2005 Sharygin Geometry Olympiad

Part of Sharygin Geometry Olympiad

Subcontests

(41)

1

lines intersect on the inscribed sphere on a tetrahedron, concurrency

The sphere inscribed in the tetrahedron

ABCD

touches its faces at points

A',B',C',D'

AA'

BB'

intersect, and the point of their intersection lies on the inscribed sphere. Prove that the segments

CC'

DD'

also intersect on the inscribed sphere.

1

existence of a point that any line passing through it bisects an unfixed angle

The angle and the point

K

inside it are given on the plane. Prove that there is a point

M

with the following property: if an arbitrary line passing through intersects the sides of the angle at points

A

B

MK

is the bisector of the angle

AMB

1

find chord of one circle tangent to another one, incircle related

In the triangle

ABC , \angle A = \alpha, BC = a

. The inscribed circle touches the lines

AB

AC

M

P

. Find the length of the chord cut by the line

MP

in a circle with diameter

BC

1

property of intersection point of diagonals in a cyclic ABCD (distance ratio)

Inside the inscribed quadrilateral

ABCD

there is a point

K

, the distances from which to the sides

ABCD

are proportional to these sides. Prove that

K

is the intersection point of the diagonals of

ABCD

1

construction of smallest segment in ABCD, that bisects perimeter, in+ex+circles

Convex quadrilateral

ABCD

is given. Lines

BC

AD

intersect at point

O

B

lying on the segment

OC

A

OD

I

is the center of the circle inscribed in the

OAB

J

is the center of the circle exscribed in the triangle

OCD

touching the side of

CD

and the extensions of the other two sides. The perpendicular from the midpoint of the segment

IJ

BC

AD

intersect the corresponding sides of the quadrilateral (not the extension) at points

X

Y

. Prove that the segment

XY

divides the perimeter of the quadrilateral

ABCD

in half, and from all segments with this property and ends on

BC

AD

XY

has the smallest length.

1

point on concurrency of 3 lines lies on circumcircle, // passing from midpoints

A_1, B_1, C_1

are the midpoints of the sides

BC,CA,BA

respectively of an equilateral triangle

ABC

. Three parallel lines, passing through

A_1, B_1, C_1

intersect, respectively, lines

B_1C_1, C_1A_1, A_1B_1

A_2, B_2, C_2

. Prove that the lines

AA_2, BB_2, CC_2

intersect at one point lying on the circle circumscribed around the triangle

ABC

1

lines concurrency, related to circle with one end of diameter the orthocenter

H

be the orthocenter of triangle

ABC

X

be an arbitrary point. A circle with a diameter of

XH

intersects lines

AH, BH, CH

A_1, B_1, C_1

for the second time, and lines

AX BX, CX

A_2, B_2, C_2

. Prove that lines A

_1A_2, B_1B_2, C_1C_2

intersect at one point.

1

computational with intersecting circles, tangents from a point of one circle

Two circles of radius

1

intersect at points

X, Y

, the distance between which is also equal to

1

C

of one circle, tangents

CA, CB

are drawn to the other. Line

CB

will cross the first circle a second time at point

A'

. Find the distance

AA'

1

8 points, 5 equal ratios given, angle between lines A_3B_3 , A_4B_4 wanted

A_1B_1

A_2B_2

are given on the plane, with

\frac{A_2B_2}{A_1B_1} = k < 1

A_1A_2

A_3

is taken, and on the extension of this segment beyond point

A_2

A_4

\frac{A_3A_2}{A_3A_1} =\frac{A_4A_2}{A_4A_1}= k

. Similarly, point

B_3

is taken on segment

B_1B_2

, and on the extension of this the segment beyond point

B_2

B_4

\frac{B_3B_2}{B_3B_1} =\frac{B_4B_2}{B_4B_1}= k

. Find the angle between lines

A_3B_3

A_4B_4

.
(Netherlands)

1

midpoint of a radius is equidistant from parallel chords of another circle

Two parallel chords

AB

CD

are drawn in a circle with center

O

. Circles with diameters

AB

CD

intersect at point

P

. Prove that the midpoint of the segment

OP

is equidistant from lines

AB

CD

1

property of a triangle that can be cut into 3 similar triangles

A triangle can be cut into three similar triangles. Prove that it can be cut into any number of triangles similar to each other.

1

only 1 of 4 contructed vertices of # lies on the interior of convex ABCD

A convex quadrangle without parallel sides is given. For each triple of its vertices, a point is constructed that supplements this triple to a parallelogram, one of the diagonals of which coincides with the diagonal of the quadrangle. Prove that of the four points constructed, exactly one lies inside the original quadrangle.

