MathDB

compare angles between altitude-median (2012 Kyiv City MO Round2 10.4)

Source:

8/2/2020

In the triangle

ABC

with sides

BC> AC> AB

the angles between altiude and median drawn from one vertex are considered. Find out at which vertex this angle is the largest of the three.
(Rozhkova Maria)

geometryanglesgeometric inequality

concurrent or //, // chords of inters. circles (2012 Kyiv City MO Round2 11.4)

Source:

8/2/2020

The circles

{{w} _ {1}}

and

{{w} _ {2}}

intersect at points

P

and

Q

. Let

AB

and

CD

be parallel diameters of circles

{ {w} _ {1}}

and

{{w} _ {2}}

, respectively. In this case, none of the points

A, B, C, D

coincides with either

P

or

Q

, and the points lie on the circles in the following order:

A, B, P, Q

on the circle

{{w} _ {1} }

and

C, D, P, Q

on the circle

{{w} _ {2}}

. The lines

AP

and

BQ

intersect at the point

X

, and the lines

CP

and

DQ

intersect at the point

Y, X \ne Y

. Prove that all lines

XY

for different diameters

AB

and

CD

pass through the same point or are all parallel.
(Serdyuk Nazar)

geometryparallelconcurrentcircles

concurrency, <APE =<BAC, < CQF =< BCA (2013 Kyiv City MO Round2 11.4)

Source:

8/5/2020

Let

H

be the intersection point of the altitudes

AP

and

CQ

of the acute-angled triangle

ABC

. On its median

BM

marked points

E

and

F

so that

\angle APE = \angle BAC

and

\angle CQF = \angle BCA

, and the point

E

lies inside the triangle

APB

, and the point

F

lies inside the triangle

CQB

. Prove that the lines

AE

,

CF

and

BH

intersect at one point.
(Vyacheslav Yasinsky)

geometryequal anglesconcurrencyconcurrent

collinear symmetrics wrt midpoints , equilateral (2015 Kyiv City MO Round2 11.2)

Source:

9/4/2020

The line passing through the center of the equilateral triangle

ABC

intersects the lines

AB

,

BC

and

CA

at the points

{{C} _ {1}}

,

{{A} _ {1}}

and

{{B} _ {1}}

, respectively. Let

{{A} _ {2}}

be a point that is symmetric

{{A} _ {1}}

with respect to the midpoint of

BC

; the points

{{B} _ {2}}

and

{{C} _ {2}}

are defined similarly. Prove that the points

{{A} _ {2}}

,

{{B} _ {2}}

and

{{C} _ {2}}

lie on the same line tangent to the inscribed circle of the triangle

ABC

.
(Serdyuk Nazar)

geometryEquilateralcollinearSymmetric

concurrceny wanted, 3 circles related (2014 Kyiv City MO Round2 10.4 11.3)

Source:

8/16/2020

Three circles are constructed for the triangle

ABC

: the circle

{{w} _ {A}}

passes through the vertices

B

and

C

and intersects the sides

AB

and

AC

at points

{{A} _ {1}}

and

{{A} _ {2}}

respectively, the circle

{{w} _ {B}}

passes through the vertices

A

and

C

and intersects the sides

BA

and

BC

at the points

{{B} _ {1}}

and

{{B} _ {2}}

,

{{w} _ {C}}

passes through the vertices

A

and

B

and intersects the sides

CA

and

CB

at the points

{{C} _ {1}}

and

{{C} _ {2}}

. Let

{{A} _ {1}} {{A} _ {2}} \cap {{B} _ {1}} {{B} _ {2}} = {C} '

,

{{A} _ {1}} {{A} _ {2}} \cap {{C} _ {1}} {{C} _ {2}} = {B} '

ta

{ {B} _ {1}} {{B} _ {2}} \cap {{C} _ {1}} {{C} _ {2}} = {A} '

is Prove that the perpendiculars, which are omitted from the points

{A} ', \, \, {B}', \, \, {C} '

to the lines

BC

,

CA

and

AB

respectively intersect at one point.
(Rudenko Alexander)

concurrentgeometrycircles

square construction given 4 collinear (2016 Kyiv City MO Round2 10.2 11.2 )

Source:

9/7/2020

On the horizontal line from left to right are the points

P, \, \, Q, \, \, R, \, \, S

. Construct a square

ABCD

, for which on the line

AD

lies lies the point

P

, on the line

BC

lies the point

Q

, on the line

AB

lies the point

R

, on the line

CD

lies the point

S

.

geometrycollinearconstructionsquare

EF=diameter of circumcircle of AO_1O_2 (2017 Kyiv City MO Round2 10.3)

Source:

9/10/2020

Circles

w_1

and

w_2

with centers at points

O_1

and

O_2

respectively, intersect at points

A

and

B

. A line passing through point

B

, intersects the circles

w_1

and

w_2

at points

C

and

D

other than

B

. Tangents to the circles

w_1

and

w_2

at points

C

and

D

intersect at point

E

. Line

EA

intersects the circumscribed circle

w

of triangle

AO_1O_2

at point

F

. Prove that the length of the segment is

EF

is equal to the diameter of the circle

w

.
(Vovchenko V., Plotnikov M.)

