Dürer 2008-21 (Hungary) 57p

geometry problems from International Dürer Math Competition, from Hungary, with aops links

 aops posts collections here

2019-21 in English pdf with solutions

geometry problems  collected inside aops here

final round 2008 -2021

categories C (juniors), D (seniors)

2008 Dürer Math Competition C1

Given the parallelogram $ABCD$. The trisection points of side $AB$ are: $H_1, H_2$, ($AH_1 = H_1H_2 =H_2B$). The trisection points of the side $DC$ are $G_1, G_2$, ($DG_1 = G_1G_2 = G_2C$), and $AD = 1, AC = 2$. Prove that triangle $AH_2G_1$ is isosceles.

2008 Dürer Math Competition C2

Given a triangle with sides $a, b, c$ and medians $s_a, s_b, s_c$ respectively. Prove the following inequality: $a + b + c> s_a + s_b + s_c> \frac34 (a + b + c)$ 

2008 Dürer Math Competition C3

We divided a regular octagon into parallelograms. Prove that there are at least $2$ rectangles between the parallelograms.

2008 Dürer Math Competition D1

Given a square grid where the distance between two adjacent grid points is $1$.

Can the distance between two grid points be $\sqrt5, \sqrt6, \sqrt7$ or $\sqrt{2007}$ ?

2008 Dürer Math Competition D3

Let $ABC$ be a equilateral triangle and let $P$ be any point on the minor arc $AC$ of the circumcircle of $ABC$.Prove that $PB=PA+PC$

2009 Dürer Math Competition C2

The material of new ball corset of the princess is quadrilateral . The tailor must sew four decorative strips on it. Two of gold, two of silver. Two of the same color on two opposite sides and the other two on it to a midline not intersecting them. The tailor is not yet familiar with the dress final shape. However, you will definitely sew the dress to be the cheapest (i.e., the gold stripe should be shorter than the silver). For design, it would be important to know what color stripe is centered. Can you decide this without knowing the the exact shape of the dress?

2009 Dürer Math Competition C3

Dürer's $n \times m$ garden is surgically divided into $n \times m$ unit squares, and in the middle of one of these squares, he planted his favourite petunia. Dürer's gardener struggles with a mole, trying to drive him out of the magnificent garden, so he builds an underground wall on the edge of the garden. The only problem is that the mole managed to stay inside the walls.. When the mole meets a wall, it changes it's direction as if it was "reflected", that is, proceeding his route in the direction that includes the same angle with the wall as his direction before. The mole starts beneath the petunia, in a direction that includes a $45^o$ angle with the walls. Is it possible for the mole to cross the petunia in a direction perpendicular to it's original direction?

(Think in terms of $n,m$.)

2009 Dürer Math Competition D1

Fencing Ferdinand wants to fence three rectangular areas. there are fences in three types, with $4$ amount of fences of each type. You will notice that there is always at least as much area it manages to enclose a total of three by enclosing three square areas (i.e., each area fencing elements of the same size to enclose it) as if it were three different, rectangular would encircle an area (i.e., use two different elements for each of the three areas). Why is this is so? When does it not matter how he fences the rectangles, in terms of the sum of the areas?

2009 Dürer Math Competition D3

What is the area of the letter $O$ made by Dürer? The two circles have a unit radius. Their centers, or the angle of a triangle formed by an intersection point of the circles is $30^o$.

2010 Dürer Math Competition C1

Dürer explains art history to his students. The following gothic window is examined.

Where the center of the arc of $BC$ is $A$, and similarly the center of the arc of $AC$ is $B$.

The question is how much is the radius of the circle (radius marked $r$ in the figure).

2010 Dürer Math Competition C5

Let $D$ the touchpoint of the inscribed circle of triangle $ABC$ be with side $AB$ . From $A$ the perpendicular lines on the angle bisectors of vertices $B$ and $C$ intersect them at points $A_1$ and $A_2$ respectively . Prove that $A_1A_2 = AD$.

2010 Dürer Math Competition D3

Three circle of unit radius passing through the point $P$ and one of the points of $A, B$ and $C$ each. What can be the radius of the circumcircle of the triangle $ABC$?

2010 Dürer Math Competition D5

Prove that we can put in any arbitrary triangle with sidelengths $a,b,c$ such that $0\le a,b,c \le \sqrt2$ into a unit cube.

2011 Dürer Math Competition C3

Given a circle with four circles that intersect in pairs as shown in the figure. The "internal" the points of intersection are $A, B, C$ and $D$, while the ‘outer’ points of intersection are $E, F, G$ and $H$. Prove that the quadrilateral $ABCD$ is cyclic if also the quadrilateral $EFGH$ is also cyclic.

