MathDB

2023.C7

1

angle in regular pentagon 2022-23 Dürer Round1 C7 D6

ABCDE

be a regular pentagon. We drew two circles around

A

and

B

with radius

AB

. Let

F

mark the intersection of the two circles that is inside the pentagon. Let

G

mark the intersection of lines

EF

and

AD

. What is the degree measure of angle

AGE

?

2008.D1

1

3a + 4b <= 5c, right triangles inequality 2007-08 Duhrer Round1 D1

Prove the following inequality if we know that

a

and

b

are the legs of a right triangle , and

c

is the length of the hypotenuse of this triangle:

3a + 4b \le 5c.

When does equality holds?

2018.D+4

1

tangent to incircle wanted 2017-18 Dürer Round1 D+4

The center of the inscribed circle of triangle

ABC

is

I

. Let

e

be the perpendicular line on

CI

passing through

I

. The line

e

itnersects the side

AC

at

A'

and the side

BC

at point

B'

. Let

A''

be the symmetric point of

A

wrt

A'

,

B''

be the symmetric point of

B

wrt

B'

. Prove that

A''B''

is a line tangent to the incircle.

2018.D2

1

leg of isosceles triangle wanted 2017-18 Dürer Round1 D2

In an isosceles triangle, we drew one of the angle bisectors. At least one of the resulting two smaller ones triangles is similar to the original. What can be the leg of the original triangle if the length of its base is

1

unit?

2018.C5

1

beautiful convex n-gons 2017-18 Dürer Round1 C5 C+4

A convex

n

-gon is called nice if its sides are not all the same length, and the sum of the distances of any interior point to the side lines is

1

. Find all integers

n \ge 4

such that a nice

n

-gon exists .

2018.C+2

1

angles wanted, right isosceles related 2017-18 Dürer Round1 C+2

In an isosceles right-angled triangle

ABC

, the right angle is at

A

.

D

lies so on the side

BC

that

2CD = DB

. Let

E

be the projection of

B

onto

AD

. What is the measure fof angle

\angle CED

?

2018.C3

1

computational geo with isosceles, 18^o 2017-18 Dürer Round1 C3

In the isosceles triangle

ABC

,

AB = AC

. Let

E

be on side

AB

such that

\angle ACE = \angle ECB = 18^o

, and let

D

be the midpoint of side

CB

. If we know the length of

AD

is

3

units, what is the length of

CE

?

2017.D+2

1

collinear wanted, 3 similar trapezoids related 2016-17 D&uuml;rer Round1 D4 D+2

Let the trapezoids

A_iB_iC_iD_i

(

i = 1, 2, 3

A_i

and

B_i

are

60^o

and the sides

A_1B_1

,

B_2C_2

and

A_3D_3

are parallel. The lines

B_iD_{i+1}

and

C_iA_{i+1}

intersect at the point

P_i

(the indices are understood cyclically, i.e.

A_4 = A_1

and

D_4 = D_1

). Prove that the points

P_1

,

P_2

and

P_3

lie on a line.

2017.C+5

1

point inside heptagon, with 7 equal lengths 2016-17 Dürer Round1 C+5

Is there a heptagon and a point

P

inside it such that any vertex of the heptagon has its distance from

P

equal to the length of the side opposite the vertex?
A side and a vertex are said to be opposite if the side is the fourth from the vertex page (in any direction).

2017.C1

1

angle between 2 digaonals of regular 10-gon 2016-17 Dürer Round1 C1

The vertices of Durer's favorite regular decagon in clockwise order:

D_1, D_2, D_3, . . . , D_{10}

. What is the angle between the diagonals

D_1D_3

and

D_2D_5

?

2016.D+3

1

PQ _|_ RS , 2 centroids, 2 orthocenters 2015-16 Dürer Round1 D+3

Let

M

be the intersection point of the diagonals of the convex quadrilateral

ABCD

. Let

P

and

Q

be the centroids of triangles

AMD

and

BMC

respectively. Let

R

and

S

are the orthocenters of triangles

AMB

and

CMD

. Prove that the lines

P Q

and

RS

are perpendicular to each other.

2016.C+3

1

min \sqrt2 AP + BP + CP, P interior of square ABCD 2015-16 Dürer Round1 C+3

Let

ABCD

be a square with unit sides. Which interior point

P

will the expression

\sqrt2 \cdot AP + BP + CP

have a minimum value, and what is this minimum?

