MathDB

1

Vadym Koval is back. Though he never left

a > 1

, we define the sequence

(a_n)

as follows:

a_{n+1} = a_n + d(a_n) - 1

,

n \in \mathbb{N}

,

a_1 = a

, where

d(b)

denotes the smallest prime divisor of

b

. Prove that for any positive integer

k

, the sequence

d(a_n)

for

n \geq k

is not increasing, i.e. the condition

d(a_{n+1}) > d(a_n)

is not true for at least one

n \geq k

.
Proposed by Vadym Koval

9.7

1

Line them up

The All-Ukrainian Mathematical Olympiad was attended by

100

children from

10

regions of Ukraine,

10

children from each region. At the opening ceremony, they stood in one row on the stage, but not all children from the same region stood consecutively. Two neighboring children from different regions can be swapped in

1

second. (You can only swap one pair of children per second).
For which shortest time

T

is it always possible to make sure that the children are lined up by the regions? The regions can go in any order.
Proposed by Anton Trygub

9.5

1

Represent n^4-k^4 as sum of (a+b)(a^2+b^2)

Let

M

denote the set consisting of positive integers that can be represented as

(a + b)(a^2 + b^2)

, where

a, b

are some distinct positive integers. Prove that for any positive integer

n > k

, the number

n^4 - k^4

is the sum of some

n - k

distinct elements of the set

M

.
Proposed by Oleksii Masalitin

9.3

1

Find pair of indices for which inequality holds

Nonnegative integers

x_1, x_2 , \ldots, x_n

are such that

x_1^2 + x_2^2 + \ldots +x_n^2 = n

,

n > 1

. Prove that for any nonnegative real numbers

y_1, y_2 , \ldots, y_n

there exist indices

i, j \in \{1, 2, \ldots, n\}

, not necessarily different, for which the following inequality holds:

x_i + y_j - x_{j+1}y_{i+1} \geq 1

(Here index

n+1 =

index

1

).
Proposed by Nazar Serdyuk

9.2

1

Divisibility over a million?

Prove that there exist distinct positive integers

a

and

b

greater than

1000000

such that

(a^b + 1)

is divisible by

(b^a + 1)

.
Proposed by Arsenii Nikolaiev

8.8

1

Close sum of digits

You are given

k

positive integers

a_1, a_2, \ldots, a_k

. Prove that there exists a positive integer

N

, such that sums of digits of numbers

Na_1, Na_2, \ldots, Na_k

differ in less than

\frac{2021}{2020}

times. That is, if

S_{max}

is the largest sum of digits of any of these numbers, and

S_{min}

is the smallest, then

\frac{S_{max}}{S_{min}} < \frac{2021}{2020}

Proposed by Arsenii Nikolaiev

8.6

1

Deep reflections

You are given a

2021 \times 2021

square, its inner border is a mirror.

2020

of its squares have mirror borders, and all other squares have transparent borders. You are given an angle

\alpha \in (0^\circ, 90^\circ)

. A ray of light passes without change of direction through the transparent boundary, and is reflected according to the law of reflection when hitting the mirror boundary (if the ray hits exactly the corner of the mirror square, it is reflected from the horizontal side of the square, see the figure below for examples).
https://i.ibb.co/mqmTdNS/2021-UMO-8-6.png Prove that from some point on the bottom side of the square you can release a ray of light forming the chosen angle

\alpha

with this side (there are two such rays) so that the light reaches the top side of the square.
Proposed by Arsenii Nikolaiev

8.5

1

Funny representation

Find all positive integers that can be represented as

\frac{abc+ab+a}{abc+bc+c}

for some positive integers

a, b, c

.
Proposed by Oleksii Masalitin

8.3

1

Positive number

The numbers

(a_1 , a_2 , \ldots, a_{2021})

and

(b_1 , b_2 , \ldots, b_{2021})

are some different permutations of the numbers

(1, 2, \ldots, 2021)

, and the numbers

(c_1 , c_2 , \ldots, c_{2021})

are some permutation of the numbers

(2, 4, \ldots, 4042)

. Prove that the given number D is positive:

D = \frac{c_1^2 -4 a_1b_1}{a_1 + b_1 + c_1} + \frac{c_2^2 - 4a_2b_2}{a_2 + b_2 + c_2} + \ldots + \frac{c_{2021}^2 - 4a_{2021}b_{2021}}{a_{2021} + b_{2021} + c_{2021}}

Proposed by Bogdan Rublov

8.2

1

Kotoras tribe

The Kotoras alphabet has exactly

11

letters. It is known that for any positive integer

k

, there are exactly five

k

-letter bad words in the language of this tribe. Prove that it is possible to come up with a word in the Kotoras language with any number of letters that cannot be cut into two smaller words, of which at least one is considered bad.
Proposed by Arseniy Nikolaiev

8.1

1

Always divisible by 10

It is known that for some integers

a_{2021}, a_{2020}, \ldots , a_1 , a_0

and any positive integer

n

, the expression

a_{2021}n^{2021} + a_{2020}n^{2020} +\ldots +a_1n + a_0

is divisible by

10

. Is it necessary that each of the numbers

a_{2021}, a_{2020}, \ldots , a_1 , a_0

is also divisible by

10

?