2011 Dürer Math Competition C5

Given a straight line with points $A, B, C$ and $D$. Construct using $AB$ and $CD$ regular triangles (in the same half-plane). Let $E,F$ be the third vertex of the two triangles (as in the figure) . The circumscribed circles of triangles $AEC$ and $BFD$ intersect in $G$ ($G$ is is in the half plane of triangles). Prove that the angle $AGD$ is $120^o$

2011 Dürer Math Competition D2

In an right isosceles triangle $ABC$, there are two points on the hypotenuse $AB, K$ and $M$, respectively, such that $KCM$ angle is $45^o$ (point $K$ lies between $A$ and $M$). Prove that $AK^2 + MB^2 = KM^2$

2012 Dürer Math Competition C3

Given a convex quadrilateral whose opposite sides are not parallel, and giving an internal point $P$. Find a parallelogram whose vertices are on the side lines of the rectangle and whose center is $P$. Give a method by which we can construct it (provided there is one).

2012 Dürer Math Competition D5

The points of a circle of unit radius are colored in two colors. Prove that $3$ points of the same color can be chosen such that the area of the triangle they define is at least $\frac{9}{10}$.

2013 Dürer Math Competition C5

The points $A, B, C, D, P$ lie on an circle as shown in the figure such that $\angle AP B =  \angle BPC  = \angle  CPD$. Prove that the lengths of the segments are denoted by $a, b, c, d$ by $\frac{a + c}{b + d} =\frac{b}{c}$.

2013 Dürer Math Competition D3

The circle circumscribed to the triangle $ABC$ is $k$. The altitude $AT$ intersects circle $k$ at $P$. The perpendicular from $P$ on line $AB$ intersects is at $R$. Prove that line $TR$ is parallel to the tangent of the circle $k$ at point $A$.

2014 Dürer Math Competition C2

Let P be an arbitrary interior point of the equilateral triangle $ABC$. From $P$ draw parallel to the sides: $A'_1A_1 \parallel AB$, $B' _1B_1 \parallel BC$ and $C'_1C_1  \parallel CA$. Prove that the sum of legths $| AC_1 | + | BA_1 | + | CB_1 |$ is independent of the choice of point $P$.

2014 Dürer Math Competition D2

On the inner surface of a fixed circle, rolls a wheel half the radius of the circle, without slipping. We marked a point red on the wheel. Prove that while the wheel makes a turn, the point moves on a line.

2015 Dürer Math Competition C1

Can the touchpoints of the inscribed circle of a triangle with the triangle form an obtuse triangle?

2015 Dürer Math Competition C4

On a circumference of a unit radius, take points $A$ and $B$ such that section $AB$ has length one. $C$ can be any point on the longer arc of the circle between $A$ and $B$. How do we take $C$ to make the perimeter of the triangle $ABC$ as large as possible?

2015 Dürer Math Competition D1

From all three vertices of triangle $ABC$, we set perpendiculars to the exterior and interior of the other vertices angle bisectors. Prove that the sum of the squares of the segments thus obtained is exactly $2 (a^2 + b^2 + c^2)$, where $a, b$, and $c$ denote the lengths of the sides of the triangle.

2015 Dürer Math Competition D4

The projection of the vertex $C$ of the rectangle $ABCD$ on the diagonal $BD$ is $E$. The projections of $E$ on $AB$ and $AD$ are $F$ and $G$ respectively.. Prove that $$AF^{2/3} + AG^{2/3} = AC^{2/3}$$ .

2016 Dürer Math Competition C2

Show that in a triangle the altitude of the longest side is at most as long as it the the sum of the lengths of the perpendicular segments drawn from any point on the longest side on the other two sides.

2016 Dürer Math Competition C+4

The two intersections of the circles $k_i$ and $k_{i + 1}$ are $P_i$ and $Q_i$ ($1 \le  i  \le  5, k_6 = k_1$). On the circle $k_1$ lies an arbitrary point $A$. Then the points $B, C, D, E, F, G, H, I, J, K$ lie on the circles $k_2,  k_3, k_4, k_5, k_1, k_2, k_3, k_4, k_5, k_1$ respectively, such that $AP_1B, BP_2C, CP_3D, DP_4E, EP_5F, F Q_1G, GQ_2H, HQ_3I, IQ_4J, JQ_5K$ are straight line triplets. Prove that that $K = A$.

Circles can have different radii, and They can be located in different ways from the figure. We assume that during editing none neither of the two points mentioned above coincide.

