MathDB

1

Non-intersecting convex envelops

n

(n > 2)

points on the plane such that no three of them are collinear. In how many ways this set of points can be divided into two non-empty subsets with non-intersecting convex envelops?

23

1

Locus of centroids of 12 points

An arbitrary point is selected on each of twelve diagonals of the faces of a cube.The centroid of these twelve points is determined. Find the locus of all these centroids.

22

1

Circle inscribed between an arc and a chord

A circle

\omega

with center

I

is inscribed into a segment of the disk, formed by an arc and a chord

AB

. Point

M

is the midpoint of this arc

AB

, and point

N

is the midpoint of the complementary arc. The tangents from

N

touch

\omega

in points

C

and

D

. The opposite sidelines

AC

and

BD

of quadrilateral

ABCD

meet in point

X

, and the diagonals of

ABCD

meet in point

Y

. Prove that points

X, Y, I

and

M

are collinear.

21

1

Product of two segments

Two perpendicular lines pass through the orthocenter of an acute-angled triangle. The sidelines of the triangle cut on each of these lines two segments: one lying inside the triangle and another one lying outside it. Prove that the product of two internal segments is equal to the product of two external segments.
Nikolai Beluhov and Emil Kolev

20

1

Incircles and excircles

Point

D

lies on side

AB

of triangle

ABC

. Let

\omega_1

and

\Omega_1,\omega_2

and

\Omega_2

be the incircles and the excircles (touching segment

AB

) of triangles

ACD

and

BCD.

Prove that the common external tangents to

\omega_1

and

\omega_2,

\Omega_1

and

\Omega_2

meet on

AB

.

19

1

Two circles and tangents

Two circles with radii 1 meet in points

X, Y

, and the distance between these points also is equal to

1

. Point

C

lies on the first circle, and lines

CA, CB

are tangents to the second one. These tangents meet the first circle for the second time in points

B', A'

. Lines

AA'

and

BB'

meet in point

Z

. Find angle

XZY

.

18

1

Using ruler only

A triangle and two points inside it are marked. It is known that one of the triangle’s angles is equal to

58^{\circ}

, one of two remaining angles is equal to

59^{\circ}

, one of two given points is the incenter of the triangle and the second one is its circumcenter. Using only the ruler without partitions determine where is each of the angles and where is each of the centers.

17

1

Square in a circle

A square

ABCD

is inscribed into a circle. Point

M

lies on arc

BC

,

AM

meets

BD

in point

P

,

DM

meets

AC

in point

Q

. Prove that the area of quadrilateral

APQD

is equal to the half of the area of the square.

16

1

Intersection of segments on circumcircle

Given right-angled triangle

ABC

with hypothenuse

AB

. Let

M

be the midpoint of

AB

and

O

be the center of circumcircle

\omega

of triangle

CMB

. Line

AC

meets

\omega

for the second time in point

K

. Segment

KO

meets the circumcircle of triangle

ABC

in point

L

. Prove that segments

AL

and

KM

meet on the circumcircle of triangle

ACM

.

15

1

Reflections of line

Given triangle

ABC

. Consider lines

l

with the next property: the reflections of

l

in the sidelines of the triangle concur. Prove that all these lines have a common point.

14

1

Ratio of triangle areas

In a convex quadrilateral

ABCD

suppose

AC \cap BD = O

and

M

is the midpoint of

BC

. Let

MO \cap AD = E

. Prove that

\frac{AE}{ED} = \frac{S_{\triangle ABO}}{S_{\triangle CDO}}

.

13

1

Locus of concyclic points

Points

A, B

are given. Find the locus of points

C

such that

C

, the midpoints of

AC, BC

and the centroid of triangle

ABC

are concyclic.

12

1

Sum of distances

Let

O

be the circumcenter of an acute-angled triangle

ABC

. A line passing through

O

and parallel to

BC

meets

AB

and

AC

in points

P

and

Q

respectively. The sum of distances from

O

to

AB

and

AC

is equal to

OA

. Prove that

PB + QC = PQ

.

11

1

Concurrent lines

Given triangle

ABC

and point

P

. Points

A', B', C'

are the projections of

P

to

BC, CA, AB

. A line passing through

P

and parallel to

AB

meets the circumcircle of triangle

PA'B'

for the second time in point

C_{1}

. Points

A_{1}, B_{1}

AA_{1}, BB_{1}, CC_{1}

concur; b) triangles

ABC

and

A_{1}B_{1}C_{1}

are similar.

10

1

Convex quadrilateral

In a convex quadrilateral all sidelengths and all angles are pairwise different. a) Can the greatest angle be adjacent to the greatest side and at the same time the smallest angle be adjacent to the smallest side? b) Can the greatest angle be non-adjacent to the smallest side and at the same time the smallest angle be non-adjacent to the greatest side?

9

1

Restore triangle

In triangle

ABC

, given lines

l_{b}

and

l_{c}

containing the bisectors of angles

B

and

C

, and the foot

L_{1}

of the bisector of angle

A

. Restore triangle

ABC

4

4

4

4

4

4

4

2012 Sharygin Geometry Olympiad

Subcontests

Non-intersecting convex envelops

Locus of centroids of 12 points

Circle inscribed between an arc and a chord

Product of two segments

Incircles and excircles

Two circles and tangents

Using ruler only

Square in a circle

Intersection of segments on circumcircle

Reflections of line

Ratio of triangle areas

Locus of concyclic points

Sum of distances

Concurrent lines

Convex quadrilateral

Restore triangle