MathDB

1

Broken line construction with a condition

Call the intersection of two segments almost perfect if for each of the segments the distance between the midpoint of the segment and the intersection is at least

2022

times smaller than the length of the segment. Prove that there exists a closed broken line of segments such that every segment intersects at least one other segment, and every intersection of segments is almost perfect.

P7

1

Circular colored graph, minimum of colors needed

A kingdom has

2021

towns. All of the towns lie on a circle, and there is a one-way road going from every town to the next

101

towns in a clockwise order. Each road is colored in one color. Additionally, it is known that for any ordered pair of towns

A

and

B

it is possible to find a path from

A

to

B

so that no two roads of the path would have the same color. Find the minimal number of road colors in the kingdom.

P6

1

Game of picking numbers ending in a coprime pair

The numbers

1,2,3,\ldots ,n

are written in a row. Two players, Maris and Filips, take turns making moves with Maris starting. A move consists of crossing out a number from the row which has not yet been crossed out. The game ends when there are exactly two uncrossed numbers left in the row. If the two remaining uncrossed numbers are coprime, Maris wins, otherwise Filips is the winner. For each positive integer

n\ge 4

determine which player can guarantee a win.

P5

1

White and black board process to get one color

Let

n \ge 2

be a positive integer. An

n\times n

grid of squares has been colored as a chessboard. Let a move consist of picking a square from the board and then changing the colors to the opposite for all squares that lie in the same row as the chosen square, as well as for all squares that lie in the same column (the chosen square itself is also changed to the opposite color). Find all values of

n

for which it is possible to make all squares of the grid be the same color in a finite sequence of moves.

P12

1

Incentre configuration with parallel lines desired

Let

\triangle ABC

be a triangle with incentre

I

. Points

E

and

F

are the tangency points of the incircle and the sides

AC

and

AB

, respectively. Suppose that the lines

BI

and

CI

intersect the line

EF

at

Y

and

Z

, respectively. Let

M

denote the midpoint of the segment

BC

, and

N

denote the midpoint of the segment

YZ

. Prove that

AI \parallel MN

.

P11

1

Prove that two tangents intersect on BC

Let

\triangle ABC

be an acute triangle. Point

D

is arbitrarily chosen on the side

BC

. Let the circumcircle of the triangle

\triangle ADB

intersect the segment

AC

at

M

, and the circumcircle of the triangle

\triangle ADC

intersect the segment

AB

at

N

. Prove that the tangents of the circumcircle of the triangle

\triangle AMN

at

M

and

N

intersect at a point that lies on the line

BC

.

P10

1

Triangle geometry with a point satisfying length condition

Let

\triangle ABC

be a triangle satisfying

AB<AC

. Let

D

be a point on the segment

AC

such that

AB=AD

. Let then

X

be a point on the segment

BC

satisfying

BD^2=BX\cdot BC

. Let the circumcircles of the triangles

\triangle XDC

and

\triangle ABC

intersect at

M \neq C

. Prove that the line

MD

goes through the midpoint of the arc

\widehat{BAC}

of the circumcircle of

\triangle ABC

.

P9

1

Easy geometry with lines intersecting circles

Let

ABCD

be a cyclic quadrilateral inscribed in circle

\Omega

. Let the lines

AB

and

CD

intersect at

P

, and the lines

AD

and

BC

intersect at

Q

. Let then the circumcircle of the triangle

\triangle APQ

intersect

\Omega

at

R \neq A

. Prove that the line

CR

goes through the midpoint of the segment

PQ

.

P16

1

Two expressions are powers of a prime

Find all triples of positive integers

(a,b,p)

, where

p

is a prime, such that both

a+b

and

ab+1

are some powers of

p

(not necessarily the same).

P14

1

Picking numbers from a set to satisfy congruence

Let

A

be a set of

20

distinct positive integers which are all no greater than

397

. Prove that for any positive integer

n

it is possible to pick four (not necessarily distinct) elements

x_1, x_2, x_3, x_4

of

A

satisfying

x_1 \neq x_2

and

(x_1-x_2)n\equiv x_3-x_4 \pmod{397}.

P4

1

Find a polynomial with n-th power variable

Let

p(x)

be a polynomial with real coefficients and

n

be a positive integer. Prove that there exists a non-zero polynomial

q(x)

with real coefficients such that the polynomial

p(x)\cdot q(x)

has non-zero coefficients only by the powers which are multiples of

n

.

P2

1

Classical inequality

Prove that for positive real numbers

a,b,c

satisfying

abc=1

the following inequality holds:

\frac{a}{b}+\frac{b}{c}+\frac{c}{a} \ge \frac{a^2+1}{2a}+\frac{b^2+1}{2b}+\frac{c^2+1}{2c}.

P3

1

Standard looking FE

Let

\mathbb R

be the set of real numbers. Determine all functions

f:\mathbb R\to\mathbb R

that satisfy the equation

f(f(x))+yf(xy+1) = f(x-f(y)) + xf(y)^2.

for all real numbers

x

and

y

.

P1

1

System of equations with powers

Find all triplets of positive real numbers

(x,y,z)

that satisfy the following system of equations:

\begin{cases} x+y^2+z^3=3\\ y+z^2+x^3=3\\ z+x^2+y^3=3. \end{cases}

2022 Latvia Baltic Way TST

Subcontests

Broken line construction with a condition

Circular colored graph, minimum of colors needed

Game of picking numbers ending in a coprime pair

White and black board process to get one color

Incentre configuration with parallel lines desired

Prove that two tangents intersect on BC

Triangle geometry with a point satisfying length condition

Easy geometry with lines intersecting circles

Two expressions are powers of a prime

Picking numbers from a set to satisfy congruence

Find a polynomial with n-th power variable

Classical inequality

Standard looking FE

System of equations with powers