2006 Sharygin Geometry Olympiad

Part of Sharygin Geometry Olympiad

Subcontests

(44)

1

existence of a closed polygonal line

Five lines go through one point. Prove that there exists a closed five-segment polygonal line, the vertices and the middle of the segments of which lie on these lines, and each line has exactly one vertex.

1

both cyclic and tangential ABCD construction (3 points given)

A quadrangle was drawn on the board, that you can inscribe and circumscribe a circle. Marked are the centers of these circles and the intersection point of the lines connecting the midpoints of the opposite sides, after which the quadrangle itself was erased. Restore it with a compass and ruler.

1

can tetrahedron scan turn out to be a triangle with sides 3,4,5 ?

Can a tetrahedron scan turn out to be a triangle with sides

3, 4

5

(a tetrahedron can be cut only along the edges)?

1

another Sharygin concurrency proposed by A. Zaslavsky (median, circumcircle)

Lines containing the medians of the triangle

ABC

intersect its circumscribed circle for a second time at the points

A_1, B_1, C_1

. The straight lines passing through

A,B,C

parallel to opposite sides intersect it at points

A_2, B_2, C_2

. Prove that lines

A_1A_2,B_1B_2,C_1C_2

intersect at one point.

1

locus of intersection points of extarnal tangents of tangent circles

Given a circle and a point

P

inside it, different from the center. We consider pairs of circles tangent to the given internally and to each other at point

P

. Find the locus of the points of intersection of the common external tangents to these circles.

1

concyclic projections and a point symmetry lead also to concyclic projections

The projections of the point

X

onto the sides of the

ABCD

quadrangle lie on the same circle.

Y

is a point symmetric to

X

with respect to the center of this circle. Prove that the projections of the point

B

AX,XC, CY, YA

also lie on the same circle.

1

diagonals of quadrilaterals share intersection points in same ratio

A convex quadrilateral

ABC

A',B',C',D'

are the orthocenters of triangles

BCD, CDA, DAB, ABC

respectively. Prove that in the quadrilaterals

ABCP

A'B'C'D'

, the corresponding diagonals share the intersection points in the same ratio.

1

line passing through circumcenter and orthocenter bisects area and perimeter

A straight line passing through the center of the circumscribed circle and the intersection point of the heights of the non-equilateral triangle

ABC

divides its perimeter and area in the same ratio.Find this ratio.

1

non-convex hexagon with each angle is either 90 or 270, cut into hexagons

In a non-convex hexagon, each angle is either

90

270

degrees. Is it true that for some lengths of the sides it can be cut into two hexagons similar to it and unequal to each other?

1

equal ratios of sidelengths on similar triangles lead to collinear points

ABC

A_1B_1C_1

are similar and differently oriented. On the segment

AA_1

A'

is taken such that

AA' / A_1A'= BC / B_1C_1

. We similarly construct

B'

C'

A', B',C'

lie on one straight line.

1

euler circles tangent to a fixed circle

Given a circle, point

A

on it and point

M

inside it. We consider the chords

BC

passing through

M

. Prove that the circles passing through the midpoints of the sides of all the triangles

ABC

are tangent to a fixed circle.

1

property of line connecting tangent points of two circles to a third one

Given a circle of radius

K

. Two other circles, the sum of the radii of which are also equal to

K

, tangent to the circle from the inside. Prove that the line connecting the points of tangency passes through one of the common points of these circles.

1

circumcenter of triangle of projections of P on sides of ABC is interior point

ABC

P

inside it are given.

A', B', C'

are the projections of

P

onto the straight lines ot the sides

BC,CA,AB

. Prove that the center of the circle circumscribed around the triangle

A'B'C'

lies inside the triangle

ABC

1

convex polygon with diagonal equal to side

Is there a convex polygon with each side equal to some diagonal, and each diagonal equal to some side?

1

intersecting equal circles given, arc bisector wanted

Two equal circles intersect at points

A

B

P

is the point of one of the circles that is different from

A

B, X

Y

are the second intersection points of the lines of

PA, PB

with the other circle. Prove that the line passing through

P

and perpendicular to

AB

divides one of the arcs

XY

1

paralellogram and 2 circles given. equal segments wanted

A parallelogram

ABCD

is given. Two circles with centers at the vertices

A

C

B

. The straight line

\ell

that passes through

B

and crosses the circles at second time at points

X, Y

respectively. Prove that

DX = DY

1

smallest n so n-gon may be cut in triangle, quadrilateral,...,2006-gon

n

is the smallest such that “there is a

n

-gon that can be cut into a triangle, a quadrilateral, ..., a

2006

1

inscribe equilateral triangle in a semicircle with max perimeter

Inscribe the equilateral triangle of the largest perimeter in a given semicircle.

