Kyiv City MO 1984-93 - geometry

Part of Old Kyiv MO Geometry

Subcontests

(63)

1

cotA +cotB+ cotC=(a^2+b^2+c^2)/4S 1993 Kyiv City MO 10.5

Prove that for the sides

a, b, c

A, B, C

S

of the triangle holds

\cot A+ \cot B + \cot C = \frac{a^2+b^2+c^2}{4S}.

1

S <= (a^2+b^2+c^2)/ 4\sqrt3 1993 Kyiv City MO 11.4

a, b, c

be the lengths of the sides of a triangle, and let

S

be it's area. Prove that

S \le \frac{a^2+b^2+c^2}{4\sqrt3}

and the equality is achieved only for an equilateral triangle.

1

2 cubes inscribed in a sphere 1993 Kyiv City MO 11.3

Two cubes are inscribed in a sphere of radius

R

. Calculate the sum of squares of all segments connecting the vertices of one cube with the vertices of the other cube

1

a cos А + bcos В + с cos С <= р 1992 Kyiv City MO 10.4

Prove theat for an arbitrary triangle holds the inequality

a \cos A+ b \cos B + c \cos C \le p ,

a, b, c

are the sides of the triangle,

A, B, C

are the angles,

p

is the semiperimeter.

1

constant sum of squares of distances, 2n points 1993 Kyiv City MO 10.3

The diameter of a circle of radius

R

is divided into

2n

equal parts. The point

M

is taken on the circle. Prove that the sum of the squares of the distances from the point

M

to the points of division (together with the ends of the diameter) does not depend on the choice of the point

M

. Calculate this sum.

1

area chasing, line from midpoint of median to vertice 1992 Kyiv City MO 10.4

In the triangle

ABC

BD

is drawn and through its midpoint and vertex

A

\ell

. Thus the triangle

ABC

is divided into three triangles and one quadrilateral. Determine the areas of these figures if the area of triangle

ABC

S

1

<C = 135^o if (ABC)=1/4 (c^2 - a^2 - b^2) 1993 Kyiv City MO 9.2

a, b, c

be the lengths of the sides of a triangle, and let

S

be its area. We know that

S = \frac14 (c^2 - a^2 - b^2)

\angle C = 135^o

1

constant sum of squares of distances, equilateral, circle 1993 Kyiv City MO 9.3

The circle divides each side of an equilateral triangle into three equal parts. Prove that the sum of the squares of the distances from any point of this circle to the vertices of the triangle is constant.

1

constant sum of distances, circle, diameter related 1993 Kyiv City MO 8.4

The diameter of a circle of radius

R

is divided into

4

equal parts. The point

M

is taken on the circle. Prove that the sum of the squares of the distances from the point

M

to the points of division (together with the ends of the diameter) does not depend on the choice of the point

M

. Calculate this sum.

1

MB_1 = MA_1 id <ACB=60^o, angle bisectors 1993 Kyiv City MO 8.3

In the triangle

ABC

\angle .ACB = 60^o

, and the bisectors

AA_1

BB_1

intersect at the point

M

MB_1 = MA_1

1

intersection of a plane and a pyramid 1992 Kyiv City MO 11.5

The base of the pyramid is a triangle

ABC

\angle ACB= 30^o

, and the length of the median from the vertex

B

is twice less than the side

AC

and is equal to

\alpha

. All side edges of the pyramid are inclined to the plane of the base at an angle

a

. Determine the cross-sectional area of the pyramid with a plane passing through the vertex

B

parallel to the edge

AD

and inclined to the plane of the base at an angle of

\beta

1

congruent when having equal areas, in a square 1992 Kyiv City MO 9.2

Two lines divide a square into

4

figures of the same area. Prove that all these figures are congruent.

1

<=1 center of symmetry in bounded figure 1992 Kyiv City MO 9.3

Prove that a bounded figure cannot have more than one center of symmetry.

1

locus of centroids of triangles, inscribed in circle 1992 Kyiv City MO 8.3

Find the locus of the intersection points of the medians all triangles inscribed in a given circle.

1

equilateral triangle construction, right angle related 1992 Kyiv City MO 7.2

Inside a right angle is given a point

A

. Construct an equilateral triangle, one of the vertices of which is point

A

, and two others lie on the sides of the angle (one on each side).

