MathDB

Simple division trick

Source: Kyiv City MO 2021 Round 1, Problem 8.1

12/21/2023

Find all positive integers

n

that can be subtracted from both the numerator and denominator of the fraction

\frac{1234}{6789}

, to get, after the reduction, the fraction of form

\frac{a}{b}

, where

a, b

are single digit numbers.
Proposed by Bogdan Rublov

Fractionsnumber theoryalgebra

Erased digit

Source: Kyiv City MO 2021 Round 1, Problem 8.2

12/21/2023

Oleksiy writes all the digits from

0

to

9

on the board, after which Vlada erases one of them. Then he writes

10

nine-digit numbers on the board, each consisting of all the nine digits written on the board (they don't have to be distinct). It turned out that the sum of these

10

numbers is a ten-digit number, all of whose digits are distinct. Which digit could have been erased by Vlada?
Proposed by Oleksii Masalitin

number theoryDigits

Almost magin square

Source: Kyiv City MO 2021 Round 1, Problem 8.3

12/21/2023

The

1 \times 1

cells located around the perimeter of a

3 \times 3

square are filled with the numbers

1, 2, \ldots, 8

so that the sums along each of the four sides are equal. In the upper left corner cell is the number

8

, and in the upper left is the number

6

(see the figure below). https://i.ibb.co/bRmd12j/Kyiv-MO-2021-Round-1-8-2.png How many different ways to fill the remaining cells are there under these conditions?
Proposed by Mariia Rozhkova

permutationsMagic squarescombinatorics

Return of the geo king

Source: Kyiv City MO 2021 Round 1, Problem 8.4

12/21/2023

Let

BM

be the median of the triangle

ABC

with

AB > BC

. The point

P

is chosen so that

AB\parallel PC

and

PM \perp BM

. Prove that

\angle ABM = \angle MBP

.
Proposed by Mykhailo Shandenko

geometry

Irreduced to ashes, modulo $p$

Source: Kyiv City MO 2021 Round 1, Problem 8.5

12/21/2023

For a prime number

p > 3

, define the following irreducible fraction:

\frac{m}{n} = \frac{p-1}{2} + \frac{p-2}{3} + \ldots + \frac{2}{p-1} - 1

Prove that

m

is divisible by

p

.
Proposed by Oleksii Masalitin

number theoryprime numbers

Birthday paradox

Source: Kyiv City MO 2021 Round 1, Problem 9.1

12/21/2023

Before the math competition, Dmytro overheard Olena and Mykola talking about their birthdays.
О: "The day and month of my birthday are half as large as the day and month of Mykola's birthday." М: "Also, the day of Olena's birth and the month of my birth are consecutive positive integers." О: "And the sum of all these four numbers is a multiple of

17

."
Can Dmitro determine the day and month of Olena's birth?
Proposed by Olena Artemchuk and Mykola Moroz

Birthday

Make almost equal

Source: Kyiv City MO 2021 Round 1, Problem 9.2

12/21/2023

Roma wrote on the board each of the numbers

2018, 2019, 2020

,

100

times each. Let us denote by

S(n)

the sum of digits of positive integer

n

. In one action, Roma can choose any positive integer

k

and instead of any three numbers

a, b, c

written on the board write the numbers

2S(a + b) + k, 2S(b + c) + k

and

2S(c + a) + k

. Can Roma after several such actions make

299

numbers on the board equal, and the last one differing from them by

1

?
Proposed by Oleksii Masalitin

sum of digitsnumber theory

King improves upon 8.4

Source: Kyiv City MO 2021 Round 1, Problem 9.5

12/21/2023

Let

BM

be the median of triangle

ABC

in which

AB > BC

. The point

P

is chosen so that

AB\parallel PC

and

PM \perp BM

. On the line

BP

, point

Q

is chosen so that

\angle AQC = 90^\circ

, and points

B

and

Q

are on opposite sides of the line

AC

. Prove that

AB = BQ

.
Proposed by Mykhailo Shtandenko

geometry

Number of same colored numbers

Source: Kyiv City MO 2021 Round 1, Problem 9.4

12/21/2023

You are given a positive integer

k

and not necessarily distinct positive integers

a_1, a_2 , a_3 , \ldots, a_k

. It turned out that for any coloring of all positive integers from

1

to

2021

in one of the

k

colors so that there are exactly

a_1

numbers of the first color,

a_2

numbers of the second color,

\ldots

, and

a_k

numbers of the

k

-th color, there is always a number

x \in \{1, 2, \ldots, 2021\}

, such that the total number of numbers colored in the same color as

x

is exactly

x

. What are the possible values of

k

?
Proposed by Arsenii Nikolaiev

combinatorics

Beautiful inequality

Source: Kyiv City MO 2021 Round 1, Problem 10.4

12/21/2023

Positive real numbers

a, b, c

satisfy

a^2 + b^2 + c^2 + a + b + c = 6

. Prove the following inequality:

