MathDB

1

3 equal regular tetrahedrons with common center, can polyhedron faces be equal?

Three equal regular tetrahedrons have the common center. Is it possible that all faces of the polyhedron that forms their intersection are equal?

24

1

3 points A',B',C' on sides of acute ABC such that it's longest side is minimal

Given is an acute-angled triangle

ABC

. On sides

BC, CA, AB

, find points

A', B', C'

such that the longest side of triangle

A'B'C'

is minimal.

23

1

line cuts a triangle, midpoints of segments of projections of intersections

Given are triangle

ABC

and line

\ell

intersecting

BC, CA

and

AB

at points

A_1, B_1

and

C_1

respectively. Point

A'

is the midpoint of the segment between the projections of

A_1

to

AB

and

AC

. Points

B'

and

C'

are defined similarly. (a) Prove that

A', B'

and

C'

lie on some line

\ell'

. (b) Suppose

\ell

passes through the circumcenter of

\triangle ABC

. Prove that in this case

\ell'

passes through the center of its nine-points circle.
M. Marinov and N. Beluhov

22

1

concurrent related to tangents to Euler circle and circumcircle

Let

CX, CY

be the tangents from vertex

C

of triangle

ABC

to the circle passing through the midpoints of its sides. Prove that lines

XY , AB

and the tangent to the circumcircle of

ABC

at point

C

concur.

21

1

concurent lines related to midpoints of chords and midpoints of arcs

On a circle with diameter

AC

, let

B

be an arbitrary point distinct from

A

and

C

. Points

M, N

are the midpoints of chords

AB, BC

, and points

P, Q

are the midpoints of smaller arcs restricted by these chords. Lines

AQ

and

BC

meet at point

K

, and lines

CP

and

AB

meet at point

L

. Prove that lines

MQ, NP

and

KL

concur.

20

1

criterion for tangential ABCD to be cyclic related to midpoints of diagonals

Quadrilateral

ABCD

is circumscribed around a circle with center

I

. Points

M

and

N

are the midpoints of diagonals

AC

and

BD

. Prove that

ABCD

is cyclic quadrilateral if and only if

IM : AC = IN : BD

.
Nikolai Beluhov and Aleksey Zaslavsky

19

1

nonisosceles triangle with altitude = bisector = median from different vertices

Does there exist a nonisosceles triangle such that the altitude from one vertex, the bisectrix from the second one and the median from the third one are equal?

18

1

min and max no of parts that are angles so that n lines divide the plane

On the plane, given are

n

lines in general position, i.e. any two of them aren’t parallel and any three of them don’t concur. These lines divide the plane into several parts. What is a) the minimal, b) the maximal number of these parts that can be angles?

17

1

compare shortest median vs longest bisectrix, shortest bis.vs longest altitude

a) Does there exist a triangle in which the shortest median is longer that the longest bisectrix? b) Does there exist a triangle in which the shortest bisectrix is longer that the longest altitude?

16

1

concurrent lines related to reflections of a line wrt to sides of given triangle

Given are triangle

ABC

and line

\ell

. The reflections of

\ell

in

AB

and

AC

meet at point

A_1

. Points

B_1, C_1

AA_1, BB_1, CC_1

concur, b) their common point lies on the circumcircle of

ABC

c) two points constructed in this way for two perpendicular lines are opposite.

15

1

quadrilaterals with equal areas, related to tangents from a point to circle

Given a circle with center

O

and radius equal to

1

.

AB

and

AC

are the tangents to this circle from point

A

. Point

M

on the circle is such that the areas of quadrilaterals

OBMC

and

ABMC

are equal. Find

MA

.

14

1

triangle where altitude and median from A form triangle so that <A bis is median

In triangle

ABC

, the altitude and the median from vertex

A

form (together with line

BC

) a triangle such that the bisectrix of angle

A

is the median; the altitude and the median from vertex

B

form (together with line AC) a triangle such that the bisectrix of angle

B

is the bisectrix. Find the ratio of sides for triangle

ABC

.