1

concurent wanted, _|_ diagonals in convex quadr. 1991 Kyiv City MO 8.5

The diagonals of the convex quadrilateral

ABCD

are mutually perpendicular. Through the midpoint of the sides

AB

AD

draw lines, which are perpendicular to the opposite sides. Prove that they intersect on line

AC

1

given concurrent _|_ through incenters of pyramid 1991 Kyiv City MO 11.5

Lines that are drawn perpendicular to the faces of a triangular pyramid through the centers of the inscribed circles intersect at one point. Prove that the sums of the opposite edges of such a pyramid are equal to each other.

1

citerion / converse of a property of altitudes 1991 Kyiv City MO 10.2

In an acute-angled triangle

ABC

AB

BC

AC

C_1

A_1

B_1

are marked such that the segments

AA_1

BB_1

CC_1

intersect at some point

O

AA_1C

BB_1A

CC_1B

are equal. Prove that

AA_1

BB_1

CC_1

are the altitudes of the triangle.

1

angle wanted , inside a regular 8-gon pyramid 1991 Kyiv City MO 10.5

Diagonal sections of a regular 8-gon pyramid, which are drawn through the smallest and largest diagonals of the base, are equal. At what angle is the plane passing through the vertex, the pyramids and the smallest diagonal of the base inclined to the base?
[hide=original wording]Діагональні перерізи правильної 8-кутної піраміди, які Проведені через найменшу і найбільшу діагоналі основи, рівновеликі. Під яким кутом до основи нахилена площина, що проходить через вершину, піраміди і найменшу діагональ основи?

1

area # >=3/2 with lattice points at vertices 1991 Kyiv City MO 9.5

A parallelogram is constructed on the coordinate plane, the coordinates of which are integers. It is known that inside the parallelogram and on its contour there are other (except vertices) points with integer coordinates. Prove that the area of the parallelogram is not less than

3/2

1

area chasing, # inscribed in quadr. 1991 Kyiv City MO 9.4

A parallelogram is inscribed in a quadrilateral, two opposite vertices of which are the midpoints of the opposite sides of the quadrilateral. Determine the area of such a parallelogram if the area of the quadrilateral is equal to

S_o

1

area chasing in a triangle 1991 Kyiv City MO 9.3

M

is the midpoint of the median

BD

of the triangle

ABC

, the area of which is

S

AM

intersects the side

BC

K

. Determine the area of the triangle

BKM

1

square construction, 3 points known 1991 Kyiv City MO 8.4

Construct a square, if you know its center and two points that lie on adjacent sides.

1

# wanted, 4 equilaterals on sides of # 1991 Kyiv City MO 8.3

On the sides of the parallelogram

ABCD

outside it are constructed equilateral triangles

ABM

BCN

CDP

ADQ

MNPQ

is a parallelogram.

1

locus of P, ACP equilateral, C moves along a circle 1991 Kyiv City MO 7.4

Given a circle, point

C

on it and point

A

outside the circle. The equilateral triangle

ACP

is constructed on the segment

AC

C

moves along the circle. What trajectory will the point

P

1

<BCM+<DAM=? <BMC + <AMD = 180^o, rectangle 1991 Kyiv City MO 7.5

Inside the rectangle

ABCD

is taken a point

M

\angle BMC + \angle AMD = 180^o

. Determine the sum of the angles

BCM

DAM

1

sum of distances from vertices of cube >=4\sqrt3 a 1990 Kyiv City MO 11.1

Prove that the sum of the distances from any point in space from the vertices of a cube with edge

a

is not less than

4\sqrt3 a

1

AC=? AC// CD=AB=1, <ABC = 90^o, <CBD=30^o 1990 Kyiv City MO 11.3

AC

ABC

is extended at segment

CD = AB = 1

. It is known that

\angle ABC = 90^o

\angle CBD = 30^o

AC

1

max area of ABCD, intersectons of circle , y=x^2 1990 Kyiv City MO 10.5

A circle centered at a point

(0, 1)

on the coordinate plane intersects the parabola

y = x^2

at four points:

A, B, C, D.

Find the largest possible value of the area of the quadrilateral

ABCD

1

cyclic quadr. construction given 3 sides 1,4,3 1990 Kyiv City MO 10.3

Construct a quadrilateral with three sides

1

4

3

so that a circle could be circumscribed around it.