2017 Dürer Math Competition C2

The triangle $ABC$ is isosceles and has a right angle at the vertex $A$. Construct all points that simultaneously satisfy the following two conditions:

(i) are equidistant from points $A$ and $B$

(ii) heve distance exactly three times from point $C$ as far as from point $B$.

2017 Dürer Math Competition C+1

Given a plane with two circles, one with points $A$ and $B$, and the other with points $C$ and $D$ are shown in the figure. The line $AB$ passes through the center of the first circle and touches the second circle while the line $CD$ passes through the center of the second circle and touches the first circle. Prove that the lines $AD$ and $BC$ are parallel.

2017 Dürer Math Competition D4

The convex quadrilateral $ABCD$ is has angle $A$ equal to $60^o$ , angle bisector of $A$ the diagonal $AC$ and $\angle ACD= 40^o$ and $\angle ACB = 120^o$. Inside the quadrilateral the point $P$ lies such that $\angle PDA = 40^o$ and $\angle PBA = 10^o$;

a) Find the angle $\angle DPB$?

b) Prove that $P$ lies on the diagonal $AC$.

2017 Dürer Math Competition D+5

The inscribed circle of the triangle $ABC$ touches the sides $BC, CA, AB$ at points $A_1, B_1, C_1$ respectively. The points $P_b, Q_b, R_b$ are the points of the segments $BC_1, C_1A_1, A_1B$, respectively, such that $BP_bQ_bR_b$ is parallelogram. In the same way, the points $P_c, Q_c, R_c$ are the points of the sections $CB_1, B_1A_1, A_1C$, respectively such that $CP_cQ_cR_c$ is a parallelogram. The intersection of the lines $P_bR_b$ and $P_cR_c$ is $T$. Show that $TQ_b = TQ_c$.

2018 Dürer Math Competition C3

Points $A, B, C, D$ are located in the plane as follows: sections $AB$ and $CD$ are perpendicular to each other and are of equal length, moreover, D is just the trisection point of segment $AB$ closer to $A$. The perpendicular from point $D$ on segment $BC$ intersects it at $E$. Let the trisection point of segment $DE$ closer to $E$ be $H$. Prove that segments $CH$ and the sections $AE$ are perpendicular to each other.

2018 Dürer Math Competition D4

Triangle $A'B'C'$ is located inside triangle $ABC$ such that $AB  \parallel A'B'$, $BC  \parallel B'C'$ and $CA  \parallel C'A'$ , and all three sides of these parallel sides are at distance $d$ at each case. Let $O$ and $O'$ be the centers of the inscribed circles of the triangles $ABC$ and $A'B'C'$ and $K$ and $K'$ are the the centers of their circumcircles. Prove that points $O, O', K$ and $K'$ lie on a straight line.

2018 Dürer Math Competition C+1

Prove that you can select two adjacent sides of any quadrilateral and supplement them in order to create a parallelogram, the resulting parallelogram contains the original quadrilateral .

2018 Dürer Math Competition C+2

Given an $ABC$ triangle. Let $D$ be an extension of section $AB$ beyond $A$ such that that $AD = BC$ and $E$ is the extension of the section $BC$ beyond $B$ such that $BE = AC$. Prove that the circumcircle of triangle $DEB$ passes through the center of the inscribed circle of triangle $ABC$.

2019 Dürer Math Competition C5

$A, B, C, D$ are four distinct points such that triangles $ABC$ and $CBD$ are both equilateral. Find as many circles as you can, which are equidistant from the four points. How can these circles be constructed?

Remark: The distance between a point $P$ and a circle c is measured as follows: we join $P$ and the centre of the circle with a straight line, and measure how much we need to travel along thisline (starting from $P$) to hit the perimeter of the circle. If $P$ is an internal point of the circle, the distance is the length of the shorter such segment. The distance between a circle and itscentre is the radius of the circle.

2019 Dürer Math Competition D3

a) Does there exist a quadrilateral with (both of) the following properties: three of its edges are of the same length, but the fourth one is different, and three of its angles are equal, but the fourth one is different?

b) Does there exist a pentagon with (both of) the following properties: four of its edges are of the same length, but the fifth one is different, and four of its angles are equal, but the fifth one is different?

2020 Dürer Math Competition C4

Albrecht likes to draw hexagons with all sides having equal length. He calls an angle of such a hexagon nice if it is exactly $120^o$. He writes the number of its nice angles inside each hexagon. How many different numbers could Albrecht write inside the hexagons? Show examples for as many values as possible and give a reasoning why others cannot appear.