13

1

locus of centroids of triangles inscribed in a triangle, similar in tetrahedrons

a) Find the locus of centroids for triangles whose vertices lie on the sides of a given triangle (each side contains a single vertex). b) Find the locus of centroids for tetrahedrons whose vertices lie on the faces of a given tetrahedron (each face contains a single vertex).

12

1

contruct a point with equal angles related to bases of altitudes

Let

AP

and

BQ

be the altitudes of acute-angled triangle

ABC

. Using a compass and a ruler, construct a point

M

on side

AB

such that

\angle AQM = \angle BPM

.

11

1

parallel lines related to touchpoints of excircle and incircle

The excircle of right-angled triangle

ABC

(

\angle B =90^o

) touches side

BC

at point

A_1

and touches line

AC

in point

A_2

. Line

A_1A_2

meets the incircle of

ABC

for the first time at point

A'

, point

C'

is defined similarly. Prove that

AC||A'C'

.

10

1

drawing parallels to diagonals of trapezoid, passing through intersection

The diagonals of trapezoid

ABCD

meet at point

O

. Point

M

of lateral side

CD

and points

P, Q

of bases

BC

and

AD

are such that segments

MP

and

MQ

are parallel to the diagonals of the trapezoid. Prove that line

PQ

passes through point

O

.

9

1

equal segments related to AC and tangents of circumcircles of CHB,AHB

Let

H

be the orthocenter of triangle

ABC

. The tangents to the circumcircles of triangles

CHB

and

AHB

at point

H

meet

AC

at points

A_1

and

C_1

respectively. Prove that

A_1H = C_1H

4

4

4

4

4

4

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2011 Sharygin Geometry Olympiad

Subcontests

3 equal regular tetrahedrons with common center, can polyhedron faces be equal?

3 points A',B',C' on sides of acute ABC such that it's longest side is minimal

line cuts a triangle, midpoints of segments of projections of intersections

concurrent related to tangents to Euler circle and circumcircle

concurent lines related to midpoints of chords and midpoints of arcs

criterion for tangential ABCD to be cyclic related to midpoints of diagonals

nonisosceles triangle with altitude = bisector = median from different vertices

min and max no of parts that are angles so that n lines divide the plane

compare shortest median vs longest bisectrix, shortest bis.vs longest altitude

concurrent lines related to reflections of a line wrt to sides of given triangle

quadrilaterals with equal areas, related to tangents from a point to circle

triangle where altitude and median from A form triangle so that <A bis is median

locus of centroids of triangles inscribed in a triangle, similar in tetrahedrons

contruct a point with equal angles related to bases of altitudes

parallel lines related to touchpoints of excircle and incircle

drawing parallels to diagonals of trapezoid, passing through intersection

equal segments related to AC and tangents of circumcircles of CHB,AHB

2011 Sharygin Geometry Olympiad

Subcontests

3 equal regular tetrahedrons with common center, can polyhedron faces be equal?

3 points A',B',C' on sides of acute ABC such that it's longest side is minimal

line cuts a triangle, midpoints of segments of projections of intersections

concurrent related to tangents to Euler circle and circumcircle

concurent lines related to midpoints of chords and midpoints of arcs

criterion for tangential ABCD to be cyclic related to midpoints of diagonals

nonisosceles triangle with altitude = bisector = median from different vertices

min and max no of parts that are angles so that n lines divide the plane

compare shortest median vs longest bisectrix, shortest bis.vs longest altitude

concurrent lines related to reflections of a line wrt to sides of given triangle

quadrilaterals with equal areas, related to tangents from a point to circle

triangle where altitude and median from A form triangle so that &lt;A bis is median

locus of centroids of triangles inscribed in a triangle, similar in tetrahedrons

contruct a point with equal angles related to bases of altitudes

parallel lines related to touchpoints of excircle and incircle

drawing parallels to diagonals of trapezoid, passing through intersection

equal segments related to AC and tangents of circumcircles of CHB,AHB

triangle where altitude and median from A form triangle so that <A bis is median