MathDB

1

two intersecting spheres

(V.Protasov, 10--11) In the space, given two intersecting spheres of different radii and a point

A

belonging to both spheres. Prove that there is a point

B

in the space with the following property: if an arbitrary circle passes through points

A

and

B

then the second points of its meet with the given spheres are equidistant from

B

.

22

1

All vertices of a pyramid lie on the facets of a cube

(A.Khachaturyan, 10--11) a) All vertices of a pyramid lie on the facets of a cube but not on its edges, and each facet contains at least one vertex. What is the maximum possible number of the vertices of the pyramid? b) All vertices of a pyramid lie in the facet planes of a cube but not on the lines including its edges, and each facet plane contains at least one vertex. What is the maximum possible number of the vertices of the pyramid?

21

1

lenth of perpendicular bisectors

(A.Zaslavsky, B.Frenkin, 10--11) In a triangle, one has drawn perpendicular bisectors to its sides and has measured their segments lying inside the triangle. a) All three segments are equal. Is it true that the triangle is equilateral? b) Two segments are equal. Is it true that the triangle is isosceles? c) Can the segments have length 4, 4 and 3?

24

1

Inequality related with tetrahedron

(I.Bogdanov, 11) Let

h

be the least altitude of a tetrahedron, and

d

the least distance between its opposite edges. For what values of

t

the inequality

d>th

is possible?

20

1

if a line bisect perimeter then this line bisects the area

(A.Zaslavsky, 10--11) a) Some polygon has the following property: if a line passes through two points which bisect its perimeter then this line bisects the area of the polygon. Is it true that the polygon is central symmetric? b) Is it true that any figure with the property from part a) is central symmetric?

19

1

Determine the ratio

(V.Protasov, 10-11) Given parallelogram

ABCD

such that AB \equal{} a, AD \equal{} b. The first circle has its center at vertex

A

and passes through

D

, and the second circle has its center at

C

and passes through

D

. A circle with center

B

meets the first circle at points

M_1

,

N_1

, and the second circle at points

M_2

,

N_2

. Determine the ratio

M_1N_1/M_2N_2

.

18

1

Inequality with S

(A.Abdullayev, 9--11) Prove that the triangle having sides

a

,

b

,

c

and area

S

satisfies the inequality a^2\plus{}b^2\plus{}c^2\minus{}\frac12(|a\minus{}b|\plus{}|b\minus{}c|\plus{}|c\minus{}a|)^2\geq 4\sqrt3 S.

17

1

By this ruler only

(A.Myakishev, 9--11) Given triangle

ABC

and a ruler with two marked intervals equal to

AC

and

BC

. By this ruler only, find the incenter of the triangle formed by medial lines of triangle

ABC

.

16

1

Determine the locus of intersections of lines

(A.Zaslavsky, 9--11) Given two circles. Their common external tangent is tangent to them at points

A

and

B

. Points

X

,

Y

on these circles are such that some circle is tangent to the given two circles at these points, and in similar way (external or internal). Determine the locus of intersections of lines

AX

and

BY

.

15

1

Draw a line

(M.Volchkevich, 9--11) Given two circles and point

P

not lying on them. Draw a line through

P

which cuts chords of equal length from these circles.

14

1

The Euler line parallel to the bisector

(V.Protasov, 9--10) The Euler line of a non-isosceles triangle is parallel to the bisector of one of its angles. Determine this angle (There was an error in published condition of this problem).

13

1

Prove that point lies on the incircle.

(A.Myakishev, 9--10) Given triangle

ABC

. One of its excircles is tangent to the side

BC

at point

A_1

and to the extensions of two other sides. Another excircle is tangent to side

AC

at point

B_1

. Segments

AA_1

and

BB_1

meet at point

N

. Point

P

is chosen on the ray

AA_1

so that AP\equal{}NA_1. Prove that

P

lies on the incircle.

12

1

Prove that lines are parallel

(A.Myakishev, 9--10) Given a triangle

ABC

. Point

A_1

is chosen on the ray

BA

so that segments

BA_1

and

BC

are equal. Point

A_2

is chosen on the ray

CA

so that segments

CA_2

and

BC

are equal. Points

B_1

,

B_2

and

C_1

,

C_2

are chosen similarly. Prove that lines

A_1A_2

,

B_1B_2

,

C_1C_2

are parallel.

11

1

Common chords of pairs of circles

(A.Zaslavsky, 9--10) Given four points

A

,

B

,

C

,

D

. Any two circles such that one of them contains

A

and

B

, and the other one contains

C

and

D

, meet. Prove that common chords of all these pairs of circles pass through a fixed point.

10

1

crcumscribed quadrilateral

(A.Zaslavsky, 9--10) Quadrilateral

ABCD

is circumscribed arounda circle with center

I

. Prove that the projections of points

B

and

D

to the lines

IA

and

IC

lie on a single circle.

9

1

The reflections of diagonal in the bissector

(A.Zaslavsky, 9--10) The reflections of diagonal

BD

of a quadrilateral

ABCD

in the bisectors of angles

B

and

D

pass through the midpoint of diagonal

AC

. Prove that the reflections of diagonal

AC

in the bisectors of angles

A

and

C

pass through the midpoint of diagonal

BD

(There was an error in published condition of this problem).

4

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4

4

4

4

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2008 Sharygin Geometry Olympiad

Subcontests

two intersecting spheres

All vertices of a pyramid lie on the facets of a cube

lenth of perpendicular bisectors

Inequality related with tetrahedron

if a line bisect perimeter then this line bisects the area

Determine the ratio

Inequality with S

By this ruler only

Determine the locus of intersections of lines

Draw a line

The Euler line parallel to the bisector

Prove that point lies on the incircle.

Prove that lines are parallel

Common chords of pairs of circles

crcumscribed quadrilateral

The reflections of diagonal in the bissector