MathDB

2

Sasha has $10$ cards with numbers $1, 2, 4, 8,\ldots, 512$

10

cards with numbers

1, 2, 4, 8,\ldots, 512

. He writes the number

0

on the board and invites Dima to play a game. Dima tells the integer

0 < p < 10, p

can vary from round to round. Sasha chooses

p

cards before which he puts a “

+

” sign, and before the other cards he puts a “

-

" sign. The obtained number is calculated and added to the number on the board. Find the greatest absolute value of the number on the board Dima can get on the board after several rounds regardless Sasha’s moves.

Find all $n$ such that after $n-1$ moves it is possible to obtain $0$.

Numbers

1, 2,\ldots, n

are written on the board. By one move, we replace some two numbers

a, b

with the number

a^2-b{}

. Find all

n{}

such that after

n-1

moves it is possible to obtain

0

.

6

2

\gcd(a, b) + lcm(a, b) = \gcd(a, c) + lcm(a, c)

Let

a, b, c

be positive integers such that

\gcd(a, b) + \text{lcm}(a, b) = \gcd(a, c) + \text{lcm}(a, c).

Does it follow from this that

b = c

?

Find the greatest $n$

Let

n \leq 100

be an integer. Hare puts real numbers in the cells of a

100 \times 100

table. By asking Hare one question, Wolf can find out the sum of all numbers of a square

n \times n

, or the sum of all numbers of a rectangle

1 \times (n - 1)

(or

(n - 1) \times 1

). Find the greatest

n{}

such that, after several questions, Wolf can find the numbers in all cells, with guarantee.

5

1

ne can find a rectangle $1\times k$ in which the sum of the numbers equals $0$

Is it possible to fill a table

1\times n

with pairwise distinct integers such that for any

k = 1, 2,\ldots, n

one can find a rectangle

1\times k

in which the sum of the numbers equals

0

if a)

n= 11

; b)

n= 12

?

3

1

The numbers $1, 2, 3,\ldots, 2\underbrace{00\ldots0}_{100 \text{ zeroes}}2$ are

The numbers

1, 2, 3,\ldots, 2\underbrace{00\ldots0}_{100 \text{ zeroes}}2

are written on the board. Is it possible to paint half of them red and remaining ones blue, so that the sum of red numbers is divisible by the sum of blue ones?

2

1

Prove that this hexagon can be covered by a triangle with perimeter at most $3$.

In a convex hexagon the value of each angle is

120^{\circ}

. The perimeter of the hexagon equals

2

. Prove that this hexagon can be covered by a triangle with perimeter at most

3

.

1

2

the product of any $n$ odd consecutive positive integers is divisible by $45$.

Determine the least positive integer

n{}

for which the following statement is true: the product of any

n{}

odd consecutive positive integers is divisible by

45

.

Prove that if $q+q^{'} =r +r^{'}$, then $2n$ is a perfect square.

Let

n{}

and

m

be positive integers,

n>m>1

. Let

n{}

divided by

m

have partial quotient

q

and remainder

r

(so that

n = qm + r

, where

r\in\{0,1,...,m-1\}

). Let

n-1

divided by

m

have partial quotient

q^{'}

and remainder

r^{'}

. a) It appears that

q+q^{'} =r +r^{'} = 99

. Find all possible values of

n{}

. b) Prove that if

q+q^{'} =r +r^{'}

, then

2n

is a perfect square.

8

2

Prove that the triangle $A^{'}B^{'}C^{'}$ is equilateral.

Let

ABC

be an equilateral triangle with the side length equals

a+ b+ c

. On the side

AB{}

of the triangle

ABC

points

C_1

and

C_2

are chosen, on the side

BC

points

A_1

and

A_2

, arc chosen, and on the side

CA

points

B_1

and

B_2

are chosen such that

A_1A_2 = CB_1 = BC_2 = a, B_1B_2 = AC_1 = CA_2 = b, C_1C_2 = BA_1 = AB_2 = c

. Let the point

A^{’}

be such that the triangle

A^{'} B_2C_1

is equilateral, and the points

A

and

A^{'}

lie on different sides of the line

B_2C_1

. Similarly, the points

B^{’}

and

C^{'}

are constructed (the triangle

B^{'} C_2A_1

is equilateral, and the points

B

and

B^{’}

lie on different sides of the line

C_2A_1

; the triangle

C^{'} A_2B_1

is equilateral, and the points

C

and

C^{'}

lie on different sides of the line

A_2B_1

). Prove that the triangle

A^{'}B^{'}C^{'}

is equilateral.

Cute geometry

Let

ABC

be an acute-angled triangle, and let

AA_1, BB_1, CC_1

be its altitudes. Points

A', B', C'

are chosen on the segments

AA_1, BB_1, CC_1

, respectively, so that

\angle BA'C = \angle AC'B = \angle CB'A = 90^{o}

. Let segments

AC'

and

CA'

intersect at

B"

; points

A", C"

A'B"C'A"B'C"

is circumscribed.

4

1

Russian graph with minimal number of edges satisfying a specific condition

Let

n>k>1

be positive integers and let

G

be a graph with

n

vertices such that among any

k

vertices, there is a vertex connected to the rest

k-1

vertices. Find the minimal possible number of edges of

G

.
Proposed by V. Dolnikov

2023 Caucasus Mathematical Olympiad

Subcontests

Sasha has $10$ cards with numbers $1, 2, 4, 8,\ldots, 512$

Find all $n$ such that after $n-1$ moves it is possible to obtain $0$.

\gcd(a, b) + lcm(a, b) = \gcd(a, c) + lcm(a, c)

Find the greatest $n$

ne can find a rectangle $1\times k$ in which the sum of the numbers equals $0$

The numbers $1, 2, 3,\ldots, 2\underbrace{00\ldots0}_{100 \text{ zeroes}}2$ are

Prove that this hexagon can be covered by a triangle with perimeter at most $3$.

the product of any $n$ odd consecutive positive integers is divisible by $45$.

Prove that if $q+q^{'} =r +r^{'}$, then $2n$ is a perfect square.

Prove that the triangle $A^{'}B^{'}C^{'}$ is equilateral.

Cute geometry

Russian graph with minimal number of edges satisfying a specific condition

2023 Caucasus Mathematical Olympiad

Subcontests

Sasha has $10$ cards with numbers $1, 2, 4, 8,\ldots, 512$

Find all $n$ such that after $n-1$ moves it is possible to obtain $0$.

\gcd(a, b) + lcm(a, b) = \gcd(a, c) + lcm(a, c)

Find the greatest $n$

ne can find a rectangle $1\times k$ in which the sum of the numbers equals $0$

The numbers $1, 2, 3,\ldots, 2\underbrace{00\ldots0}_{100 \text{ zeroes}}2$ are

Prove that this hexagon can be covered by a triangle with perimeter at most $3$.

the product of any $n$ odd consecutive positive integers is divisible by $45$.

Prove that if $q+q^{&#039;} =r +r^{&#039;}$, then $2n$ is a perfect square.

Prove that the triangle $A^{&#039;}B^{&#039;}C^{&#039;}$ is equilateral.

Cute geometry

Russian graph with minimal number of edges satisfying a specific condition

Prove that if $q+q^{'} =r +r^{'}$, then $2n$ is a perfect square.

Prove that the triangle $A^{'}B^{'}C^{'}$ is equilateral.