1

tangent points of bases (circles) of 4 cones are concyclic

Four cones are given with a common vertex and the same generatrix, but with, generally speaking, different radii of the bases. Each of them is tangent to two others. Prove that the four tangent points of the circles of the bases of the cones lie on the same circle.

1

ratio of edges when dihedral angles are given, in a tetrahedron

In the tetrahedron

ABCD

, the dihedral angles at the

BC, CD

DA

edges are equal to

\alpha

, and for the remaining edges equal to

\beta

. Find the ratio

AB / CD

1

locus of projections of a fixed point on a line / plane created by perp. rays

a) Two perpendicular rays are drawn through a fixed point

P

inside a given circle, intersecting the circle at points

A

B

. Find the geometric locus of the projections of

P

AB

. b) Three pairwise perpendicular rays passing through the fixed point

P

inside a given sphere intersect the sphere at points

A, B, C

. Find the geometrical locus of the projections

P

ABC

1

HG: GO = 2: 1, NG: GI = 2: 1, special points in cyclic / tangential ABCD

ABCD

is a convex quadrangle,

G

is its center of gravity as a homogeneous plate (i.e., the intersection point of two lines, each of which connects the centroids of triangles having a common diagonal). a) Suppose that around

ABCD

we can circumscribe a circle centered on

O

H

G

, taking orthocenters instead of centroids. Then the points of

H, G, O

lie on the same line and

HG: GO = 2: 1

. b) Suppose that in

ABCD

we can inscribe a circle centered on

I

. The Nagel point N of the circumscribed quadrangle is the intersection point of two lines, each of which passes through points on opposite sides of the quadrangle that are symmetric to the tangent points of the inscribed circle relative to the midpoints of the sides. (These lines divide the perimeter of the quadrangle in half). Then

N, G, I

lie on one straight line, with

NG: GI = 2: 1

1

locus of circumcenters, fixed circle and point not on circle

A, B

on a circle and a point

P

not lying on the circle.

X

is an arbitrary point of the circle,

Y

is the intersection point of lines

AX

BP

. Find the locus of the centers of the circles circumscribed around the triangles

PXY

1

(cevian) triangle area inequality with products, = iff AA',BB'CC' concurrent

AB, BC, CA

ABC

C', A', B'

are taken. Prove that for the areas of the corresponding triangles, the inequality holds:

S_{ABC}S^2_{A'B'C'}\ge 4S_{AB'C'}S_{BC'A'}S_{CA'B'}

and equality is achieved if and only if the lines

AA', BB', CC'

intersect at one point.

1

concurrent circles passing through midpoints of triangle's sides (orthocenter)

Four points are given

A, B, C, D

A_1, B_1, C_1,D_1

are orthocenters of the triangles

BCD, CDA, DAB, ABC

A_2, B_2, C_2,D_2

are orthocenters of the triangles

B_1C_1D_1, C_1D_1A_1, D_1A_1B_1,A_1B_1C_1

etc. Prove that the circles passing through the midpoints of the sides of all the triangles intersect at one point.

1

circumcenter is midpoint of segment of incenter-orthocenter of another triangle

Through the midpoints of the sides of the triangle

T

, straight lines are drawn perpendicular to the bisectors of the opposite angles of the triangle. These lines formed a triangle

T_1

. Prove that the center of the circle circumscribed about

T_1

is in the midpoint of the segment formed by the center of the inscribed circle and the intersection point of the heights of triangle

T

1

perpendiculars from orthocenter, parallel lines lead to collinear

Two perpendicular lines are drawn through the orthocenter

H

ABC

, one of which intersects

BC

X

, and the other intersects

AC

Y

AZ, BZ

are parallel, respectively with

HX

HY

. Prove that the points

X, Y, Z

lie on the same line.

1

intersecting circles, parallel chords, intersections, parallel wanted

In two circles intersecting at points

A

B

, parallel chords

A_1B_1

A_2B_2

are drawn. The lines

AA_1

BB_2

intersect at the point

X, AA_2

BB_1

intersect at the point

Y

XY // A_1B_1

1

equilateral triangles on sides of a triangle create a big equilateral

Regular triangles are built on the sides of the triangle

ABC

. It turned out that their vertices form a regular triangle. Is the original triangle regular also?

