geometry problems from QEDMO, a German Math Flight with aops links
aops post collections: QEDMO
(inside zip)
a few words by Darij Grinberg
QEDMO stands for QED Mathematical Olympiad. The QED is an organization of German mathematical olympiad participants based in Bavaria who organize meetings and seminars for each other. From late 2005 on, some of these meetings feature a math fight (an oral mathematical contest, with two teams solving questions and debating the solutions in front of the blackboard) called the QEDMO. The problems are of varying difficulty (some very basic problems occur on every QEDMO, but a few of the problems have the level of an IMO problem 3).
Darij Grinberg was the one responsible for the selection of the geometry problems.
13/15 editions, 2005 - 2012, 2015-17
2003: 12th or 13th only in aops here
(I do not know if it is 12th or 13th)
2005 1st QEDMO p2 (G2) by Darij Grinberg
Let $ABC$ be a triangle. Let $C^{\prime}$ and $A^{\prime}$ be the reflections of its vertices $C$ and $A$, respectively, in the altitude of triangle $ABC$ issuing from $B$. The perpendicular to the line $BA^{\prime}$ through the point $C^{\prime}$ intersects the line $BC$ at $U$; the perpendicular to the line $BC^{\prime}$ through the point $A^{\prime}$ intersects the line $BA$ at $V$. Prove that $UV \parallel CA$.
2005 1st QEDMO p5 (G1) by Darij Grinberg
Let $ABC$ be a triangle, and let $C^{\prime}$ and $A^{\prime}$ be the feet of its altitudes issuing from the vertices $C$ and $A$, respectively. Denote by $P$ the midpoint of the segment $C^{\prime}A^{\prime}$. The circumcircles of triangles $AC^{\prime}P$ and $CA^{\prime}P$ have a common point apart from $P$; denote this common point by $Q$. Prove that:
(a) The point $Q$ lies on the circumcircle of the triangle $ABC$.
(b) The line $PQ$ passes through the point $B$.
(c) We have $\frac{AQ}{CQ}=\frac{AB}{CB}$.
2004 1st QEDMO p9 (G3) by Darij Grinberg
Let $ABC$ be a triangle with $AB\neq CB$. Let $C^{\prime}$ be a point on the ray $[AB$ such that $AC^{\prime}=CB$. Let $A^{\prime}$ be a point on the ray $[CB$ such that $CA^{\prime}=AB$. Let the circumcircles of triangles $ABA^{\prime}$ and $CBC^{\prime}$ intersect at a point $Q$ (apart from $B$). Prove that the line $BQ$ bisects the segment $CA$.
2005 1st QEDMO p14 (G4) by Darij Grinberg
Let $ABCDE$ be a convex pentagon. Let $A^{\prime}=BD\cap CE$, $B^{\prime}=CE\cap DA$, $C^{\prime}=DA\cap EB$, $D^{\prime}=EB\cap AC$ and $E^{\prime}=AC\cap BD$. Furthermore, let $A^{\prime\prime}=AA^{\prime}\cap EB$, $B^{\prime\prime}=BB^{\prime}\cap AC$, $C^{\prime\prime}=CC^{\prime}\cap BD$, $D^{\prime\prime}=DD^{\prime}\cap CE$ and $E^{\prime\prime}=EE^{\prime}\cap DA$.
Prove that:
\[ \frac{EA^{\prime\prime}}{A^{\prime\prime}B}\cdot\frac{AB^{\prime\prime}}{B^{\prime\prime}C}\cdot\frac{BC^{\prime\prime}}{C^{\prime\prime}D}\cdot\frac{CD^{\prime\prime}}{D^{\prime\prime}E}\cdot\frac{DE^{\prime\prime}}{E^{\prime\prime}A}=1. \]
2006 2nd QEDMO p4 by Darij Grinberg
Let $ABCD$ be a cyclic quadrilateral. Let $X$ be the foot of the perpendicular from the point $A$ to the line $BC$, let $Y$ be the foot of the perpendicular from the point $B$ to the line $AC$, let $Z$ be the foot of the perpendicular from the point $A$ to the line $CD$, let $W$ be the foot of the perpendicular from the point $D$ to the line $AC$.
