MathDB

1

NT function with primes

f:\mathbb{N} \to \mathbb{N}

is given. If

a,b

are coprime, then

f(ab)=f(a)f(b)

. Also, if

m,k

are primes (not necessarily different), then

f(m+k-3)=f(m)+f(k)-f(3).

Find all possible values of

f(11)

.

P15

1

Most trivial TST NT ever seen

Denote by

s(n)

the sum of all natural divisors of

n

which are smaller than

n

. Does there exist a positive integer

a

such that the equation

s(n)=a+n

has infinitely many solutions in positive integers?

P14

1

Infinitely many solutions to floor equation

Prove that there exist infinitely many triples of positive integers

(a,b,c)

so that

a,b,c

are pairwise coprime and

\bigg \lfloor \frac{a^2}{2021} \bigg \rfloor + \bigg \lfloor \frac{b^2}{2021} \bigg \rfloor = \bigg \lfloor \frac{c^2}{2021} \bigg \rfloor.

P13

1

Standard technique power NT

Does there exist a natural number

a

so that: a)

\Big ((a^2-3)^3+1\Big) ^a-1

is a perfect square? b)

\Big ((a^2-3)^3+1\Big) ^{a+1}-1

is a perfect square?

P12

1

Weird length conditions

Five points

A,B,C,P,Q

are chosen so that

A,B,C

aren't collinear. The following length conditions hold:

\frac{AP}{BP}=\frac{AQ}{BQ}=\frac{21}{20}

and

\frac{BP}{CP}=\frac{BQ}{CQ}=\frac{20}{19}

. Prove that line

PQ

goes through the circumcentre of

\triangle ABC

.

P11

1

Incentre with perpendicularity

Incircle of

\triangle ABC

has centre

I

and touches sides

AC, AB

at

E,F

, respectively. The perpendicular bisector of segment

AI

intersects side

AC

at

P

. On side

AB

a point

Q

is chosen so that

QI \perp FP

. Prove that

EQ \perp AB

.

P10

1

Circle with lots of symmetry

Circle

\omega

with centre

M

and diameter

XY

is given. Point

A

is chosen on

\omega

so that

AX<AY

. Points

B, C

are chosen on segments

XM, YM

, respectively, in a way that

BM=CM

. A parallel line to

AB

is constructed through

C

; the line intersects

\omega

at

P

so that

P

lies on the smaller arc

\widehat{AY}

. Similarly, a parallel line to

AC

is constructed through

B

; the line intersects

\omega

at

Q

so that

Q

lies on the smaller arc

\widehat{XA}

. Lines

PQ

and

XY

intersect at

S

. Prove that

AS

is tangent to

\omega

.

P9

1

Cyclic pentagon

Pentagon

ABCDE

with

CD\parallel BE

is inscribed in circle

\omega

. Tangent to

\omega

through

B

intersects line

AC

at

F

in a way that

A

lies between

C

and

F

. Lines

BD

and

AE

intersect at

G

. Prove that

FG

is tangent to the circumcircle of

\triangle ADG

.

P4

1

Inequality with lattice points

Determine the smallest positive constant

k

such that no matter what

3

lattice points we choose the following inequality holds:

L_{\max} - L_{\min} \ge \frac{1}{\sqrt{k} \cdot L_{max}}

where

L_{\max}

,

L_{\min}

is the maximal and minimal distance between chosen points.

P7

1

Football training

22

football players took part in the football training. They were divided into teams of equal size for each game (

11:11

). It is known that each football player played with each other at least once in opposing teams. What is the smallest possible number of games they played during the training.

P5

1

Lines in the space

Six lines are drawn in the plane. Determine the maximum number of points, through which at least

3

lines pass.

P8

1

Easy process problem

Initially on the blackboard eight zeros are written. In one step, it is allowed to choose numbers

a,b,c,d

, erase them and replace them with the numbers

a+1

,

b+2

,

c+3

,

d+3

. Determine: a) the minimum number of steps required to achieve

8

consecutive integers on the board b) whether it is possible to achieve that sum of the numbers is

2021

c) whether it is possible to achieve that product of the numbers is

2145

P6

1

Combi construction problem

Let's call

1 \times 2

rectangle, which can be a rotated, a domino. Prove that there exists polygon, who can be covered by dominoes in exactly

2021

different ways.

P3

1

System of equations

Find all triplets of real numbers

(x,y,z)

such that the following equations are satisfied simultaneously: \begin{align*} x^3+y=z^2 \\ y^3+z=x^2 \\ z^3+x =y^2 \end{align*}

P1

1

Easy inequality

Prove that for positive real numbers

a,b,c

satisfying

abc=1

the following inequality holds:

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

P2

1

Functional inequality

Determine all functions

f: \mathbb{R} \backslash \{0 \} \rightarrow \mathbb{R}

such that, for all nonzero

x

:

f(\frac{1}{x}) \ge 1 -f(x) \ge x^2f(x)

2021 Latvia Baltic Way TST

Subcontests

NT function with primes

Most trivial TST NT ever seen

Infinitely many solutions to floor equation

Standard technique power NT

Weird length conditions

Incentre with perpendicularity

Circle with lots of symmetry

Cyclic pentagon

Inequality with lattice points

Football training

Lines in the space

Easy process problem

Combi construction problem

System of equations

Easy inequality

Functional inequality