1

OA_1=OB_1 if <C=60^o, O incenter 1990 Kyiv City MO 9.3

The angle bisectors

AA_1

BB_1

of the triangle ABC intersect at point

O

. Prove that when the angle

C

60^0

OA_1=OB_1

1

exists triangle with sidelenths sinA, sinB, sinC 1990 Kyiv City MO 9.4

\alpha, \beta, \gamma

be the angles of some triangle. Prove that there is a triangle whose sides are equal to

\sin \alpha

\sin \beta

\sin \gamma

1

angles, a/(b+c)=c/(a+b), 120^o 1990 Kyiv City MO 7.3

Given a triangle with sides

a, b, c

\frac{a}{b+c}=\frac{c}{a+b}

. Determine the angles of this triangle, if you know that one of them is equal to

120^0

1

angle for min length of tangent, incircle of equilateral 1990 Kyiv City MO 8.2

A line passes through the center

O

of an equilateral triangle

ABC

and intersects the side

BC

. At what angle wrt

BC

should this line be drawn this line so that its segment inside the triangle has the smallest possible length?

1

sum m_a/h_a <1 + R/r 1989 Kyiv City MO 8.2

h_a,h_b,h_c

be the altitudes, and let

m_a,m_b,m_c

be the medians of the acute triangle drawn to the sides

a, b, c

respectively. Let

r

R

be the radii of the inscribed and circumscribed circles. Prove that

\frac{m_a}{h_a}+\frac{m_b}{h_b}+\frac{m_c}{h_c} <1+\frac{R}{r}.

1

concyclic projections of a point on faces of pyramid 1989 Kyiv City MO 10.5

The base of the quadrangular pyramid

SABCD

is a quadrilateral

ABCD

, the diagonals of which are perpendicular. The apex of the pyramid is projected at intersection point

O

of the diagonals of the base. Prove that the feet of the perpendiculars drawn from point

O

to the side faces of the pyramid lie on one circle.

1

area wanted, incircle related 1989 Kyiv City MO 9.1

The perimeter of the triangle

ABC

2p

, the length of the side

AC

b

ABC

\beta

. A circle with center at point

O

, inscribed in this triangle, touches the side

BC

K

. Calculate the area of the triangle

BOK

1

construction wanted, equilateral inscribed right triangle 1989 Kyiv City MO 8.5

The student drew a right triangle

ABC

on the board with a right angle at the vertex

B

and inscribed in it an equilateral triangle

KMP

such that the points

K, M, P

lie on the sides

AB, BC, AC

, respectively, and

KM \parallel AC

. Then the picture was erased, leaving only points

A, P

C

. Restore erased points and lines.

1

triangle construction given 2 midpoints 1989 Kyiv City MO 7.3

The student drew a triangle

ABC

on the board, in which

AB>BC

AB

D

BD = AC

E

F

be the midpoints of the segments

AD

BC

respectively. Then the whole picture was erased, leaving only dots

E

F

. Restore triangle

ABC

1

2 triplets of edges of tetrahedron form a triangle 1988 Kyiv City MO 10.2

Given an arbitrary tetrahedron. Prove that its six edges can be divided into two triplets so that from each triple it was possible to form a triangle.

1

unusual distance <15 problem, each side <20 in quadr. 1988 Kyiv City MO 9.1

Each side of a convex quadrilateral is less than

20

cm. Prove that you can specify the vertex of the quadrilateral, the distance from which to any point

Q

inside the quadrilateral is less than

15

1

circumcenter of AKC is also circumcenter of ABC 1988 Kyiv City MO 8.5

In the triangle

ABC

, the angle bisector

AK

is drawn. The center of the circle inscribed in the triangle

AKC

coincides with the center of the circle, circumscribed around the triangle

ABC

. Determine the angles of triangle

ABC

1

2 isosceles by diagonall of isosceles trapezoid 1988 Kyiv City MO 7.1

An isosceles trapezoid is divided by each diagonal into two isosceles triangles. Determine the angles of the trapezoid.

1

n tangent spheres in a right circular cone 1987 Kyiv City MO 10.3

In a right circular cone with the radius of the base

R

h

n

spheres of the same radius

r

n \ge 3

). Each ball touches the base of the cone, its side surface and other two balls. Determine

r

1

tangential 1987-gon with consecutive sides 1,...,1987 - 1987 Kyiv City MO 10.1

1987

-gon with consecutive sides lengths

1, 2, 3,..., 1986, 1987

, in which you can fit a circle?

1

triangle construction given circumcircle 1987 Kyiv City MO 9.4

Inscribe a triangle in a given circle, if its smallest side is known, as well as the point of intersection of altitudes lying outside the circle.