Prove that $XY\parallel ZW$.
2006 2nd QEDMO p7, by Darij Grinberg (combining other problems in one)
Let $H$ be the orthocenter of a triangle $ABC$, and let $D$ be the midpoint of the segment $AH$.
The altitude $BH$ of triangle $ABC$ intersects the perpendicular to the line $AB$ through the point $A$ at the point $M$. The altitude $CH$ of triangle $ABC$ intersects the perpendicular to the line $CA$ through the point $A$ at the point $N$. The perpendicular bisector of the segment $AB$ intersects the perpendicular to the line $BC$ through the point $B$ at the point $U$.
The perpendicular bisector of the segment $CA$ intersects the perpendicular to the line $BC$ through the point $C$ at the point $V$. Finally, let $E$ be the midpoint of the side $BC$ of triangle $ABC$.
Prove that the points $D$, $M$, $N$, $U$, $V$ all lie on one and the same perpendicular to the line $AE$.
Extensions. In other words, we have to show that the points $M$, $N$, $U$, $V$ lie on the perpendicular to the line $AE$ through the point $D$. Additionally, one can find two more points on this perpendicular:
(a) The nine-point circle of triangle $ABC$ is known to pass through the midpoint $E$ of its side $BC$. Let $D^{\prime}$ be the point where this nine-point circle intersects the line $AE$ apart from $E$. Then, the point $D^{\prime}$ lies on the perpendicular to the line $AE$ through the point $D$.
(b) Let the tangent to the circumcircle of triangle $ABC$ at the point $A$ intersect the line $BC$ at a point $X$. Then, the point $X$ lies on the perpendicular to the line $AE$ through the point $D$.
(most parts of this problem are due to Victor Thebault, 1950, but got rediscovered by manyothers - e. g. part of this problem was in Balkan MO 2003)
2006 2nd QEDMO p10 by Darij Grinberg
Let $X_1$, $Z_2$, $Y_1$, $X_2$, $Z_1$, $Y_2$ be six points lying on the periphery of a circle (in this order). Let the chords $Y_1Y_2$ and $Z_1Z_2$ meet at a point $A$; let the chords $Z_1Z_2$ and $X_1X_2$ meet at a point $B$; let the chords $X_1X_2$ and $Y_1Y_2$ meet at a point $C$.
Prove that $\left( BX_2-CX_1\right) \cdot BC+\left( CY_2-AY_1\right) \cdot CA+\left( AZ_2-BZ_1\right) \cdot AB=0$.
2006 2nd QEDMO p14 (Moscow MO 2005)
On the sides $BC$, $CA$, $AB$ of an acute-angled triangle $ABC$, we erect (outwardly) the squares $BB_aC_aC$, $CC_bA_bA$, $AA_cB_cB$, respectively. On the sides $B_cB_a$ and $C_aC_b$ of the triangles $BB_cB_a$ and $CC_aC_b$, we erect (outwardly) the squares $B_cB_vB_uB_a$ and $C_aC_uC_vC_b$.
Prove that $B_uC_u\parallel BC$
2006 3rd QEDMO p1 by Engel
Peter is a pentacrat and spends his time drawing pentagrams.
With the abbreviation $\left|XYZ\right|$ for the area of an arbitrary triangle $XYZ$, he notes that any convex pentagon $ABCDE$ satisfies the equality
$\left|EAC\right|\cdot\left|EBD\right|=\left|EAB\right|\cdot\left|ECD\right|+\left|EBC\right|\cdot\left|EDA\right|$.
Guess what you are supposed to do and do it.
(P151 in the German periodical Praxis der Mathematik)
2006 3rd QEDMO p6 by Darij Grinberg
The incircle of a triangle $ABC$ touches its sides $BC$, $CA$, $AB$ at the points $X$, $Y$, $Z$, respectively. Let $X^{\prime}$, $Y^{\prime}$, $Z^{\prime}$ be the reflections of these points $X$, $Y$, $Z$ in the external angle bisectors of the angles $CAB$, $ABC$, $BCA$, respectively. Show that $Y^{\prime}Z^{\prime}\parallel BC$, $Z^{\prime}X^{\prime}\parallel CA$ and $X^{\prime}Y^{\prime}\parallel AB$.