8.7

1

Game: putting numbers in a grid

Andriy and Bogdan take turns putting integers from

1

to

n

inclusive in the cells of an

n \times n

board in such a way that no row or column contains two equal numbers. Andriy begins, and the player who cannot make a move loses. Depending on

n

, which player has a winning strategy?
Proposed by Fedir Yudin

9.4

1

Triangle with 60&deg; angle, projection of othocenter onto median

Let

ABC

be a triangle with

\angle A = 60^\circ

. Points

P

and

Q

are chosen on sides

AB

and

AC

, respectively, such that

BP = PQ = QC

. Prove that the circumcircle of

\triangle APQ

passes through the projection of the orthocenter of

ABC

onto its

A

-median.
Proposed by Fedir Yudin

8.4

1

Two points inside isosceles triangle

Let

ABC

be an isosceles triangle with

AB = AC

. Points

P

and

Q

inside

\triangle ABC

are such that

\angle BPC = \frac{3}{2} \angle BAC

,

BP = AQ

and

AP = CQ

. Prove that

AP = PQ

.
Proposed by Fedir Yudin

11.8

1

max no of grams that can weigh 101 weights, each from 1 to 2020 grams

There are

101

not necessarily different weights, each of which weighs an integer number of grams from

1

g to

2020

g. It is known that at any division of these weights into two heaps, the total weight of at least one of the piles is no more than

2020

. What is the largest number of grams can weigh all

101

weights?
(Bogdan Rublev)

11.6

1

B symmedian bisects B_1k, 3 altitudes , projection of B

The altitudes

AA_1, BB_1

and

CC_1

were drawn in the triangle

ABC

. Point

K

is a projection of point

B

on

A_1C_1

. Prove that the symmmedian

\vartriangle ABC

from the vertex

B

divides the segment

B_1K

in half.
(Anton Trygub)

11.5

1

m^m+ n^n=k^k

Are there natural numbers

(m,n,k)

that satisfy the equation

m^m+ n^n=k^k

?

10.8

1

\phi (n)+1 (not necessarily different) divisors n with the sum n

Given a natural number

n

. Prove that you can choose

\phi (n)+1

(not necessarily different) divisors

n

with the sum

n

.
Here

\phi (n)

denotes the number of natural numbers less than

n

that are coprime with

n

.
(Fedir Yudin)

10.7

1

integer sequence, 2 strictly increasing sequences

The sequence

a_1,a_2, ..., a_{2n}

of integers is such that each number occurs in no more than

n

times. Prove that there are two strictly increasing sequences of indices

b_1,b_2, ..., b_{n}

and

c_1,c_2, ..., c_{n}

are such that every positive integer from the set

\{1,2,...,2n\}

occurs exactly in one of these two sequences, and for each

1\le i \le n

is true the condition

a_{b_i} \ne a_{c_i}

. (Anton Trygub)

9.6

1

center of circle tangent internally to 2 intersecting circles, lies on bisector

Circles

w_1

and

w_2

intersect at points

P

and

Q

and touch a circle

w

with center at point

O

internally at points

A

and

B

, respectively. It is known that the points

A,B

and

Q

lie on one line. Prove that the point

O

lies on the external bisector

\angle APB

.
(Nazar Serdyuk)

10.5

1

all sets of n consecutive integers {a+1,a+2,...,a+n} in which one =sum of others

Find all sets of

n\ge 2

consecutive integers

\{a+1,a+2,...,a+n\}

where

a\in Z

, in which one of the numbers is equal to the sum of all the others.
(Bogdan Rublev)

9.1

1

Alexey and Bogdan, 2 piles of stones game, 2021 stones totally

Alexey and Bogdan play a game with two piles of stones. In the beginning, one of the piles contains

2021

stones, and the second is empty. In one move, each of the guys has to pick up an even number of stones (more than zero) from an arbitrary pile, then transfer half of the stones taken to another pile, and the other half - to remove from the game. Loses the one who cannot make a move. Who will win this game if both strive to win, and Bogdan begins?
(Oleksii Masalitin)

10.3

1

sum (a^2+b^2)/(a+b) >= (a+b+c)+ (a-c)^2/(a+b+c) for a>= b>= c >0

For arbitrary positive reals

a\ge b \ge c

prove the inequality:

\frac{a^2+b^2}{a+b}+\frac{a^2+c^2}{a+c}+\frac{c^2+b^2}{c+b}\ge (a+b+c)+ \frac{(a-c)^2}{a+b+c}

(Anton Trygub)

10.2

1

naturals (k,d) exists n, for any polynomial p in P^(n) , P(k) is divisible by d

Denote by

P^{(n)}

the set of all polynomials of degree

n

the coefficients of which is a permutation of the set of numbers

\{2^0, 2^1,..., 2^n\}

. Find all pairs of natural numbers

(k,d)

for which there exists a

n

such that for any polynomial

p \in P^{(n)}

, number

P(k)

is divisible by the number

d

.
(Oleksii Masalitin)

11.1

1

a_{2021}n^{2021}+a_{2020}n^{2020}+...+a_1n+a_0 divisible by 2021

It is known that for some integers

a_{2021},a_{2020},...,a_1,a_0

the expression

a_{2021}n^{2021}+a_{2020}n^{2020}+...+a_1n+a_0

is divisible by

2021

for any arbitrary integer

n

. Is it required that each of the numbers

a_{2021},a_{2020},...,a_1,a_0

also divisible by

2021

?