geometrycircumcirclediametercircles

ABTN is cyclic iff AB = AK, KT = KC,CN = BK (2017 Kyiv City MO Round2 11.2)

Source:

9/10/2020

The median

CM

is drawn in the triangle

ABC

intersecting bisector angle

BL

at point

O

. Ray

AO

intersects side

BC

at point

K

, beyond point

K

draw the segment

KT = KC

. On the ray

BC

beyond point

C

draw a segment

CN = BK

. Prove that is a quadrilateral

ABTN

is cyclic if and only if

AB = AK

.
(Vladislav Yurashev)

geometrycyclic quadrilateralConcyclicequal segments

circumcircle of HXY equidistant from B,C (2018 Kyiv City MO Round2 10.3)

Source:

9/14/2020

In the acute triangle

ABC

the orthocenter

H

and the center of the circumscribed circle

O

were noted. The line

AO

intersects the side

BC

at the point

D

. A perpendicular drawn to the side

BC

at the point

D

intersects the heights from the vertices

B

and

C

of the triangle

ABC

at the points

X

and

Y

respectively. Prove that the center of the circumscribed circle

\Delta HXY

is equidistant from the points

B

and

C

.
(Danilo Hilko)

geometrycircumcircleequal segments

<TSC=<BAC wanted, parallelogram, circumcircle (2020 Kyiv City MO Round2 11.2)

Source:

9/21/2020

A point

P

was chosen on the smaller arc

BC

of the circumcircle of the acute-angled triangle

ABC

. Points

R

and

S

on the sides

AB

and

AC

are respectively selected so that

CPRS

is a parallelogram. Point

T

on the arc

AC

of the circumscribed circle of

\vartriangle ABC

such that

BT \parallel CP

. Prove that

\angle TSC = \angle BAC

.
(Anton Trygub)

geometryparallelogramcircumcircleequal angles

right angle wanted, AB=BC, circumcircle related (2018 Kyiv City MO Round2 11.2)

Source:

9/14/2020

In the quadrilateral

ABCD

,

AB = BC

, the point

K

is the midpoint of the side

CD

, the rays

BK

and

AD

intersect at the point

M

, the circumscribed circle

\Delta ABM

intersects the line

AC

for the second time at the point

P

. Prove that

\angle BKP = 90 {} ^ \circ

.
(Anton Trygub)

circumcirclegeometryright angle

<T_A T_B T_C= 90^o - 1/2 <ABC if OI//AC (2019 Kyiv City MO Round2 10.3)

Source:

9/18/2020

Denote in the triangle

ABC

by

T_A,T_B,T_C

the touch points of the exscribed circles of

\vartriangle ABC

, tangent to sides

BC, AC

and

AB

respectively. Let

O

be the center of the circumcircle of

\vartriangle ABC

, and

I

is the center of it's inscribed circle. It is known that

OI\parallel AC

. Prove that

\angle T_A T_B T_C= 90^o - \frac12 \angle ABC

.
(Anton Trygub)

geometryexcentersexcenterangles

AR = QR wanted inside a regular pentagon (2019 Kyiv City MO Round2 10.3.1)

Source:

9/18/2020

Let

ABCDE

be a regular pentagon with center

M

. Point

P \ne M

is selected on segment

MD

. The circumscribed circle of triangle

ABP

intersects the line

AE

for second time at point

Q

, and a line that is perpendicular to the

CD

and passes through

P

, for second time at the point

R

. Prove that

AR = QR

.

geometryequal segmentspentagonregular pentagon

circumcenter lies on midline of other triangle (2019 Kyiv City MO Round2 11.3)

Source:

9/19/2020

The line

\ell

is perpendicular to the side

AC

of the acute triangle

ABC

and intersects this side at point

K

, and the circumcribed circle

\vartriangle ABC

at points

P

and

T

(point P on the other side of line

AC

, as the vertex

B

). Denote by

P_1

and

T_1

- the projections of the points

P

and

T

on line

AB

, with the vertices

A, B

belong to the segment

P_1T_1

. Prove that the center of the circumscribed circle of the

\vartriangle P_1KT_1

lies on a line containing the midline

\vartriangle ABC

, which is parallel to the side

AC

.
(Anton Trygub)

geometryCircumcentermidline

centroid is incenter of other triangle, (2019 Kyiv City MO Round2 11.3.1)

Source:

9/19/2020

It is known that in the triangle

ABC

the smallest side is

BC

. Let

X, Y, K

and

L

- points on the sides

AB, AC

and on the rays

CB, BC

, respectively, are such that

BX = BK = BC =CY =CL

. The line

KX

intersects the line

LY

at the point

M

. Prove that the intersection point of the medians

\vartriangle KLM

coincides with the center of the inscribed circle

\vartriangle ABC

.