2016.C1

1

2 points and 3 projections concyclic 2015-16 Dürer Round1 C1 D1

Let

P

be an arbitrary point of the side line

AB

of the triangle

ABC

. Mark the perpendicular projection of

P

on the side lines

AC

and

BC

as

A_1

and

B_1

respectively. Denote

C_1

he foot of the alttiude starting from

C

. Prove that the points

A_1

,

B_1

,

C_1

,

C

and

P

lie on a circle.

2015.D4

1

sum AA_2/ AA_1=4, altitude, circumcircle 2014-15 Dürer Round1 D4

The altitude of the acute triangle

ABC

drawn from

A

, intersects the side

BC

at

A_1

and the circumscribed circle at

A_2

(different from

A

). Similarly, we get the points

B_1

,

B_2

,

C_1

,

C_2

. Prove that

\frac{AA_2}{AA_1}+\frac{BB_2}{BB_1}+\frac{CC_2}{CC_1}= 4.

2015.C2

1

<APD =<DPC, ABCD rectangle 2014-15 Dürer Round1 C2

Given a rectangle

ABCD

, side

AB

is longer than side

BC

. Find all the points

P

of the side line

AB

from which the sides

AD

and

DC

are seen from the point

P

at an equal angle (i.e.

\angle APD = \angle DPC

)

2008.C3

1

HIJK is square if ABCD , DEFG are also 2007-08 Dürer Round1 C3

Given the squares

ABCD

and

DEFG

, whose only common point is

D

. Let the midpoints of segments

AG

,

GE

,

EC

, and

CA

be

H, I, J

, and

K

respectively . Prove that

HIJK

is a square. https://cdn.artofproblemsolving.com/attachments/f/d/c3313e5bbf581977a74ea2b114d14950e38605.png

2014.D3

1

computational geo with regular octagon 2013-14 Dürer Round1 D3

ABCDEF GH

is a regular octagon with

10

units side . The circle with center

A

and radius

AC

intersects the circle with center

D

and radius

CD

at point

I

, different from

C

. What is the length of the segment

IF

?

2014.C2

1

P Q =// 2 RS wanted, 4 semicircles on a line 2013-14 Dürer Round1 C2

Above the segments

AB

and

BC

we drew a semicircle at each.

F_1

bisects

AB

and

F_2

bisects

BC

. Above the segments

AF_2

and

F_1C

we also drew a semicircle at each. Segments

P Q

and

RS

touch the corresponding semicircles as shown in the figure. Prove that

P Q \parallel RS

and

|P Q| = 2 \cdot |RS|

. https://cdn.artofproblemsolving.com/attachments/8/2/570e923b91e9e630e3880a014cc6df4dc33aa2.png

2014.C4

1

pentagon area AB=CD=EA=1, <ABC=<DEA=90^o, BC+DE=1 2013-14 Durer Round1 C4

ABCDE

is a convex pentagon with

AB = CD = EA = 1

,

\angle ABC = \angle DEA = 90^o

, and

BC + DE = 1

. What is the area of the pentagon?

2013.D3

1

area chasing, midpoints, 1/3 ratio points 2012-13 D&uuml;rer Round1 D3

The area of the triangle

ABC

shown in the figure is

1

unit. Points

D

and

E

lie on sides

AC

and

BC

respectively, and also are its ''one third'' points closer to

C

. Let

F

be that

AE

and

G

are the midpoints of segment

BD

. What is the area of the marked quadrilateral

ABGF

? https://cdn.artofproblemsolving.com/attachments/4/e/305673f429c86bbc58a8d40272dd6c9a8f0ab2.png

2013.C1

1

ratio of areas extending by equal segment each side 2012-13 Durer Round1 C1

Each side of a triangle is extended in the same clockwise direction by the length of the given side as shown in the figure. How many times the area of the triangle, obtained by connecting the endpoints, is the area of the original triangle? https://cdn.artofproblemsolving.com/attachments/1/c/a169d3ab99a894667caafee6dbf397632e57e0.png

2012.D2

1

sum of gray ares is half of the triangles' area 2011-12 Durer Round1 D2

Durer drew a regular triangle and then poked at an interior point. He made perpendiculars from it sides and connected it to the vertices. In this way,

6

small triangles were created, of which (moving clockwise) all the second one is painted gray, as shown in figure. Show that the sum of the gray areas is just half the area of the triangle. https://cdn.artofproblemsolving.com/attachments/e/7/a84ad28b3cd45bd0ce455cee2446222fd3eac2.png

2012.D3

1

distance of // planes, cube related 2011-12 Dürer Round1 D3

Show that the planes

ACG

and

BEH

defined by the vertices of the cube shown in Figure are parallel. What is their distance if the edge length of the cube is

1

meter? https://cdn.artofproblemsolving.com/attachments/c/9/21585f6c462e4289161b4a29f8805c3f63ff3e.png

2012.C5

1

IG // side of triangle 2011-12 Dürer Round1 C5

In a triangle, the line between the center of the inscribed circle and the center of gravity is parallel to one of the sides. Prove that the sidelengths form an arithmetic sequence.

