MathDB

2

principal diagonal in convex hexagon <= 2 /\sqrt3

The length of each side and each non-principal diagonal of a convex hexagon does not exceed

1

. Prove that this hexagon contains a principal diagonal whose length does not exceed

\frac{2}{\sqrt3}

.

computational in tetrahedron DABC, <ACB = <ADB, (CD) _|_ (ABC)

In the tetrahedron

DABC

:

\angle ACB = \angle ADB

,

(CD) \perp (ABC)

. In triangle

ABC

, the altitude

h

drawn to the side

AB

and the distance

d

from the center of the circumscribed circle to this side are given. Find the length of the

CD

.

5

2

PK = LQ wanted, 2 equilateral triangles outside convex cyclic ABCD

The diagonals of the inscribed quadrilateral

ABCD

meet at the point

M

,

\angle AMB = 60^o

. Equilateral triangles

ADK

and

BCL

are built outward on sides

AD

and

BC

. Line

KL

meets the circle circumscribed ariound

ABCD

at points

P

and

Q

. Prove that

PK = LQ

.

cyclic quadr. by projections of point og circumcircle of trapezoid on sides

Trapezoid

ABCD

with bases

AB

and

CD

is inscribed in a circle. Prove that the quadrilateral formed by orthogonal projections of any point of this circle onto lines

AC, BC, AD

and

BD

is inscribed.

4

2

line connecting projections of intersections of tangents perpendicular to median

Triangle

ABC

is inscribed in a circle. Through points

A

and

B

tangents to this circle are drawn, which intersect at point

P

. Points

X

and

Y

are orthogonal projections of point

P

onto lines

AC

and

BC

. Prove that line

XY

is perpendicular to the median of triangle

ABC

from vertex

C

.

circumcenter O equiditsant from AB,AC iff OM//BC, M is centroid

In triangle

ABC

,

M

is the intersection point of the medians,

O

is the center of the inscribed circle. Prove that if the line

OM

is parallel to the side

BC

, then the point

O

is equidistant from the sides

AB

and

AC

.

3

2

locus with equal angles inside a square

Given a square

ABCD

. Find the locus of points

M

such that

\angle AMB = \angle CMD

.

bicentric quadrilateral construction with a ruler and no more than 6 lines

On the board was drawn a circle with a marked center, a quadrangle inscribed in it, and a circle inscribed in it, also with a marked center. Then they erased the quadrilateral (keeping one vertex) and the inscribed circle (keeping its center). Restore any of the erased vertices of the quadrilateral using only a ruler and no more than six lines.

2

a triangle construction, <A and B,C- medians

Construct a triangle

ABC

given angle

A

and the medians drawn from vertices

B

and

C

.

spacial closed self-intersecting broken line in space

Is there a closed self-intersecting broken line in space that intersects each of its links exactly once, and in its midpoint ?

1

2

half triangle area wanted, midpoints in convex quadrilateral

In a convex quadrilateral

ABCD

,

E

is the midpoint of

CD

,

F

is midpoint of

AD

,

K

is the intersection point of

AC

with

BE

. Prove that the area of triangle

BKF

is half the area of triangle

ABC

.

line connecting midpoints BE,AD divide diagonals AC,BC in the same ratio

E

and

F

are the midpoints of the sides

BC

and

AD

of the convex quadrilateral

ABCD

. Prove that the segment

EF

divides the diagonals

AC

and

BD

in the same ratio.

2004 Oral Moscow Geometry Olympiad

Subcontests

principal diagonal in convex hexagon <= 2 /\sqrt3

computational in tetrahedron DABC, <ACB = <ADB, (CD) _|_ (ABC)

PK = LQ wanted, 2 equilateral triangles outside convex cyclic ABCD

cyclic quadr. by projections of point og circumcircle of trapezoid on sides

line connecting projections of intersections of tangents perpendicular to median

circumcenter O equiditsant from AB,AC iff OM//BC, M is centroid

locus with equal angles inside a square

bicentric quadrilateral construction with a ruler and no more than 6 lines

a triangle construction, <A and B,C- medians

spacial closed self-intersecting broken line in space

half triangle area wanted, midpoints in convex quadrilateral

line connecting midpoints BE,AD divide diagonals AC,BC in the same ratio

2004 Oral Moscow Geometry Olympiad

Subcontests

principal diagonal in convex hexagon &lt;= 2 /\sqrt3

computational in tetrahedron DABC, &lt;ACB = &lt;ADB, (CD) _|_ (ABC)

PK = LQ wanted, 2 equilateral triangles outside convex cyclic ABCD

cyclic quadr. by projections of point og circumcircle of trapezoid on sides

line connecting projections of intersections of tangents perpendicular to median

circumcenter O equiditsant from AB,AC iff OM//BC, M is centroid

locus with equal angles inside a square

bicentric quadrilateral construction with a ruler and no more than 6 lines

a triangle construction, &lt;A and B,C- medians

spacial closed self-intersecting broken line in space

half triangle area wanted, midpoints in convex quadrilateral

line connecting midpoints BE,AD divide diagonals AC,BC in the same ratio

principal diagonal in convex hexagon <= 2 /\sqrt3

computational in tetrahedron DABC, <ACB = <ADB, (CD) _|_ (ABC)

a triangle construction, <A and B,C- medians