Albrecht can also draw concave hexagons

2020 Dürer Math Competition D2

Let $ABC$ be an acute triangle where $AC > BC$. Let$T$ denote the foot of the altitude from vertex $C$, denote the circumcentre of the triangle by $O$. Show that quadrilaterals $ATOC$ and $BTOC$ have equal area.

2021 Dürer Math Competition C3

In the isosceles triangle $ABC$ we have $AC = BC$. Let $X$ be an arbitrary point of the segment $AB$. The line parallel to $BC$ and passing through $X$ intersects the segment $AC$ in $N$, and the line parallel to $AC$ and passing through $BC$ intersects the segment $BC$ in $M$. Let $k_1$ be the circle with center $N$ and radius $NA$. Similarly, let $k_2$ be the circle with center $M$ and radius $MB$. Let $T$ be the intersection of the circles $k_1$ and $k_2$ different from $X$. Show that the angles $\angle NCM$ and $\angle NTM$ are equal.

2021 Dürer Math Competition D3

Given a semicirle with center $O$ an arbitrary inner point of the diameter divides it into two segments. Let there be semicircles above the two segments as visible in the below figure. The line $\ell$ passing through the point $A$ intersects the semicircles in $4$ points: $B, C, D$ and $E$. Show that the segments $BC$ and $DE$ have the same length.

2022 Dürer Math Competition C3

To the exterior of side $AB$ of square $ABCD$, we have drawn the regular triangle $ABE$. Point $A$ reflected on line $BE$ is $F$, and point $E$ reflected on line $BF$ is $G$. Let the perpendicular bisector of segment $FG$ meet segment $AD$ at $X$. Show that the circle centered at $X$ with radius $XA$ touches line$FB$.

2022 Dürer Math Competition D4

The longer base of trapezoid $ABCD$ is $AB$, while the shorter base is $CD$. Diagonal $AC$ bisects the interior angle at $A$. The interior bisector at $B$ meets diagonal $AC$ at $E$. Line $DE$ meets segment $AB$ at $F$. Suppose that $AD = FB$ and $BC = AF$. Find the interior angles of quadrilateral $ABCD$, if we know that $\angle BEC = 54^o$.

International Categories 2019-20 Round 1 + Finals Day 1

2019 Dürer Math Competition E Round1 p5

Let $ABC$ be a non-right-angled triangle, with $AC\ne BC$. Let $F$ be the midpoint of side $BC$. Let $D$ be a point on line $AB$ satisfying$CA=CD$,and let $E$ be a point on line $BC$ satisfying $EB = ED$. The line passing through $A$ and parallel to $ED$ meets line $FD$ at point $I$. Line $AF$ meets line $ED$ at point $J$. Prove that points $C$, $I$ and $J$ are collinear.

2019 Dürer Math Competition E+ Round1 p5

Let $ABC$ and $A'B'C'$ be similar triangles with different orientation such that their orthocenters coincide. Show that lines $AA′, BB′, CC′ are concurrent or parallel.

2019 Dürer Math Competition E+ Finals 1.4

Let $ABC$ be an acute-angled triangle having angles $\alpha,\beta,\gamma$ at vertices $A, B, C$ respectively. Let isosceles triangles $BCA_1, CAB_1, ABC_1$ be erected outwards on its sides, with apex angles $2\alpha ,2\beta ,2\gamma$ respectively. Let $A_2$ be the intersection point of lines $AA_1$ and $B_1C_1$ and let us define points $B_2$ and $C_2$ analogously. Find the exact value of the expression $$\frac{AA_1}{A_2A_1}+\frac{BB_1}{B_2B_1}+\frac{CC_1}{C_2C_1}$$

2019 Dürer Math Competition E+ Finals 1. 5

Let $ABC$ be an acute triangle and let $X, Y , Z$ denote the midpoints of the shorter arcs $BC, CA, AB$ of its circumcircle, respectively. Let $M$ be an arbitrary point on side $BC$. The line through $M$, parallel to the inner angular bisector of $\angle CBA$ meets the outer angular bisector of $\angle BCA$ at point $N$. The line through $M$, parallel to the inner angular bisector of $\angle BCA$ meets the outer angular bisector of $\angle CBA$ at point $P$. Prove that lines $XM, Y N, ZP$ pass through a single point.

2020 Dürer Math Competition E Round1 p4

Let $ABC$ be an acute triangle with side $AB$ of length $1$. Say we reflect the points $A$ and $B$ across the midpoints of $BC$ and $AC$, respectively to obtain the points $A’$ and $B’$ . Assume that the orthocenters of triangles $ABC$, $A’BC$ and $B’AC$ form an equilateral triangle.

a) Prove that triangle $ABC$ is isosceles.

b) What is the length of the altitude of $ABC$ through $C$?