2009.C3

1

triangle construction, knowing 3 altitudes 2008-09 Dürer Round1 C3

We know the lengths of the

3

altitudes of a triangle. Construct the triangle.

2011.C4

1

fair grid points in 2010 x 1340 rectangle 2010-11 Dürer Round1 C4

Given a grid rectangle of size

2010 \times 1340

. A grid point is called fair if the

2

axis-parallel lines passing through it from the upper left and lower right corners of the large rectangle cut out a rectangle of equal area (such a point is shown in the figure). How many fair grid points lie inside the rectangle? https://cdn.artofproblemsolving.com/attachments/1/b/21d4fb47c94b774994ac1c3aae7690bb98c7ae.png

2011.D5

1

circumcircle cover convex n-gon 2010-11 Dürer Round1 D5

Is it true that in every convex polygon

3

adjacent vertices can be selected such that their circumcirscribed circle can cover the entire polygon?

2010.D3

1

quad is orthodiagonal if midpoints are concyclic 2009-10 Dürer Round1 D3

Prove that the diagonals of a quadrilateral are perpendicular to each other if and only if the midpoints of it's sides lie on a circle.

2010.C3

1

ball angle 45^o in pool table 3x4 2009-10 Dürer Round1 C3

The sides of a pool table are

3

and

4

meters long.We push a ball with an angle of

45^o

at the sides. Is it true that it returns to where it started no matter where we started it from?

2021.D4

1

AC + OC = AB if <A=30^o, <B=50^o 2020-21 Dürer Round1 D4

In the triangle

ABC

we have

30^o

at the vertex

A

, and

50^o

at the vertex

B

. Let

O

be the center of inscribed circle. Show that

AC + OC = AB

.

2021.C3

1

two foils overlap on rectangular window 2020-21 Dürer Round1 C3

Csenge has a yellow and a red foil on her rectangular window which look beautiful in the morning light. Where the two foils overlap, they look orange. The window is

80

cm tall,

120

cm wide and its corners are denoted by

A, B, C

and

D

in the figure. The two foils are triangular and both have two of their vertices at the two bottom corners of the window, A and

B

. The third vertex of the yellow foil is

S

, the trisecting point of side

DC

closer to

D

, whereas the third vertex of the red foil is

P

, which is one fourth on the way on segment

SC

, closer to

C

. The red region (i.e. triangle

BPE

) is of area

16

dm

^2

. What is the total area of the regions not covered by foil? https://cdn.artofproblemsolving.com/attachments/b/c/ea371aeafde6968506da6f3456e88fa0bddc6d.png

2019.D4

1

<CBD=? right isosceles ABC, AB = BD 2018-19 Dürer Round1 D4

Let

ABC

be an isosceles right-angled triangle, having the right angle at vertex

C

. Let us consider the line through

C

which is parallel to

AB

and let

D

be a point on this line such that

AB = BD

and

D

is closer to

B

than to

A

. Find the angle

\angle CBD

.

2019.C3

1

3 equal partd from 30x30 pizza 2018-19 Dürer Round1 C3

The best parts of grandma’s

30

cm

\times 30

cm square shaped pie are the edges. For this reason grandma’s three grandchildren would like to split the pie between each other so that everyone gets the same amount (of the area) of the pie, but also of the edges. Can they cut the pie into three connected pieces like that?

2022.C4

1

rectangle inscribed in triangle 2021-22 Dürer Round1 C4 D3

We inscribed in triangle

ABC

the rectangle

DEFG

such that

D

and

E

fall on side

AB

,

F

on side

BC

, and

G

on side

AC

. We know that

AF

bisects angle

\angle BAC

, and that

\frac{AD}{DE} = \frac12

. What is the measure of angle

\angle CAB

?

2009.D4

1

smallest area of lattice triangle 2008-09 Dürer Round1 D4

If all vertices of a triangle on the square grid are grid points, then the triangle is called a lattice triangle. What is the area of the lattice triangle with (one) of the smallest area, if one grid has area

1

square unit?