2(\frac{1}{a^2} + \frac{1}{b^2} + \frac{1}{c^2}) \geq 3 + \frac{1}{a} + \frac{1}{b} + \frac{1}{c}

Proposed by Oleksii Masalitin

inequalities

Hardest problem on Kyiv City MO 2021. Possibly ever

Source: Kyiv City MO 2021 Round 1, Problem 10.1

12/21/2023

Prove the following inequality:

\sin{1} + \sin{3} + \ldots + \sin{2021} > \frac{2\sin{1011}^2}{\sqrt{3}}

Proposed by Oleksii Masalitin

inequalities

Circles and parallelogram

Source: Kyiv City MO 2021 Round 1, Problem 10.3

12/21/2023

Circles

\omega_1

and

\omega_2

with centers at points

O_1

and

O_2

intersect at points

A

and

B

. Let point

C

be such that

AO_2CO_1

is a parallelogram. An arbitrary line is drawn through point

A

, which intersects the circles

\omega_1

and

\omega_2

at points

X

and

Y

, respectively. Prove that

CX = CY

.
Proposed by Oleksii Masalitin

geometryparallelogram

Almost magic square with a twist

Source: Kyiv City MO 2021 Round 1, Problem 10.2

12/21/2023

The

1 \times 1

cells located around the perimeter of a

4 \times 4

square are filled with the numbers

1, 2, \ldots, 12

so that the sums along each of the four sides are equal. In the upper left corner cell is the number

1

, in the upper right - the number

5

, and in the lower right - the number

11

.
https://i.ibb.co/PM0ry1D/Kyiv-City-MO-2021-Round-1-10-2.png
Under these conditions, what number can be located in the last corner cell?
Proposed by Mariia Rozhkova

permutationsconstructioncombinatorics

Skew knights

Source: Kyiv City MO 2021 Round 1, Problem 11.2

12/21/2023

Chess piece called skew knight, if placed on the black square, attacks all the gray squares.
https://i.ibb.co/HdTDNjN/Kyiv-MO-2021-Round-1-11-2.png
What is the largest number of such knights that can be placed on the

8\times 8

chessboard without them attacking each other?
Proposed by Arsenii Nikolaiev

combinatoricsChessboard

Cossacks gossip

Source: Kyiv City MO 2021 Round 1, Problem 11.1

12/21/2023

N

cossacks split into

3

groups to discuss various issues with their friends. Cossack Taras moved from the first group to the second, cossack Andriy moved from the second to the third, and cossack Ostap - from the third group to the first. It turned out that the average height of the cossacks in the first group decreased by

8

cm, while in the second and third groups it increased by

5

cm and

8

cm, respectively.
What is

N

, if it is known that there were

9

cossacks in the first group?

algebra

Largest power of $3$ dividing a_101

Source: Kyiv City MO 2021 Round 1, Problem 10.5

12/21/2023

The sequence

(a_n)

is such that

a_{n+1} = (a_n)^n + n + 1

for all positive integers

n

, where

a_1

is some positive integer. Let

k

be the greatest power of

3

by which

a_{101}

is divisible. Find all possible values of

k

.
Proposed by Kyrylo Holodnov

number theorySequence

Another amazing inequality

Source: Kyiv City MO 2021 Round 1, Problem 11.4

12/21/2023

For positive real numbers

a, b, c

with sum

\frac{3}{2}

, find the smallest possible value of the following expression:

\frac{a^3}{bc} + \frac{b^3}{ca} + \frac{c^3}{ab} + \frac{1}{abc}

Proposed by Serhii Torba

inequalitiesalgebrainequalities proposed

Sum of powers

Source: Kyiv City MO 2021 Round 1, Problem 11.5

12/21/2023

For positive integers

m, n

define the function

f_n(m) = 1^{2n} + 2^{2n} + 3^{2n} + \ldots +m^{2n}

. Prove that there are only finitely many pairs of positive integers

(a, b)

such that

f_n(a) + f_n(b)

is a prime number.
Proposed by Nazar Serdyuk

number theoryprime numbers