2020 Dürer Math Competition E+ Round1 p5

Suppose that you are given the foot of the altitude from vertex $A$ of a scalene triangle $ABC$, the midpoint of the arc with endpoints $B$ and $C$, not containing $A$ of the circumscribed circle of $ABC$, and also a third point $P$. Construct the triangle from these three points if $P$ is the

a) orthocenter

b) centroid

c) incenter

of the triangle.

2020 Dürer Math Competition E Finals 1.1

Let $ABC$ be an acute triangle where $AC > BC$. Let$T$ denote the foot of the altitude from vertex $C$, denote the circumcentre of the triangle by $O$. Show that quadrilaterals $ATOC$ and $BTOC$ have equal area.

2020 Dürer Math Competition E+ Finals 1. 4

Let $ABC$ be a scalene triangle and its incentre $I$. Denote by $F_A$ the intersection of the line $BC$ and the perpendicular to the angle bisector at $A$ through $I$. Let us define points $F_B$ and $F_C$ in a similar manner. Prove that points $F_A, F_B$ and $F_C$ are collinear.

2021 Dürer Math Competition E Round1 p3

The floor plan of a contemporary art museum is a (not necessarily convex) polygon and its walls are

solid. The security guard guarding the museum has two favourite spots (points $A$ and $B$) because

one can see the whole area of the museum standing at either point. Is it true that from any point of the

$AB$ section one can see the whole museum? 2021 Dürer Math Competition E+ Round1 p3

Let $k_1$ and $k_2$ be two circles that are externally tangent at point $C$. We have a point $A$ on

$k_1$ and a point $B$ on $k_2$ such that $C$ is an interior point of segment $AB$. Let $k_3$ be a

circle that passes through points $A$ and $B$ and intersects circles $k_1$ and $k_2$ another time at

points $M$ and $N$ respectively. Let $k_4$ be the circumscribed circle of triangle $CMN$. Prove

that the centres of circles $k_1, k_2, k_3$ and $k_4$ all lie on the same circle

2021 Dürer Math Competition E Finals 1.3

Let $A$ and $B$ different points of a circle $k$ centered at $O$ in such a way such that $AB$ is not a diagonal of $k$. Furthermore, let $X$ be an arbitrary inner point of the segment $AB$. Let $k_1$ be the circle that passes through the points $A$ and $X$, and $A$ is the only common point of $k$ and $k_1$. Similarly, let $k_2$ be the circle that passes through the points $B$ and $X$, and $B$ is the only common point of $k$ and $k_2$. Let $M$ be the second intersection point of $k_1$ and $k_2$. Let $Q$ denote the center of circumscribed circle of the triangle $AOB$. Let $O_1$ and $O_2$ be the centers of $k_1$ and $k_2$. Show that the points $M,O,O_1,O_2,Q$ are on a circle.

2022 Dürer Math Competition E Round1 p2

In the acute triangle $ABC$ the circle through $B$ touching the line $AC$ at $A$ has centre $P$, the circle through $A$ touching the line $BC$ at $B$ has centre $Q$. Let $R$ and $O$ be the circumradius and circumcentre of triangle $ABC$, respectively. Show that $R^2 = OP  \cdot OQ$.

2022 Dürer Math Competition E+ Round1 p2

Determine all triangles that can be split into two congruent pieces by one cut. A cut consists of segments $P_1P_2$, $P_2P_3$, . . . , $P_{n-1}P_n$ where points $P_1, P_2, . . . , P_n$ are distinct, points $P_1$ and $P_n$ lie on the perimeter of the triangle and the rest of the points lie in the interior of the triangle such that the segments are disjoint except for the endpoints.

2022 Dürer Math Competition E+ Round1 p4

Let $ABC$ be an acute triangle, and let $F_A$ and $F_B$ be the midpoints of sides $BC$ and $CA$, respectively. Let $E$ and $F$ be the intersection points of the circle centered at $F_A$ and passing through $A$ and the circle centered at $F_B$ and passing through $B$. Prove that if segments $CE$ and $CF$ have midpoints $N$ and $M$, respectively, then the intersection points of the circle centered at $M$ and passing through $E$ and the circle centered at $N$ and passing through $F$ lie on the line $AB$.

International Categories 2019-20 Finals Day 2

source:  https://durerinfo.hu/archivum/feladatsorok/