geometryincenterCentroidequal segments

<PBM+<CBM=<PCA,<BM=90^o, <ABC+<APC=180^o (2020 Kyiv City MO Round2 10.2)

Source:

9/21/2020

Let

M

be the midpoint of the side

AC

of triangle

ABC

. Inside

\vartriangle BMC

was found a point

P

such that

\angle BMP = 90^o

,

\angle ABC+ \angle APC =180^o

. Prove that

\angle PBM + \angle CBM = \angle PCA

.
(Anton Trygub)

geometryanglesright ange

product of radii, common tangents to circles (2021 Kyiv City MO Round2 11.3.1)

Source:

2/15/2021

Two circles

k_1

and

k_2

with radii

r_1

and

r_2

have no common points. The line

AB

is a common internal tangent, and the line

CD

is a common external tangent to these circles, where

A, C \in k_1

and

B, D \in k_2

. Knowing that

AB=12

and

CD =16

, find the value of the product

r_1r_2

.

geometryTangentscircles

concyclic, <OAD+<OBC= <ODA + <OCB = 90^o (2021 Kyiv City MO Round2 10.4)

Source:

2/14/2021

Inside the quadrilateral

ABCD

marked a point

O

such that

\angle OAD+ \angle OBC = \angle ODA + \angle OCB = 90^o

. Prove that the centers of the circumscribed circles around triangles

OAD

and

OBC

as well as the midpoints of the sides

AB

and

CD

lie on one circle.
(Anton Trygub)

geometryConcyclicangles

OK// AB wanted in isosceles trapezoid (2021 Kyiv City MO Round2 10.4.1)

Source:

2/15/2021

Let

ABCD

be an isosceles trapezoid,

AD=BC

,

AB \parallel CD

. The diagonals of the trapezoid intersect at the point

O

, and the point

M

is the midpoint of the side

AD

. The circle circumscribed around the triangle

BCM

intersects the side

AD

at the point

K

. Prove that

OK \parallel AB

.

geometrytrapezoidisoscelesparallel

circle tangent to incircle, <BKA=45^o (2021 Kyiv City MO Round2 11.3)

Source:

2/15/2021

In the triangle

ABC

, the altitude

BH

and the angle bisector

BL

are drawn, the inscribed circle

w

touches the side of the

AC

at the point

K

. It is known that

\angle BKA = 45^o

. Prove that the circle with diameter

HL

touches the circle

w

.
(Anton Trygub)

geometrytangent circles

Slowly growing product

Source: Kyiv City MO 2021 Round 1, Problem 9.3

12/21/2023

Let

a_n = 1 + \frac{2}{n} - \frac{2}{n^3} - \frac{1}{n^4}

. For which smallest positive integer

n

does the value of

P_n = a_2a_3a_4 \ldots a_n

exceed

100

?

algebra

Magic rectangle (almost)

Source: Kyiv City MO 2021 Round 1, Problem 7.4

12/21/2023

A rectangle

3 \times 5

is divided into

15

1 \times 1

cells. The middle

3

cells that have no common points with the border of the rectangle are deleted. Is it possible to put in the remaining

12

cells numbers

1, 2, \ldots, 12

in some order, so that the sums of the numbers in the cells along each of the four sides of the rectangle are equal?
Proposed by Mariia Rozhkova

permutationsrectangleMagic squares

We balling

Source: Kyiv City MO 2021 Round 1, Problem 7.1

12/21/2023

Mom brought Andriy and Olesya

4

balls with the numbers

1, 2, 3

and

4

written on them (one on each ball). She held

2

balls in each hand and did not know which numbers were written on the balls in each hand. The mother asked Andriy to take a ball with a higher number from each hand, and then to keep the ball with the lower number from the two balls he took. After that, she asked Olesya to take two other balls, and out of these two, keep the ball with the higher number. Does the mother know with certainty, which child has the ball with the higher number?
Proposed by Bogdan Rublov

combinatoricsgameballs

Cutting rectangle game

Source: Kyiv City MO 2021 Round 1, Problem 7.2

12/21/2023

Andriy and Olesya take turns (Andriy starts) in a

2 \times 1

rectangle, drawing horizontal segments of length

2

or vertical segments of length

1

, as shown in the figure below.
https://i.ibb.co/qWqWxgh/Kyiv-MO-2021-Round-1-7-2.png
After each move, the value

P

is calculated - the total perimeter of all small rectangles that are formed (i.e., those inside which no other segment passes). The winner is the one after whose move

P

is divisible by

2021

for the first time. Who has a winning strategy?
Proposed by Bogdan Rublov

gamecombinatoricscuttinggeometryrectangle

Small factoring

Source: Kyiv City MO 2021 Round 1, Problem 7.3

12/21/2023

Petryk factored the number

10^6 = 1000000

as a product of

7

distinct positive integers. Among all such factorings, find the one in which the largest of these

7

factors is the smallest possible.
Proposed by Bogdan Rublov

Factoringnumber theory