Durer Math Competition CD 1st Round - geometry

Subcontests

angle in regular pentagon 2022-23 Dürer Round1 C7 D6

3a + 4b <= 5c, right triangles inequality 2007-08 Duhrer Round1 D1

tangent to incircle wanted 2017-18 Dürer Round1 D+4

leg of isosceles triangle wanted 2017-18 Dürer Round1 D2

beautiful convex n-gons 2017-18 Dürer Round1 C5 C+4

angles wanted, right isosceles related 2017-18 Dürer Round1 C+2

computational geo with isosceles, 18^o 2017-18 Dürer Round1 C3

collinear wanted, 3 similar trapezoids related 2016-17 D&uuml;rer Round1 D4 D+2

point inside heptagon, with 7 equal lengths 2016-17 Dürer Round1 C+5

angle between 2 digaonals of regular 10-gon 2016-17 Dürer Round1 C1

PQ _|_ RS , 2 centroids, 2 orthocenters 2015-16 Dürer Round1 D+3

min \sqrt2 AP + BP + CP, P interior of square ABCD 2015-16 Dürer Round1 C+3

2 points and 3 projections concyclic 2015-16 Dürer Round1 C1 D1

sum AA_2/ AA_1=4, altitude, circumcircle 2014-15 Dürer Round1 D4

<APD =<DPC, ABCD rectangle 2014-15 Dürer Round1 C2

HIJK is square if ABCD , DEFG are also 2007-08 Dürer Round1 C3

computational geo with regular octagon 2013-14 Dürer Round1 D3

P Q =// 2 RS wanted, 4 semicircles on a line 2013-14 Dürer Round1 C2

pentagon area AB=CD=EA=1, <ABC=<DEA=90^o, BC+DE=1 2013-14 Durer Round1 C4

area chasing, midpoints, 1/3 ratio points 2012-13 D&uuml;rer Round1 D3

ratio of areas extending by equal segment each side 2012-13 Durer Round1 C1

sum of gray ares is half of the triangles' area 2011-12 Durer Round1 D2

distance of // planes, cube related 2011-12 Dürer Round1 D3

IG // side of triangle 2011-12 Dürer Round1 C5

triangle construction, knowing 3 altitudes 2008-09 Dürer Round1 C3

fair grid points in 2010 x 1340 rectangle 2010-11 Dürer Round1 C4

circumcircle cover convex n-gon 2010-11 Dürer Round1 D5

quad is orthodiagonal if midpoints are concyclic 2009-10 Dürer Round1 D3

ball angle 45^o in pool table 3x4 2009-10 Dürer Round1 C3

AC + OC = AB if <A=30^o, <B=50^o 2020-21 Dürer Round1 D4

two foils overlap on rectangular window 2020-21 Dürer Round1 C3

<CBD=? right isosceles ABC, AB = BD 2018-19 Dürer Round1 D4

3 equal partd from 30x30 pizza 2018-19 Dürer Round1 C3

rectangle inscribed in triangle 2021-22 Dürer Round1 C4 D3

smallest area of lattice triangle 2008-09 Dürer Round1 D4

Durer Math Competition CD 1st Round - geometry

Subcontests

angle in regular pentagon 2022-23 D&uuml;rer Round1 C7 D6

3a + 4b &lt;= 5c, right triangles inequality 2007-08 Duhrer Round1 D1

tangent to incircle wanted 2017-18 D&uuml;rer Round1 D+4

leg of isosceles triangle wanted 2017-18 D&uuml;rer Round1 D2

beautiful convex n-gons 2017-18 D&uuml;rer Round1 C5 C+4

angles wanted, right isosceles related 2017-18 D&uuml;rer Round1 C+2

computational geo with isosceles, 18^o 2017-18 D&uuml;rer Round1 C3

collinear wanted, 3 similar trapezoids related 2016-17 D&amp;uuml;rer Round1 D4 D+2