2006 3rd QEDMO p9 by Darij Grinberg
Let $ABC$ be a triangle, and $C^{\prime}$ and $A^{\prime}$ the midpoints of its sides $AB$ and $BC$. Consider two lines $g$ and $g^{\prime}$ which both pass through the vertex $A$ and are symmetric to each other with respect to the angle bisector of the angle $CAB$. Further, let $Y$ and $Y^{\prime}$ be the orthogonal projections of the point $B$ on these lines $g$ and $g^{\prime}$.
Show that the points $Y$ and $Y^{\prime}$ are symmetric to each other with respect to the line $C^{\prime}A^{\prime}$.
2007 4th QEDMO p1 by V. Alekseev, V. Galkin, V. Panferov, V. Tarasov
Let $ ABCD$ be a trapezoid with $ BC\parallel AD$, and let $ O$ be the point of intersection of its diagonals $ AC$ and $ BD$. Prove that $ \left\vert ABCD\right\vert =\left( \sqrt{\left\vert BOC\right\vert }+\sqrt{\left\vert DOA\right\vert }\right) ^{2}$.
(exercise 8 in: V. Alekseev, V. Galkin, V. Panferov, V. Tarasov,
Zadachi o trapezijah, Kvant 6/2000, pages 37-4.)
2007 4th QEDMO p5 by Engel
Let $ ABC$ be a triangle, and let $ X$, $ Y$, $ Z$ be three points on the segments $ BC$, $ CA$, $ AB$, respectively. Denote by $ X^{\prime}$, $ Y^{\prime}$, $ Z^{\prime}$ the reflections of these points $ X$, $ Y$, $ Z$ in the midpoints of the segments $ BC$, $ CA$, $ AB$, respectively. Prove that $ \left\vert XYZ\right\vert =\left\vert X^{\prime}Y^{\prime}Z^{\prime}\right\vert$.
(P144 in the German periodical Praxis der Mathematik)
2007 4th QEDMO p10 by John Sturgeon Mackay
Let $ ABC$ be a triangle. The $ A$-excircle of triangle $ ABC$ has center $ O_{a}$ and touches the side $ BC$ at the point $ A_{a}$. The $ B$-excircle of triangle $ ABC$ touches its sidelines $ AB$ and $ BC$ at the points $ C_{b}$ and $ A_{b}$. The $ C$-excircle of triangle $ ABC$ touches its sidelines $ BC$ and $ CA$ at the points $ A_{c}$ and $ B_{c}$. The lines $ C_{b}A_{b}$ and $ A_{c}B_{c}$ intersect each other at some point $ X$. Prove that the quadrilateral $ AO_{a}A_{a}X$ is a parallelogram.
(Theorem (88) in: John Sturgeon Mackay, The Triangle and its Six Scribed Circles, Proceedings of the Edinburgh Mathematical Society 1 (1883), pages 4-128 and drawings at the end of the volume.)
2007 4th QEDMO p12 by rohitsingh0812 in aops
Let $ABC$ be a triangle, and let $D$, $E$, $F$ be the points of contact of its incircle $\omega$ with its sides $BC$, $CA$, $AB$, respectively. Let $K$ be the point of intersection of the line $AD$ with the incircle $\omega$ different from $D$, and let $M$ be the point of intersection of the line $EF$ with the line perpendicular to $AD$ passing through $K$. Prove that $AM$ is parallel to $BC$.
2007 5th QEDMO p2 by Darij Grinberg
Let $ ABCD$ be a (not self-intersecting) quadrilateral satisfying $ \measuredangle DAB = \measuredangle BCD\neq 90^{\circ}$. Let $ X$ and $ Y$ be the orthogonal projections of the point $ D$ on the lines $ AB$ and $ BC$, and let $ Z$ and $ W$ be the orthogonal projections of the point $ B$ on the lines $ CD$ and $ DA$.
