2022 Balkan MO Shortlist

Part of Balkan MO Shortlist

Subcontests

(17)

1

Interesting NT

A hare and a tortoise run in the same direction, at constant but different speeds, around the base of a tall square tower. They start together at the same vertex, and the run ends when both return to the initial vertex simultaneously for the first time. Suppose the hare runs with speed 1, and the tortoise with speed less than 1. For what rational numbers

q{}

is it true that, if the tortoise runs with speed

q{}

, the fraction of the entire run for which the tortoise can see the hare is also

q{}

1

Product of digits NT

For every natural number

x{}

P(x)

be the product of the digits of the number

x{}

. Is there a natural number

n{}

such that the numbers

P(n)

P(n^2)

are non-zero squares of natural numbers, where the number of digits of the number

n{}

is equal to (a) 2021 and (b) 2022?

1

Sad casework NT

n{}

be a positive integer. What is the smallest sum of digits that

5^n + 6^n + 2022^n

1

Bash geo

ABC

be a triangle with

AB < AC

D{}

be the other intersection point of the angle bisector of

\angle A

with the circumcircle of the triangle

ABC

E{}

F{}

be points on the sides

AB

AC

respectively, such that

AE = AF

P{}

be the point of intersection of

AD

EF

M{}

be the midpoint of

BC{}

AM

and the circumcircles of the triangles

AEF

PMD

pass through a common point.

1

Circle geo

ABC

be a triangle with circumcircle

\omega

O{}

, and orthocenter

H{}

K{}

be the midpoint of

AH{}

. The perpendicular to

OK{}

K{}

AB{}

AC{}

P{}

Q{}

, respectively. The lines

BK

CK

\omega

X{}

Y{}

, respectively. Prove that the second intersection of the circumcircles of triangles

KPY

KQX

\omega

.
Stefan Lozanovski

1

Tangency geo

ABC

be a triangle and let the tangent at

B{}

to its circumcircle meet the internal bisector of the angle

A{}

P{}

. The line through

P{}

AC

AB

Q{}

Q{}

lies in the interior of segment

AB

and let the line through

Q{}

BC

AC

X{}

PC

Y{}

PX

is tangent to the circumcircle of the triangle

XYC

1

Long geo

ABC

a triangle and let

\omega

be its circumcircle. Let

E{}

be the midpoint of the minor arc

BC

\omega

M{}

the midpoint of

BC

V

be the other point of intersection of

AM

\omega

F{}

the point of intersection of

AE

BC

X{}

the other point of intersection of the circumcircle of

FEM

\omega

X'

the reflection of

V{}

with respect to

M{}

A'{}

the foot of the perpendicular from

A{}

BC

S{}

the other point of intersection of

XA'

\omega

Z \in \omega

Z\neq X

AX = AZ

, then prove that

S, X'

Z{}

1

Incenter geo

ABC

be a triangle with

AB > AC

I{}

. The internal bisector of the angle

BAC

BC

D{}

M{}

the midpoint of the segment

AD{}

F{}

be the second intersection point of

MB

with the circumcircle of the triangle

BIC

AF

is perpendicular to

FC

1

Basically a graph

Find the largest positive integer

k{}

for which there exists a convex polyhedron

\mathcal{P}

with 2022 edges, which satisfies the following properties:
[*]The degrees of the vertices of

\mathcal{P}

don’t differ by more than one, and [*]It is possible to colour the edges of

\mathcal{P}

k{}

colours such that for every colour

c{}

, and every pair of vertices

(v_1, v_2)

\mathcal{P}

, there is a monochromatic path between

v_1

v_2

c{}

.
Viktor Simjanoski, Macedonia

1

Beautiful combinatorial geometry

Alice is drawing a shape on a piece of paper. She starts by placing her pencil at the origin, and then draws line segments of length one, alternating between vertical and horizontal segments. Eventually, her pencil returns to the origin, forming a closed, non-self-intersecting shape. Show that the area of this shape is even if and only if its perimeter is a multiple of eight.

1

Operations on 100 positive integers

There are 100 positive integers written on a board. At each step, Alex composes 50 fractions using each number written on the board exactly once, brings these fractions to their irreducible form, and then replaces the 100 numbers on the board with the new numerators and denominators to create 100 new numbers.
Find the smallest positive integer

n{}

such that regardless of the values of the initial 100 numbers, after

n{}

steps Alex can arrange to have on the board only pairwise coprime numbers.

1

Strengthened Putnam 2009

Determine all functions

f : \mathbb{R}^2 \to\mathbb {R}

f(A)+f(B)+f(C)+f(D)=0,

A,B,C,D

are the vertices of a square with side-length one.
Ilir Snopce

1

Wrapped FE

Find all functions

f : \mathbb{R} \to\mathbb{R}

f(0)\neq 0

f(f(x)) + f(f(y)) = f(x + y)f(xy),

x, y \in\mathbb{R}

1

Inequality with strange condition

a, b, c, d

be non-negative real numbers such that

\frac{1}{a+1}+\frac{1}{b+1}+\frac{1}{c+1}+\frac{1}{d+1}=3.

3(ab+bc+ca+ad+bd+cd)+\frac{4}{a+b+c+d}\leqslant 5.

Vasile Cîrtoaje and Leonard Giugiuc

1

Inequality with n terms

k > 1{}

be a real number,

n\geqslant 3

be an integer, and

x_1 \geqslant x_2\geqslant\cdots\geqslant x_n

be positive real numbers. Prove that

\frac{x_1+kx_2}{x_2+x_3}+\frac{x_2+kx_3}{x_3+x_4}+\cdots+\frac{x_n+kx_1}{x_1+x_2}\geqslant\frac{n(k+1)}{2}.

Ilija Jovcheski

1

Simple FE

Find all functions

f : \mathbb{R} \to \mathbb{R}

f(x(x + f(y))) = (x + y)f(x),

x, y \in\mathbb{R}

1

Tiling a cube

Given is a cube of side length

2021

. In how many different ways is it possible to add somewhere on the boundary of this cube a

1\times 1\times 1

cube in such a way that the new shape can be filled in with

1\times 1\times k

shapes, for some natural number

k

k\geq 2