Διασκεδαστικά Μαθηματικά : ▪ Russia Sharygin Geometry Olympiad 2012

Κυριακή 6 Μαΐου 2012

▪ Russia Sharygin Geometry Olympiad 2012

1. In triangle

point

is the midpoint of side

, and point

is the foot of altitude

. Prove that

if and only if

2. A cyclic

-gon is divided by non-intersecting (inside the

-gon) diagonals to

triangles. Each of these triangles is similar to at least one of the remaining ones. For what

this is possible?

3. A circle with center

touches sides

of triangle

in points

. Lines

meet

in points

respectively. Prove that

4. Given triangle

. Point

is the midpoint of side

, and point

is the projection of

to the perpendicular bisector of segment

. Line

meets

in point

. Prove that triangle

is isosceles.

5. On side

of triangle

an arbitrary point is selected

. The tangent in

to the circumcircle of triangle

meets

in point

; point

is defined similarly. Prove that

6. Point

of hypothenuse

of a right-angled triangle

is such that

. Point

on cathetus

is such that

; point

is defined similarly. Find angle

, where

is the midpoint of

7. In a non-isosceles triangle

the bisectors of angles

and

are inversely proportional to the respective sidelengths. Find angle

8. Let

be the median of right-angled triangle

. The incircle of triangle

touches sides

in points

; points

are defined similarly. Prove that lines

and

meet on the bisector of angle

9. In triangle

, given lines

and

containing the bisectors of angles

and

, and the foot

of the bisector of angle

. Restore triangle

10. 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?

11. Given triangle

and point

. Points

are the projections of

. A line passing through

and parallel to

meets the circumcircle of triangle

for the second time in point

. Points

are defined similarly. Prove that

a) lines

concur;

b) triangles

and

are similar.

12. Let

be the circumcenter of an acute-angled triangle

. A line passing through

and parallel to

meets

and

in points

and

respectively. The sum of distances from

and

is equal to

. Prove that

13. Points

are given. Find the locus of points

such that

, the midpoints of

and the centroid of triangle

are concyclic.

14. In a convex quadrilateral

suppose

and

is the midpoint of

. Let

. Prove that

15. Given triangle

. Consider lines

with the next property: the reflections of

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

16. Given right-angled triangle

with hypothenuse

. Let

be the midpoint of

and

be the center of circumcircle

of triangle

. Line

meets

for the second time in point

. Segment

meets the circumcircle of triangle

in point

. Prove that segments

and

meet on the circumcircle of triangle

17. A square

is inscribed into a circle. Point

lies on arc

meets

in point

meets

in point

. Prove that the area of quadrilateral

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

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

, one of two remaining angles is equal to

, 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.

19. Two circles with radii 1 meet in points

, and the distance between these points also is equal to

. Point

lies on the first circle, and lines

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

. Lines

and

meet in point

. Find angle

20. Point

lies on side

of triangle

. Let

and

be the incircles and the excircles (touching segment

) of triangles

and

Prove that the common external tangents to

and

meet on

21. 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.

22. A circle

with center

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

. Point

is the midpoint of this arc

, and point

is the midpoint of the complementary arc. The tangents from

touch

in points

and

. The opposite sidelines

and

of quadrilateral

meet in point

, and the diagonals of

meet in point

. Prove that points

and

are collinear.

23. 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.

24. Given are

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?