1

trapezoid construction given 4 points 1987 Kyiv City MO 8.2

Construct a trapezoid given the midpoints of the legs, the point of intersection of the diagonals and the foot of the perpendicular, drawn from this point on the larger base.

1

AK > diameter of incircle of ABC 1987 Kyiv City MO 7.1

The circle inscribed in the triangle

ABC

touches the side BC at point

K

. Prove that the segment

AK

is longer than the diameter of the circle.

1

max area of MKCE inscribed in square ABCD 1986 Kyiv City MO 10.5

E

be a point on the side

AD

ABCD

. Find such points

M

K

AB

BC

respectively, such that the segments

MK

EC

are parallel, and the quadrilateral

MKCE

has the largest area.

1

triangle of area 1 inside polygon of // sides , area 1 1986 Kyiv City MO 9.5

Prove that inside any convex hexagon with pairs of parallel sides of area

1

, you can draw a triangle of area

1/2

1

rhombuses, equal #, faces of convex polyhedron 1986 Kyiv City MO 9.2

The faces of a convex polyhedron are congruent parallelograms. Prove that they are all rhombuses.

1

sum of angles of polyline ABC on surface of cube 1985 Kyiv City MO 10.2

AB

on the surface of the cube is the shortest polyline on the surface that connects

A

B

ABC

consisted of such segments

AB, BC,CA

. What may be the sum of angles of such triangle if none of the vertex is on the edge of the cube ?

1

vertices of rectangles inscribed in # lying on AB 1986 Kyiv City MO 8.2

A rectangle is said to be inscribed in a parallelogram if its vertices lie one on each side of the parallelogram. On the larger side

AB

of the parallelogram

ABCD

, find all those points

K

that are the vertices of the rectangles inscribed in

ABCD

1

diagonal's sum < sides' sum in ABCD 1986 Kyiv City MO 7.5

Prove that the sum of the lengths of the diagonals of an arbitrary quadrilateral is less than the sum of the lengths of its sides.

1

equilateral wanted, 2 equilaterals on sides of # 1985 Kyiv City MO 9.5

Outside the parallelogram

ABCD

AB

BC

are constructed equilateral triangles

ABK

BCM

. Prove that the triangle

KMD

is equilateral.

1

geo ineq, given max diagonal of hexagon 1985 Kyiv City MO 8.3

The longest diagonal of a convex hexagon is

2

. Is there necessarily a side or diagonal in this hexagon whose length does not exceed

1

1

product of areas AOB,BOC,COD,DOA doesn't end in 1985 Kyiv City MO 7.3

O

is the point of intersection of the diagonals of the convex quadrilateral

ABCD

. It is known that the areas of triangles

AOB, BOC, COD

DOA

are expressed in natural numbers. Prove that the product of these areas cannot end in

1985

1

S_{B_1..B_6}<= 3/2 S_{A_1...A_6} , hexagons 1984 Kyiv City MO 10.5

The vertices of a regular hexagon

A_1,A_2,...,A_6

lie respectively on the sides

B_1B_2

B_2B_3

B_3B_4

B_4B_5

B_5B_6

B_6B_1

of a convex hexagon

B_1B_2B_3B_4B_5B_6

S_{B_1B_2B_3B_4B_5B_6} \le \frac32 S_{A_1A_2A_3A_4A_5A_6}.

1

right triangle construction, gicen 2 medians - 1984 Kyiv City MO 8.3

Construct a right triangle given the lengths of segments of the medians

m_a,m_b

corresponding on its legs.

1

compare perimeters after folding / gluing 1984 Kyiv City MO 9.2

P

, cut out of paper, is bent in a straight line and both halves are glued. Can the perimeter of the polygon

Q

obtained by gluing be larger than the perimeter of the polygon

P

1

connect 2 points, small ruler and compass - 1984 Kyiv City MO 9.5

Using a ruler with a length of

20

cm and a compass with a maximum deviation of

10

cm to connect the segment given two points lying at a distance of

1

1

area of quadr. independent of P, symmetric wrt midpoints - 1984 Kyiv City MO 8.1

Inside the convex quadrilateral

ABCD

'M

. Reflect it symmetrically with respect to the midpoints of the sides of the quadrilateral and connect the obtained points so that they form a convex quadrilateral. Prove that the area of this quadrilateral does not depend on the choice of the point

M

1

<ABM obtuse or right, CM=BC - 1984 Kyiv City MO 7.3

On the extension of the largest side

AC

of the triangle

ABC

set aside the segment

CM

CM = BC

. Prove that the angle

ABM

is obtuse or right.