point inside heptagon, with 7 equal lengths 2016-17 D&uuml;rer Round1 C+5

angle between 2 digaonals of regular 10-gon 2016-17 D&uuml;rer Round1 C1

PQ _|_ RS , 2 centroids, 2 orthocenters 2015-16 D&uuml;rer Round1 D+3

min \sqrt2 AP + BP + CP, P interior of square ABCD 2015-16 D&uuml;rer Round1 C+3

2 points and 3 projections concyclic 2015-16 D&uuml;rer Round1 C1 D1

sum AA_2/ AA_1=4, altitude, circumcircle 2014-15 D&uuml;rer Round1 D4

&lt;APD =&lt;DPC, ABCD rectangle 2014-15 D&uuml;rer Round1 C2

HIJK is square if ABCD , DEFG are also 2007-08 D&uuml;rer Round1 C3

computational geo with regular octagon 2013-14 D&uuml;rer Round1 D3

P Q =// 2 RS wanted, 4 semicircles on a line 2013-14 D&uuml;rer Round1 C2

pentagon area AB=CD=EA=1, &lt;ABC=&lt;DEA=90^o, BC+DE=1 2013-14 Durer Round1 C4

area chasing, midpoints, 1/3 ratio points 2012-13 D&amp;uuml;rer Round1 D3

ratio of areas extending by equal segment each side 2012-13 Durer Round1 C1

sum of gray ares is half of the triangles' area 2011-12 Durer Round1 D2

distance of // planes, cube related 2011-12 D&uuml;rer Round1 D3

IG // side of triangle 2011-12 D&uuml;rer Round1 C5

triangle construction, knowing 3 altitudes 2008-09 D&uuml;rer Round1 C3

fair grid points in 2010 x 1340 rectangle 2010-11 D&uuml;rer Round1 C4

circumcircle cover convex n-gon 2010-11 D&uuml;rer Round1 D5

quad is orthodiagonal if midpoints are concyclic 2009-10 D&uuml;rer Round1 D3

ball angle 45^o in pool table 3x4 2009-10 D&uuml;rer Round1 C3

AC + OC = AB if &lt;A=30^o, &lt;B=50^o 2020-21 D&uuml;rer Round1 D4

two foils overlap on rectangular window 2020-21 D&uuml;rer Round1 C3

&lt;CBD=? right isosceles ABC, AB = BD 2018-19 D&uuml;rer Round1 D4

3 equal partd from 30x30 pizza 2018-19 D&uuml;rer Round1 C3

rectangle inscribed in triangle 2021-22 D&uuml;rer Round1 C4 D3

smallest area of lattice triangle 2008-09 D&uuml;rer Round1 D4

angle in regular pentagon 2022-23 Dürer Round1 C7 D6

3a + 4b <= 5c, right triangles inequality 2007-08 Duhrer Round1 D1

tangent to incircle wanted 2017-18 Dürer Round1 D+4

leg of isosceles triangle wanted 2017-18 Dürer Round1 D2

beautiful convex n-gons 2017-18 Dürer Round1 C5 C+4

angles wanted, right isosceles related 2017-18 Dürer Round1 C+2

computational geo with isosceles, 18^o 2017-18 Dürer Round1 C3

collinear wanted, 3 similar trapezoids related 2016-17 Dürer Round1 D4 D+2

point inside heptagon, with 7 equal lengths 2016-17 Dürer Round1 C+5

angle between 2 digaonals of regular 10-gon 2016-17 Dürer Round1 C1

PQ _|_ RS , 2 centroids, 2 orthocenters 2015-16 Dürer Round1 D+3

min \sqrt2 AP + BP + CP, P interior of square ABCD 2015-16 Dürer Round1 C+3

2 points and 3 projections concyclic 2015-16 Dürer Round1 C1 D1

sum AA_2/ AA_1=4, altitude, circumcircle 2014-15 Dürer Round1 D4

<APD =<DPC, ABCD rectangle 2014-15 Dürer Round1 C2

HIJK is square if ABCD , DEFG are also 2007-08 Dürer Round1 C3

computational geo with regular octagon 2013-14 Dürer Round1 D3

P Q =// 2 RS wanted, 4 semicircles on a line 2013-14 Dürer Round1 C2

pentagon area AB=CD=EA=1, <ABC=<DEA=90^o, BC+DE=1 2013-14 Durer Round1 C4

area chasing, midpoints, 1/3 ratio points 2012-13 Dürer Round1 D3

distance of // planes, cube related 2011-12 Dürer Round1 D3

IG // side of triangle 2011-12 Dürer Round1 C5

triangle construction, knowing 3 altitudes 2008-09 Dürer Round1 C3

fair grid points in 2010 x 1340 rectangle 2010-11 Dürer Round1 C4

circumcircle cover convex n-gon 2010-11 Dürer Round1 D5

quad is orthodiagonal if midpoints are concyclic 2009-10 Dürer Round1 D3

ball angle 45^o in pool table 3x4 2009-10 Dürer Round1 C3

AC + OC = AB if <A=30^o, <B=50^o 2020-21 Dürer Round1 D4

two foils overlap on rectangular window 2020-21 Dürer Round1 C3

<CBD=? right isosceles ABC, AB = BD 2018-19 Dürer Round1 D4

3 equal partd from 30x30 pizza 2018-19 Dürer Round1 C3

rectangle inscribed in triangle 2021-22 Dürer Round1 C4 D3

smallest area of lattice triangle 2008-09 Dürer Round1 D4