MathDB

NT for beginners

Source: Kyiv City MO 2023 7.3

5/14/2023

Prove that there don't exist positive integer numbers

k

and

n

which satisfy equation

n^n+(n+1)^{n+1}+(n+2)^{n+2} = 2023^k

. Proposed by Mykhailo Shtandenko

number theory

Equal signs needed easy

Source: Kyiv City MO 2023 Round 1, Problem 7.2

12/16/2023

You are given

n \geq 3

distinct real numbers. Prove that one can choose either

3

numbers with positive sum, or

2

numbers with negative sum.
Proposed by Mykhailo Shtandenko

algebra

Close integer

Source: Kyiv City MO 2023 Round 1, Problem 8.1

12/16/2023

Find the integer which is closest to the value of the following expression:

((7 + \sqrt{48})^{2023} + (7 - \sqrt{48})^{2023})^2 - ((7 + \sqrt{48})^{2023} - (7 - \sqrt{48})^{2023})^2

algebraexpression

Hedgehogs going WILD

Source: Kyiv City MO 2023 Round 1, Problem 8.3

12/16/2023

A hedgehog is a circle without its boundaries. The diameter of the hedgehog is the diameter of the corresponding circle. We say that the hedgehog sits at the at the point where the center of the circle is located.
We are given a triangle with sides

a, b, c

, with hedgehogs sitting at its vertices. It is known that inside the triangle there is a point from which you can reach any side of the triangle by walking along a straight line without hitting any hedgehog. What is the largest possible sum of the diameters of these hedgehogs?
Proposed by Oleksiy Masalitin

geometry

Easy algebra for 8-graders

Source: Kyiv City MO 2023 8.2

5/14/2023

Positive integers

k

and

n

are given such that

3 \le k \le n

.Prove that among any

n

pairwise distinct real numbers one can choose either

k

numbers with positive sum, or

k-1

numbers with negative sum. Proposed by Mykhailo Shtandenko

algebra

Similar numbers

Source: Kyiv City MO 2023 Round 1, Problem 8.4

12/16/2023

Let's call a pair of positive integers

\overline{a_1a_2\ldots a_k}

and

\overline{b_1b_2\ldots b_k}

k

-similar if all digits

a_1, a_2, \ldots, a_k , b_1 , b_2, \ldots, b_k

are distinct, and there exist distinct positive integers

m, n

, for which the following equality holds:

a_1^m + a_2^m + \ldots + a_k^m = b_1^n + b_2^n + \ldots + b_k^n

For which largest

k

do there exist

k

-similar numbers?
Proposed by Oleksiy Masalitin

number theoryalgebraDigits

Convex quadrilaterals don't exist??

Source: Kyiv City MO 2023 Round 1, Problem 8.5

12/16/2023

You are given a square

n \times n

. The centers of some of some

m

of its

1\times 1

cells are marked. It turned out that there is no convex quadrilateral with vertices at these marked points. For each positive integer

n \geq 3

, find the largest value of

m

for which it is possible.
Proposed by Oleksiy Masalitin, Fedir Yudin

combinatoricscombinatorial geometrysquare grid

Inequality with number $2023$

Source: Kyiv City MO 2023 Round 1, Problem 10.1

12/16/2023

Find all positive integers

n

that satisfy the following inequalities:

-46 \leq \frac{2023}{46-n} \leq 46-n

inequalities

Equal signs needed

Source: Kyiv City MO 2023 9.2

5/14/2023

Non-zero real numbers

a, b

and

c

are given such that

ab+bc+ac=0

. Prove that numbers

a+b+c

and

\dfrac{1}{a+b}+\dfrac{1}{b+c}+\dfrac{1}{a+c}

are either both positive or both negative. Proposed by Mykhailo Shtandenko

algebra

Integer convex equilateral 2023-gon

Source: Kyiv City MO 2023 Round 1, Problem 9.5

12/16/2023

Does there exist on the Cartesian plane a convex

2023

-gon with vertices at integer points, such that the lengths of all its sides are equal?
Proposed by Anton Trygub

geometrynumber theory

Arc midpoints in the right triangle

Source: Kyiv City MO 2023 Round 1, Problem 9.3

12/16/2023

You are given a right triangle

ABC

with

\angle ACB = 90^\circ

. Let

W_A , W_B

respectively be the midpoints of the smaller arcs

BC

and

AC

of the circumcircle of

\triangle ABC

, and

N_A , N_B

respectively be the midpoints of the larger arcs

BC

and

AC

of this circle. Denote by

P

and

Q

the points of intersection of segment

AB

with the lines

N_AW_B

and

N_BW_A

, respectively. Prove that

AP = BQ

.
Proposed by Oleksiy Masalitin

geometrycircumcircle

System error

Source: Kyiv City MO 2023 Round 1, Problem 10.2

12/16/2023

For any given real

a, b, c

solve the following system of equations:

\left\{\begin{array}{l}ax^3+by=cz^5,\\az^3+bx=cy^5,\\ay^3+bz=cx^5.\end{array}\right.

