the triangles $ACD$, $AEF$ and $AMN$ are similar to each other.

2006 German Federal (BWM) Round 2 p3
A point $P$ is given inside an acute-angled triangle $ABC$. Let $A',B',C'$ be the orthogonal projections of $P$ on sides $BC, CA, AB$ respectively. Determine the locus of points $P$ for which $\angle BAC = \angle B'A'C'$ and $\angle CBA = \angle C'B'A'$

2007 German Federal (BWM) Round 2 p3
A set $E$ of points in the 3D space let $L(E)$ denote the set of all those points which lie on lines composed of two distinct points of $E.$ Let $T$ denote the set of all vertices of a regular tetrahedron. Which points are in the set $L(L(T))?$

2008 German Federal (BWM) Round 2 p3
Through a point in the interior of a sphere we put three pairwise perpendicular planes. Those planes dissect the surface of the sphere in eight curvilinear triangles. Alternately the triangles are coloured black and wide to make the sphere surface look like a checkerboard. Prove that exactly half of the sphere's surface is coloured black.

2009 German Federal (BWM) Round 2 p3

Given a triangle $ABC$ and a point $P$ on the side $AB$ . Let $Q$ be the intersection of the straight line $CP$ (different from $C$) with the circumcicle of the triangle. Prove the inequality $\frac{\overline{PQ}}{\overline{CQ}} \le \left(\frac{\overline{AB}}{\overline{AC}+\overline{CB}}\right)^2$and that equality holds if and only if the $CP$ is bisector of the angle $ACB$ 

2010 German Federal (BWM) Round 2 p1

Let $a, b, c$ be the side lengths of an non-degenerate triangle with $a \le b \le c$. With $t (a, b, c)$ denote the minimum of the quotients $\frac{b}{a}$ and $\frac{c}{b}$ . Find all values that $t (a, b, c)$ can take.

2010 German Federal (BWM) Round 2 p3

Given an acute-angled triangle $ABC$. Let CB be the altitude and $E$ a random point on the line $CD$. Finally, let $P, Q, R$ and $S$ are the projections of $D$ on the straight lines $AC, AE, BE$ and $BC$. Prove that the points $P, Q, R$ and $S$ lie either on a circle or on one straight line.

2011 German Federal (BWM) Round 2 p4

Let $ABCD$ be a tetrahedron that is not degenerate and not necessarily regular, where sides $AD$ and $BC$ have the same length $a$, sides $BD$ and $AC$ have the same length $b$, side $AB$ has length $c_1$ and the side $CD$ has length $c_2$. There is a point $P$ for which the sum of the distances to the corner points of the tetrahedron is minimal. Determine this sum depending on the quantities $a, b, c_1$ and $c_2$.

2012 German Federal (BWM) Round 2 p3

The incircle of the triangle $ABC$ touches the sides $BC, CA$ and $AB$ in points $A_1, B_1$ and $C_1$ respectively. $C_1D$ is a diameter of the incircle. Finally, let $E$ be the intersection of the lines $B_1C_1$ and $A_1D$. Prove that the segments $CE$ and $CB_1$ have equal length.

2013 German Federal (BWM) Round 2 p3
Let $ABCDEF$ be a convex hexagon whose vertices lie on a circle. Suppose that $AB\cdot CD\cdot EF = BC\cdot DE\cdot FA$. Show that the diagonals $AD, BE$ and $CF$ are concurrent

2014 German Federal (BWM) Round 2 p4
Three non-collinear points $A_1, A_2, A_3$ are given in a plane. For  $n = 4, 5, 6, \ldots$, $A_n$ be the centroid of the triangle $A_{n-3}A_{n-2}A_{n-1}$.
a) Show that there is exactly one point
$S$, which lies in the interior of the triangle $A_{n-3}A_{n-2}A_{n-1}$ for all $n\ge 4$.
b) Let $T$ be the intersection of the line $A_1A_2$ with $SA_3$. Determine the two ratios, $A_1T : TA_2$ and $TS : SA_3$.

2015 German Federal (BWM) Round 2 p4
Let $ABC$ be a triangle, such that its incenter $I$ and circumcenter $U$ are distinct. For all points $X$ in the interior of the triangle let $d(X)$ be the sum of distances from $X$ to the three (possibly extended) sides of the triangle. Prove: If two distinct points $P,Q$ in the interior of the triangle $ABC$ satisfy $d(P)=d(Q)$, then $PQ$ is perpendicular to $UI$.

2017 German Federal (BWM) Round 2 p3
Given is a triangle with side lengths $a,b$ and $c$, incenter $I$ and centroid $S$.
Prove: If $a+b=3c$, then $S \neq I$ and line $SI$ is perpendicular to one of the sides of the triangle.

2018 German Federal (BWM) Round 2 p3
Let $T$ be a point on a line segment $AB$ such that $T$ is closer to $B$ than to $A$. Show that for each point $C \ne T$ on the line through $T$ perpendicular to $AB$ there is exactly one point $D$ on the line segment $AC$ with $\angle CBD=\angle BAC$. Moreover, show that the line through $D$ perpendicular to $AC$ intersects the line $AB$ in a point $E$ which is independent of the position of $C$.

2019 German Federal (BWM) Round 2 p3
Let $ABC$ be atriangle with  $\overline{AC}> \overline{BC}$ and incircle $k$. Let $M,W,L$ be the intersections of the median, angle bisector and altitude from point $C$ respectively. The tangent to $k$ passing through $M$, that is different from $AB$, touch $k$ in $T$. Prove that the angles $\angle MTW$ and $\angle TLM$ are equal.

2020 German Federal (BWM) Round 2 p3

Two lines $m$ and $n$ intersect in a unique point $P$. A point $M$ moves along $m$ with constant speed, while another point $N$ moves along $n$ with the same speed. They both pass through the point $P$, but not at the same time. Show that there is a fixed point $Q \ne P$ such that the points $P,Q,M$ and $N$ lie on a common circle all the time.

2021 German Federal (BWM) Round 2 p3

We are given a circle $k$ and a point $A$ outside of $k$. Next we draw three lines through $A$: one secant intersecting the circle $k$ at points $B$ and $C$, and two tangents touching the circle$k$ at points $D$ and $E$. Let $F$ be the midpoint of $DE$.

Show that the line $DE$ bisects the angle $\angle BFC$.

2022 German Federal (BWM) Round 2 p3

In an acute triangle $ABC$ with $AC<BC$, lines $m_a$ and $m_b$ are the perpendicular bisectors of sides $BC$ and $AC$, respectively. Further, let $M_c$ be the midpoint of side $AB$. The Median $CM_c$ intersects $m_a$ in point $S_a$ and $m_b$ in point $S_b$; the lines $AS_b$ und $BS_a$ intersect in point $K$. Prove: $\angle ACM_c = \angle KCB$.

source:
http://www.bundeswettbewerb-mathematik.de/