Challenging Recreational Mathematics: ▪ Russia Sharygin Geometry Olympiad 2013

Τρίτη 9 Απριλίου 2013

▪ Russia Sharygin Geometry Olympiad 2013

1 Let

be an isosceles triangle with

. Point

lies on the side

, and

is the perpendicular from

. It is known that

. Find

2 Let

be an isosceles triangle (

) with

. The bisectors of angles

and

meet the opposite sides at points

and

respectively. Prove that the triangle

(where

is the circumcenter of

) is regular.

3 Let

be a right-angled triangle (

). The excircle inscribed into the angle

touches the extensions of the sides

at points

respectively; points

are defined similarly. Prove that the perpendiculars from

respectively, concur.

4 Let

be a non isosceles triangle. Point

is its circumcenter, and the point

is the center of the circumcircle

of triangle

. The altitude of

from

meets

at a point

. The line

intersects the circumcircle of

at points

and

. Prove that one of the segments

and

is equal to the segment

5 Four segments drawn from a given point inside a convex quadrilateral to its vertices, split the quadrilateral into four equal triangles. Can we assert that this quadrilateral is a rhombus?

6 Diagonals

and

of a trapezoid

meet at

. The circumcircles of triangles

and

intersect the line

for the second time at points

and

respectively. Let

be the midpoint of segment

. Prove that

7 Let

be a bisector of triangle

. Points

are the incenters of triangles

respectively. The line

meets

in point

. Prove that

8 Let

be an arbitrary point inside the circumcircle of a triangle

. The lines

and

meet the circumcircle in points

and

respectively. The line

intersects

and

at points

and

respectively. Find the locus of points

such that the circumcircles of triangles

and

touch.

9 Let

and

be the points of tangency of the excircles of a triangle

with its sides

and

respectively. It is known that the reflection of the incenter of

across the midpoint of

lies on the circumcircle of triangle

. Find

10 The incircle of triangle

touches the side

at point

; the incircle of triangle

touches the sides

and

at points

; the incircle of triangle

touches the sides

and

at points

. Prove that the lines

, and

concur.

11 a) Let

be a convex quadrilateral and

be the radii of the incircles of triangles

. Can the inequality

hold?

b) The diagonals of a convex quadrilateral

meet in point

. Let

be the radii of the incircles of triangles

. Can the inequality

hold?

12 On each side of triangle

, two distinct points are marked. It is known that these points are the feet of the altitudes and of the bisectors.

a) Using only a ruler determine which points are the feet of the altitudes and which points are the feet of the bisectors.

b) Solve p.a) drawing only three lines.

13 Let

and

be the tangency points of the incircle of triangle

with

and

respectively,

and

be the tangency points of the excircle inscribed into the angle

with the extensions of

and

respectively. Prove that the orthocenter

of triangle

lies on

if and only if the lines

and

are orthogonal.

14 Let

be the midpoints of diagonals

of a right-angled trapezoid

(

The circumcircles of triangles

meet the line

in the points

Prove that the distances from

to the midpoint of

are equal.

15 (a) Triangles

and

are inscribed into triangle

so that

. Prove that these triangles are equal.

(b) Points

lie inside a triangle

so that

is on segment

, and the angles

are equal. Prove that the triangles

and

are equal.

16 The incircle of triangle

touches

at points

, respectively. The perpendicular from the incenter

to the median from vertex

meets the line

in point

. Prove that

is parallel to

17 An acute angle between the diagonals of a cyclic quadrilateral is equal to

. Prove that an acute angle between the diagonals of any other quadrilateral having the same sidelengths is smaller than

18 Let

be a bisector of triangle

. Points

and

are projections of

and

respectively to

. The circle with diameter

intersects

at points

and

. Prove that

19 a) The incircle of a triangle

touches

and

at points

and

respectively. The bisectors of angles

and

meet the perpendicular bisector to the bisector

in points

and

respectively. Prove that the lines

and

concur.

b) Let

be the bisector of a triangle

. Points

and

are the circumcenters of triangles

and

respectively. Points

and

are the projections of

and

to the bisectors of angles

and

respectively. Prove that the lines

and

concur.

c) Prove that the two points obtained in pp. a) and b) coincide.

20 Let

be an arbitrary point on the side

of triangle

. Points

and

on the rays

and

are such that

. The lines

and

meet in point

. Prove that all the lines

have a common point.

21 Chords

and

of circle

meet at point

. The line through

parallel to

meets

again at

, and

meets

again at

. Let

and let

be the reflection of

over

. Show that

passes through the midpoint of

22 The common perpendiculars to the opposite sidelines of a nonplanar quadrilateral are mutually orthogonal. Prove that they intersect.

23 Two convex polytopes

and

do not intersect. The polytope

has exactly

planes of symmetry. What is the maximal number of symmetry planes of the union of

and

, if

has a)

, b)

symmetry planes?

c) What is the answer to the question of p.b), if the symmetry planes are replaced by the symmetry axes?

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Challenging Recreational Mathematics

Τρίτη 9 Απριλίου 2013

▪ Russia Sharygin Geometry Olympiad 2013

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