Proposed by Oleksiy Masalitin, Bogdan Rublov

algebrasystem of equations

Cutting grid into corners squared

Source: Kyiv City MO 2023 Round 1, Problem 10.4

12/16/2023

Positive integers

m, n

are such that

mn

is divisible by

9

but not divisible by

27

. Rectangle

m \times n

is cut into corners, each consisting of three cells. There are four types of such corners, depending on their orientation; you can see them on the figure below. Could it happen that the number of corners of each type was the exact square of some positive integer?
Proposed by Oleksiy Masalitin
https://i.ibb.co/Y8QSHyf/Kyiv-MO-2023-10-4.png

gridcombinatorics

This one was misplaced

Source: Kyiv City MO 2023 Round 1, Problem 11.3

12/16/2023

Let

I

be the incenter of triangle

ABC

with

AB < AC

. Point

X

is chosen on the external bisector of

\angle ABC

such that

IC = IX

. Let the tangent to the circumscribed circle of

\triangle BXC

at point

X

intersect the line

AB

at point

Y

. Prove that

AC = AY

.
Proposed by Oleksiy Masalitin

circumcirclegeometry

Pairs with close sums

Source: Kyiv City MO 2023 Round 1, Problem 11.2

12/16/2023

You are given

n\geq 4

positive real numbers. Consider all

\frac{n(n-1)}{2}

pairwise sums of these numbers. Show that some two of these sums differ in at most

\sqrt[n-2]{2}

times.
Proposed by Anton Trygub

algebra

Grid geometry

Source: Kyiv City MO 2023 Round 1, Problem 10.3

12/16/2023

Consider all pairs of distinct points on the Cartesian plane

(A, B)

with integer coordinates. Among these pairs of points, find all for which there exist two distinct points

(X, Y)

with integer coordinates, such that the quadrilateral

AXBY

is convex and inscribed.
Proposed by Anton Trygub

analytic geometrygridgeometrycyclic quadrilateral

Compare A and B

Source: Kyiv City MO 2023 Round 1, Problem 11.1

12/16/2023

Which number is larger:

A = \frac{1}{9} : \sqrt[3]{\frac{1}{2023}}

, or

B = \log_{2023} 91125

?

algebraCompare

AoPS suggested tag 3D geometry for this number theory

Source: Kyiv City MO 2023 Round 1, Problem 11.4

12/16/2023

Find all pairs

(m, n)

of positive integers, for which number

2^n - 13^m

is a cube of a positive integer.
Proposed by Oleksiy Masalitin

number theory

Graph wars

Source: Kyiv City MO 2023 Round 1, Problem 11.5

12/16/2023

In a galaxy far, far away there are

225

inhabited planets. Between some pairs of inhabited planets there is a bidirectional space connection, and it is possible to reach any planet from any other (possibly with several transfers). The influence of a planet is the number of other planets with which this planet has a direct connection. It is known that if two planets are not connected by a direct space flight, they have different influences. What is the smallest number of connections possible under these conditions?
Proposed by Arsenii Nikolaev, Bogdan Rublov

combinatoricsgraph theory

equal circumradii, diagonals of quadr. (Kyiv City Olympiad 2003 9.4)

Source:

6/27/2021

The diagonals of a convex quadrilateral divide it into four triangles. The radii of the circles circumscribed around these triangles are equal. Can such a property have a quadrilateral other than: a) parallelogram, b) rhombus?
(Sharygin Igor)

geometryparallelogramrhombusinradius

angle chasing, isosceles and angle bisectors (2015 Kyiv City MO 8.3)

Source:

9/2/2020

In the isosceles triangle

ABC

,

(AB = BC)

the bisector

AD

was drawn, and in the triangle

ABD

the bisector

DE

was drawn. Find the values of the angles of the triangle

ABC

, if it is known that the bisectors of the angles

ABD

and

AED

intersect on the line

AD

.
(Fedak Ivan)

geometryangle bisectoranglesisoscelesAngle Chasing

trapezoid construction (Kyiv City Olympiad 2003 8.5)

Source:

6/27/2021

Three segments

2

cm,

5

cm and

12

cm long are constructed on the plane. Construct a trapezoid with bases of

2

cm and

5

cm, the sum of the sides of which is

12

cm, and one of the angles is

60^o

.
(Bogdan Rublev)

geometrytrapezoidconstruction

angles in isosceles wanted, AM=2ВК, 24^o (Kyiv City Olympiad 2004 8.7 )

Source:

6/27/2021

In an isosceles triangle

ABC

with base

AC

, on side

BC

is selected point

K

so that

\angle BAK = 24^o

. On the segment

AK

the point

M

is chosen so that

\angle ABM = 90^o

,

AM=2BK

. Find the values of all angles of triangle

ABC

.

anglesgeometryisosceles

<DOE=90^o if BD=AD, CB=CE, right triangle (Kyiv City Olympiad 2004 7.3)

Source:

6/27/2021

Given a right triangle

ABC

(

\angle A <45^o

,

\angle C = 90^o

), on the sides

AC

and

AB

which are selected points

D,E

respectively, such that

BD = AD

and

CB = CE

. Let the segments

BD

and

CE

intersect at the point

O

. Prove that

\angle DOE = 90^o

.

geometryright triangleright angle

triangle construction given 3 points (Kyiv City Olympiad 2004 9.7)

Source:

6/27/2021

The board depicts the triangle

ABC

, the altitude

AH

and the angle bisector

AL

which intersectthe inscribed circle in the triangle at the points

M

and

N, P

and

Q

, respectively. After that, the figure was erased, leaving only the points

H, M

and

Q

. Restore the triangle

ABC

.
(Bogdan Rublev)

geometryconstruction