1

segments from P to the incircle and circumcircle are equal

A circle is circumscribed around triangle

ABC

and a circle is inscribed in it, which touches the sides of the triangle

BC,CA,AB

A_1,B_1,C_1

, respectively. The line

B_1C_1

intersects the line

BC

P

M

is the midpoint of the segment

PA_1

. Prove that the segments of the tangents drawn from the point

M

to the inscribed and circumscribed circle are equal.

1

locus of orthocenters of APB where P is fixed, AB diameter of fixed circle

Given a circle and a fixed point

P

not lying on it. Find the geometrical locus of the orthocenters of the triangles

ABP

AB

is the diameter of the circle.

1

2 fixed lines and 2 fixed points, 2 parallel segments, locus of intersection

Two straight lines

a

b

are given and also points

A

B

X

slides along the line

a

Y

slides along the line

b

AX \parallel BY

. Find the locus of the intersection point of

AY

XB

1

if <A is obtuse, then AB = AC, when A-bisector =(A-altitude + A-median) /2

In the triangle

ABC

, the bisector of angle

A

is equal to the half-sum of the height and median drawn from vertex

A

. Prove that if

\angle A

is obtuse, then

AB = AC

1

concurrency related to symmetric of A,B,C wrt opposite sides, circumcenter

In the triangle

ABC, O

is the center of the circumscribed circle,

A ', B', C '

are the symmetrics of

A, B, C

with respect to opposite sides,

A_1, B_1, C_1

are the intersection points of the lines

OA'

BC, OB'

AC, OC'

AB

. Prove that the lines

A A_1, BB_1, CC_1

intersect at one point.

1

regular n-gon be cut by disjoint diagonals into (n-2) isosceles triangles

n

n

-gon be cut by disjoint diagonals into

n- 2

isosceles (including equilateral) triangles?

1

concurrency defined by angles and lines of unit circle

L(a)

is the line connecting the points of the unit circle corresponding to the angles

a

\pi - 2a

. Prove that if

a + b + c = 2\pi

, then the lines

L (a), L (b)

L (c)

intersect at one point.

1

segment divides square in 2 tangential polygons, find length in terms of inradii

AB

divides the square into two parts, in each of which a circle can be inscribed. The radii of these circles are equal to

r_1

r_2

respectively, where

r_1> r_2

. Find the length of

AB

1

2 circles construction, related to a given circle and line segment

a) Given a segment

AB

C

inside it, which is the chord of a circle of radius

R

. Inscribe in the formed segment a circle tangent to point

C

and to the circle of radius

R

. b) Given a segment

AB

C

inside it, which is the point of tangency of a circle of radius

r

A

B

a circle tangent to a circle of radius

r

1

angle chasing around a square, congruent triangles given

E

is taken inside the square

ABCD

F

is taken outside, so that the triangles

ABE

BCF

are congruent . Find the angles of the triangle

ABE

, if it is known that

EF

is equal to the side of the square, and the angle

BFD

1

fold a 10 x 10 square from a 1 x 118 rectangular strip (may be bent, not torn)

10 \times 10

1 \times 118

rectangular strip. b) Fold a

10 \times 10

1 \times (100+9\sqrt3)

rectangular strip (approximately

1\times 115.58

). The strip can be bent, but not torn.

1

find angle in a two (a) squares (b) regular pentagons formation

a) Given two squares

ABCD

DEFG

E

lying on the segment

CD

F,G

outside the square

ABCD

. Find the angle between lines

AE

BF

. b) Two regular pentagons

OKLMN

OPRST

are given, and the point

P

lies on the segment

ON

, and the points

R, S, T

are outside the pentagon

OKLMN

. Find the angle between straight lines

KP

MS

1

find the circumcenter of a triangle that it's vertsices lie outside the paper

The map shows sections of three straight roads connecting the three villages, but the villages themselves are located outside the map. In addition, the fire station located at an equal distance from the three villages is not indicated on the map, although its location is within the map. Is it possible to find this place with the help of a compass and a ruler, if the construction is carried out only within the map?

1

perp. bisector of AB passes through fixed point as A,B move along 2 circles

A, B

move with equal speeds along two equal circles. Prove that the perpendicular bisector of

AB

passes through a fixed point.

1

2 lines intersecting an angle 46^o are axes of symmetry, min no of axes

Two straight lines intersecting at an angle of

46^o

are the axes of symmetry of the figure

F

on the plane. What is the smallest number of axes of symmetry this figure can have?