1

impossible triangle construction given, height, angle bisector, median, <A >135

It is given that for no side of the triangle from the height drawn to it, the bisector and the median it is impossible to make a triangle. Prove that one of the angles of the triangle is greater than

135^o

1

midpoint wanted, telated to 4 orthocenters in ABCD

P

be the intersection point of the diagonals of the quadrangle

ABCD

M

the intersection point of the lines connecting the midpoints of its opposite sides,

O

the intersection point of the perpendicular bisectors of the diagonals,

H

the intersection point of the lines connecting the orthocenters of the triangles

APD

BCP

APB

CPD

M

is the midpoint of

OH

1

set of midpoints where ends lie on different arcs AB of the same circle

Given a circle and points

A, B

on it. Draw the set of midpoints of the segments, one of the ends of which lies on one of the arcs

AB

, and the other on the second.

1

find isosceles triangles that cannot be cut into 3 isosceles with same sides

Find all isosceles triangles that cannot be cut into three isosceles triangles with the same sides.

1

perpendicular diagonals criterion for a cyclic quadrilateral

ABCD

is inscribed in a circle whose center

O

lies inside it. Prove that if

\angle BAO = \angle DAC

, then the diagonals of the quadrilateral are perpendicular.

1

in space exists a point that all sides of the triangle are visible at angle \phi

A triangle is given, all the angles of which are smaller than

\phi

\phi <2\pi / 3

. Prove that in space there is a point from which all sides of the triangle are visible at an angle

\phi

1

decompose a cube into 5 convex pentagons of equal areas

Envelop the cube in one layer with five convex pentagons of equal areas.

1

projections of 3 perpendiculars on 4th face concurrent, in tetrahedron

Perpendiculars at their centers of gravity (points of intersection of medians) are restored to the faces of the tetrahedron. Prove that the projections of the three perpendiculars to the fourth face intersect at one point.

1

tsunami in a regular tetrahedron planet, epicenter, propagation speed given

The planet Tetraincognito covered by ocean has the shape of a regular tetrahedron with an edge of

900

km. What area of the ocean will the tsunami' cover

2

hours after the earthquake with the epicenter in a) the center of the face, b) the middle of the edge, if the tsunami propagation speed is

300

1

circumsphere lies entirely inside another circumsphere, tetrahedra related

I

be the center of the sphere inscribed in the tetrahedron

ABCD, A ', B', C ', D'

be the centers of the spheres circumscribed around the tetrahedra

IBCD, ICDA, IDAB, IABC

, respectively. Prove that the sphere circumscribed around

ABCD

lies entirely inside the circumscribed around

A'B'C'D '

1

flying saucer jumps in the plane of the lunar orbit

As you know, the moon revolves around the earth. We assume that the Earth and the Moon are points, and the Moon rotates around the Earth in a circular orbit with a period of one revolution per month. The flying saucer is in the plane of the lunar orbit. It can be jumped through the Moon and the Earth - from the old place (point

A

), it instantly appears in the new (at point

A '

) so that either the Moon or the Earth is in the middle of segment

AA'

. Between the jumps, the flying saucer hangs motionless in outer space. 1) Determine the minimum number of jumps a flying saucer will need to jump from any point inside the lunar orbit to any other point inside the lunar orbit. 2) Prove that a flying saucer, using an unlimited number of jumps, can jump from any point inside the lunar orbit to any other point inside the lunar orbit for any period of time, for example, in a second.

1

concurrency of midpoints of symmetric points wrt 3 lines forming triangle

On the plane are three straight lines

\ell_1, \ell_2,\ell_3

, forming a triangle, and the point

O

is marked, the center of the circumscribed circle of this triangle. For an arbitrary point X of the plane, we denote by

X_i

the point symmetric to the point X with respect to the line

\ell_i, i = 1,2,3

. a) Prove that for an arbitrary point

M

the straight lines connecting the midpoints of the segments

O_1O_2

M_1M_2, O_2O_3

M_2M_3, O_3O_1

M_3M_1

intersect at one point, b) where can this intersection point lie?

1

ruler construction, point such that a circle is tangent to the incircle

A circle is inscribed in the triangle

ABC

and it's center

I

and the points of tangency

P, Q, R

BC

C A

AB

are marked, respectively. With a single ruler, build a point

K

at which the circle passing through the vertices B and

C

touches (internally) the inscribed circle.