Establish the following facts:
a) The quadrilateral $ XYZW$ is an isosceles trapezoid such that $ XY\parallel ZW$.
b) Let $ M$ be the midpoint of the segment $ AC$. Then, the lines $ XZ$ and $ YW$ pass through the point $ M$.
c) Let $ N$ be the midpoint of the segment $ BD$, and let $ X^{\prime}$, $ Y^{\prime}$, $ Z^{\prime}$, $ W^{\prime}$ be the midpoints of the segments $ AB$, $ BC$, $ CD$, $ DA$. Then, the point $ M$ lies on the circumcircles of the triangles $ W^{\prime}X^{\prime}N$ and $ Y^{\prime}Z^{\prime}N$.
2007 5th QEDMO p8 by Darij Grinberg
Let $ A$, $ B$, $ C$, $ A^{\prime}$, $ B^{\prime}$, $ C^{\prime}$, $ X$, $ Y$, $ Z$, $ X^{\prime}$, $ Y^{\prime}$, $ Z^{\prime}$ and $ P$ be pairwise distinct points in space such that
$ A^{\prime} \in BC;\ B^{\prime}\in CA;\ C^{\prime}\in AB;\ X^{\prime}\in YZ;\ Y^{\prime}\in ZX;\ Z^{\prime}\in XY;$
$ P \in AX;\ P\in BY;\ P\in CZ;\ P\in A^{\prime}X^{\prime};\ P\in B^{\prime}Y^{\prime};\ P\in C^{\prime}Z^{\prime}$.
Prove that
$ \frac {BA^{\prime}}{A^{\prime}C}\cdot\frac {CB^{\prime}}{B^{\prime}A}\cdot\frac {AC^{\prime}}{C^{\prime}B} = \frac {YX^{\prime}}{X^{\prime}Z}\cdot\frac {ZY^{\prime}}{Y^{\prime}X}\cdot\frac {XZ^{\prime}}{Z^{\prime}Y}$.
Let there be a finite number of straight lines in the plane, none of which are three in one point to cut.
Show that the intersections of these straight lines can be colored with $3$ colors so that that no two
points of the same color are adjacent on any of the straight lines. (Two points of intersection are called
adjacent if they both lie on one of the finitely many straight lines and there is no other such intersection
on their connecting line.)
Let $A, B, C, A', B', C'$ be six pairs of different points. Prove that the Circles $BCA'$, $CAB'$ and
$ABC'$ have a common point, then the Circles $B'C'A, C'A'B$ and $A'B'C$ also share a common point.
Note: For three pairs of different points $X, Y$ and $Z$ we define the Circle $XYZ$ as the circumcircle
of the triangle $XYZ$, or - in the case when the points $X, Y$ and $Z$ lie on a straight line - this straight
line.
Albatross and Frankinfueter both own a circle. Frankinfueter also has stolen from Prof. Trugweg a ruler. Before that, Trugweg had two points with a distance of $1$ drawn his (infinitely large) board. For a natural number $n$, let A $(n)$ be the number of the construction steps that Albatross needs at least to create two points with a distance of $n$ to construct. Similarly, Frankinfueter needs at least $F(n)$ steps for this.
How big can $\frac{A (n)}{F (n)}$ become?
There are only the following three construction steps:
a) Mark an intersection of two straight lines, two circles or a straight line with one circle.
b) Pierce at a marked point $P$ and draw a circle around $P$ through one marked point .
c) Draw a straight line through two marked points (this implies possession of a ruler ahead!).
The inscribed circle of a triangle $ABC$ has the center $O$ and touches the triangle sides $BC, CA$ and
$AB$ at points $X, Y$ and $Z$, respectively. The parallels to the straight lines $ZX, XY$ and $YZ$ the
straight lines $BC, CA$ and $AB$ (in this order!) intersect through the point $O$. Points $K, L$ and $M$.
Then the parallels to the straight lines $CA, AB$ and $BC$ intersect through the points $K, L$ and $M$
in one point.