11.4

1

f(xf(x+y))+f((x+y)f(y))=(x+y)^2

Find all the following functions

f:R\to R

, which for arbitrary valid

x,y

holds equality:

f(xf(x+y))+f((x+y)f(y))=(x+y)^2

(Vadym Koval)

11.2

1

3 vertices from n into n-gon, are sidelengths of a triangle

Find all natural numbers

n \ge 3

for which in an arbitrary

n

-gon one can choose

3

vertices dividing its boundary into three parts, the lengths of which can be the lengths of the sides of some triangle.
(Fedir Yudin)

10.4

1

circumcenter of OIH lies on line DE if OI// BC, incenter, orthocenter

Let

O, I, H

be the circumcenter, the incenter, and the orthocenter of

\triangle ABC

. The lines

AI

and

AH

intersect the circumcircle of

\triangle ABC

for the second time at

D

and

E

, respectively. Prove that if

OI \parallel BC

, then the circumcenter of

\triangle OIH

lies on

DE

.
(Fedir Yudin)

2021 Ukraine National Mathematical Olympiad

Subcontests

Vadym Koval is back. Though he never left

Line them up

Represent n^4-k^4 as sum of (a+b)(a^2+b^2)

Find pair of indices for which inequality holds

Divisibility over a million?

Close sum of digits

Deep reflections

Funny representation

Positive number

Kotoras tribe

Always divisible by 10

Game: putting numbers in a grid

Triangle with 60&deg; angle, projection of othocenter onto median

Two points inside isosceles triangle

max no of grams that can weigh 101 weights, each from 1 to 2020 grams

B symmedian bisects B_1k, 3 altitudes , projection of B

m^m+ n^n=k^k

\phi (n)+1 (not necessarily different) divisors n with the sum n

integer sequence, 2 strictly increasing sequences

center of circle tangent internally to 2 intersecting circles, lies on bisector

all sets of n consecutive integers {a+1,a+2,...,a+n} in which one =sum of others

Alexey and Bogdan, 2 piles of stones game, 2021 stones totally

sum (a^2+b^2)/(a+b) >= (a+b+c)+ (a-c)^2/(a+b+c) for a>= b>= c >0

naturals (k,d) exists n, for any polynomial p in P^(n) , P(k) is divisible by d

a_{2021}n^{2021}+a_{2020}n^{2020}+...+a_1n+a_0 divisible by 2021

f(xf(x+y))+f((x+y)f(y))=(x+y)^2

3 vertices from n into n-gon, are sidelengths of a triangle

circumcenter of OIH lies on line DE if OI// BC, incenter, orthocenter

2021 Ukraine National Mathematical Olympiad

Subcontests

Vadym Koval is back. Though he never left

Line them up

Represent n^4-k^4 as sum of (a+b)(a^2+b^2)

Find pair of indices for which inequality holds

Divisibility over a million?

Close sum of digits

Deep reflections

Funny representation

Positive number

Kotoras tribe

Always divisible by 10

Game: putting numbers in a grid

Triangle with 60&amp;deg; angle, projection of othocenter onto median

Two points inside isosceles triangle

max no of grams that can weigh 101 weights, each from 1 to 2020 grams

B symmedian bisects B_1k, 3 altitudes , projection of B

m^m+ n^n=k^k

\phi (n)+1 (not necessarily different) divisors n with the sum n

integer sequence, 2 strictly increasing sequences

center of circle tangent internally to 2 intersecting circles, lies on bisector

all sets of n consecutive integers {a+1,a+2,...,a+n} in which one =sum of others

Alexey and Bogdan, 2 piles of stones game, 2021 stones totally

sum (a^2+b^2)/(a+b) &gt;= (a+b+c)+ (a-c)^2/(a+b+c) for a&gt;= b&gt;= c &gt;0

naturals (k,d) exists n, for any polynomial p in P^(n) , P(k) is divisible by d

a_{2021}n^{2021}+a_{2020}n^{2020}+...+a_1n+a_0 divisible by 2021

f(xf(x+y))+f((x+y)f(y))=(x+y)^2

3 vertices from n into n-gon, are sidelengths of a triangle

circumcenter of OIH lies on line DE if OI// BC, incenter, orthocenter

Triangle with 60° angle, projection of othocenter onto median

sum (a^2+b^2)/(a+b) >= (a+b+c)+ (a-c)^2/(a+b+c) for a>= b>= c >0