1

find angles given O, I, G, H, not knowing which is who

We took a non-equilateral acute-angled triangle and marked

4

wonderful points in it: the centers of the inscribed and circumscribed circles, the center of gravity (the point of intersection of the medians) and the intersection point of altitudes. Then the triangle itself was erased. It turned out that it was impossible to establish which of the centers corresponds to each of the marked points. Find the angles of the triangle

1

existence of circle of minimum radius, lattice points related

Given a circle centered at the origin. Prove that there is a circle of smaller radius that has no less points with integer coordinates.

1

\sqrt{S_1 S_2} \le S \le \sqrt{S_2 S_3} area cevian triangle inequality

P

be an arbitrary point inside the triangle

ABC

A_1, B_1

C_1

denote the intersection points of the straight lines

AP, BP

CP

, respectively, with the sides

BC, CA

AB

. We order the areas of the triangles

AB_1C_1,A_1BC_1,A_1B_1C

. Denote the smaller by

S_1

, the middle by

S_2

, and the larger by

S_3

\sqrt{S_1 S_2} \le S \le \sqrt{S_2 S_3}

S

is the area of the triangle

A_1B_1S_1

1

minimum length segment constructions, parallel related

ABC

\ell_1, \ell_2

are given. Through an arbitrary point

D

AB

, a line parallel to

\ell_1

AC

E

and a line parallel to

\ell_2

BC

F

. Construct a point

D

for which the segment

EF

has the smallest length.

1

quadrilateral construction given sidelengths and distance of mid-diagonals

Construct a quadrangle along the given sides

a, b, c

d

and the distance

I

between the midpoints of its diagonals.

1

square was cut in n rectangles with sides a_i x b_j, min n for different a_i,b

The square was cut into

n^2

rectangles with sides

a_i \times b_j, i , j= 1,..., n

. For what is the smallest

n

\{a_1, b_1, ..., a_n, b_n\}

all the numbers can be different?

1

cut random triangle into 4 similar triangles, not all the same

Cut the non-equilateral triangle into four similar triangles, among which not all are the same.

1

perpendiculars from P on equilateral ABC, centroid of projections', midpoint

O

be the center of a regular triangle

ABC

. From an arbitrary point

P

of the plane, the perpendiculars were drawn on the sides of the triangle. Let

M

denote the intersection point of the medians of the triangle , having vertices the feet of the perpendiculars. Prove that

M

is the midpoint of the segment

PO

1

ABCD inscribed in 3 rectangles, 2o of which are squares, is 3rd square also?

Around the convex quadrilateral

ABCD

, three rectangles are circumscribed . It is known that two of these rectangles are squares. Is it true that the third one is necessarily a square? (A rectangle is circumscribed around the quadrilateral

ABCD

if there is one vertex

ABCD

on each side of the rectangle).

1

concentric circles, A on larger, midpoint of BC on smaller, ABC equilateral

Two circles with radii

1

2

have a common center at the point

O

A

of the regular triangle

ABC

lies on the larger circle, and the middpoint of the base

CD

lies on the smaller one. What can the angle

BOC

1

n equal parts in AB, n+1 equal parts in AC, coloring triangles, area wanted

AB

ABC

was divided into

n

equal parts (dividing points

B_0 = A, B_1, B_2, ..., B_n = B

AC

of this triangle was divided into

(n + 1)

equal parts (dividing points

C_0 = A, C_1, C_2, ..., C_{n+1} = C

). Colored are the triangles

C_iB_iC_{i+1}

i = 1,2, ..., n

). What part of the area of the triangle is painted over?

1

locus of (a) orthocenters (b) centroids (c) circumcenters

There are two parallel lines

p_1

p_2

A

B

p_1

C

p_2

. We will move the segment

BC

parallel to itself and consider all the triangles

AB'C '

thus obtained. Find the locus of the points in these triangles: a) points of intersection of heights, b) the intersection points of the medians, c) the centers of the circumscribed circles.

1

convex n-gon with equal sines of angles and sidelegths all different

At what smallest

n

is there a convex

n

-gon for which the sines of all angles are equal and the lengths of all sides are different?

1

locus of midpoints of chords, 2 intersecting equal circles related

Given a circle and a point

K

inside it. An arbitrary circle equal to the given one and passing through the point

K

has a common chord with the given circle. Find the geometric locus of the midpoints of these chords.

1

3 polygons, equal in area and perimeter, made from 5 equal-square cross

Cut a cross made up of five identical squares into three polygons, equal in area and perimeter.

1

perpendicular non intersecting chords wanted

AC

BD

of the circle intersect at point

P

. The perpendiculars to

AC

BD

C

D

, respectively, intersect at point

Q

. Prove that the lines

AB

PQ

are perpendicular.