Let $ABCD$ and $A'B'C'D'$ be two squares, both are oriented clockwise. In addition, it is assumed
that all points are arranged as shown in the figure.Then it has to be shown that the sum of the areas of
the quadrilaterals $ABB'A'$ and $CDD'C'$ equal to the sum of the areas of the quadrilaterals $BCC'B'$
and $DAA'D'$
Let $ABC$ be a triangle. Let $x_1$ and $x_2$ be two congruent circles, which touch each other and
the segment $BC$, and which both lie within triangle $ABC$, and for which it also holds that $x_1$
touches the segment $CA$, and that $x_2$ is the segment $AB$. Let $X$ be the contact point of these
two circles $x_1$ and $x_2$. Let $y_1$ and $y_2$ two congruent circles that touch each other and the
segment $CA$, and both within of triangle $ABC$, and for which it also holds that $y_1$ touches the
segment $AB$, and that $y_2$ the segment $BC$. Let $Y$ be the contact point of these two circles
$y_1$ and $y_2$. Let $z_1$ and $z_2$ be two congruent circles that touch each other and the segment
$AB$, and both within triangle $ABC$, and for which it also holds that $z_1$ touches the segment
$BC$, and that $z_2$ the segment $CA$. Let $Z$ be the contact point of these two circles $z_1$ and
$z_2$. Prove that the straight lines $AX, BY$ and $CZ$ intersect at a point.
2010 7th QEDMO p12
Let $Y$ and $Z$ be the feet of the altitudes of a triangle $ABC$ drawn from angles $B$ and $C$,
respectively. Let $U$ and $V$ be the feet of the perpendiculars from $Y$ and $Z$ on the straight line
$BC$. The straight lines $YV$ and $ZU$ intersect at a point $L$. Prove that $AL \perp BC$.
$9$ points are given in the interior of the unit square.
Prove there exists a triangle of area $\le \frac18$ whose vertices are three of the points
A synogon is a convex $2n$-gon with all sides of the same length and all opposite sides are parallel. Show that every synogon can be broken down into a finite number of rhombuses.
Albatross and Frankinfueter are playing again: each of them takes turns choosing one point in the plane with integer coordinates and paint it in his favorite color. Albatross plays first. Someone wins as soon as there is a square with all four corners in the are colored in their own color. Does anyone has a winning strategy and if so, who?
Let $P$ be a convex polygon, so have all interior angles smaller than $180^o$, and let $X$ be a point in the interior of $P$. Prove that $P$ has a side $[AB]$ such that the perpendicular from $X$ to the line $AB$ lies on the side $[AB]$.
Find for which natural numbers $n$ one can color the sides and diagonals of a regular $n$-gon with $n$ colors in such a way that for each triplet in pairs of different colors, a triangle can be found, the sides of which are sides or diagonals of $n$-gon and which is colored with exactly these three colors.
Let $n$ be a natural number and $L = Z^2$ the set of points on the plane with integer coordinates. Every point in $L$ is colored now in one of the colors red or green. Show that there are $n$ different points $x_1,...,x_n \in L$ all of which have the same color and whose center of gravity is also in $L$ and is of the same color.
In the following, a rhombus is one with edge length $1$ and interior angles $60^o$ and $120^o$ . Now let $n$ be a natural number and $H$ a regular hexagon with edge length $n$, which is covered with rhombuses without overlapping has been. The rhombuses then appear in three different orientations. Prove that whatever the overlap looks exactly, each of these three orientations can be viewed at the same time.
Prove that there are $2012$ points in the plane, none of which are three on one straight line and in pairs have integer distances .
soon 12th and 13th
Let $D$ be a regular dodecagon in the plane. How many squares are there in the plane at least two vertices in common with the vertices of $D$?
Let $ABC$ be a triangle of area $1$ with medians $s_a, s_b,s_c$. Show that there is a triangle whose sides are the same length as $s_a, s_b$, and $s_c$, and determine its area.
Let $\ell$ be a straight line and $P \notin \ell$ be a point in the plane. On $\ell$ are, in this arrangement, points $A_1, A_2,...$ such that the radii of the incircles of all triangles $P A_iA_{i + 1}$ are equal. Let $k \in N$. Show that the radius of the incircle of the triangle $P A_j A_{j + k}$ does not depend on the choice of $j \in N$ .
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