MathDB

1

Magic combi with numbered cards

300

cards with numbers from

1

to

300

written on them, each number on exactly one card. The magician then lays these cards on a

3 \times 100

rectangle in the following way - one card in each unit square so that the number cannot be seen and cards with consecutive numbers are in neighbouring squares. Afterwards, the magician turns over

k

cards of his choice. What is the smallest value of

k

for which it can happen that the opened cards definitely determine the exact positions of all other cards?

7

1

Old and classical combi

Natural numbers from

1

to

400

are divided in

100

disjoint sets. Prove that one of the sets contains three numbers which are lengths of a non-degenerate triangle's sides.

6

1

Blink and you miss the solution

For a natural number

n \ge 3

we denote by

M(n)

the minimum number of unit squares that must be coloured in a

6 \times n

rectangle so that any possible

2 \times 3

rectangle (it can be rotated, but it must be contained inside and cannot be cut) contains at least one coloured unit square. Is it true that for every natural

n \ge 3

the number

M(n)

can be expressed as

M(n)=p_n+k_n^3

, where

p_n

is a prime and

k_n

is a natural number?

5

1

Combinatorial game with number choosing

Natural numbers

1,2,...,500

are written on a blackboard. Two players

A

and

B

consecutively make moves,

A

starts. Each move a player chooses two numbers

n

and

2n

and erases them from the blackboard. If a player cannot perform a valid move, he loses. Which player can guarantee a win?

10

1

Orthocenter and reflections

Given

\triangle ABC

and it's orthocenter

H

. Point

P

is arbitrary chosen on the side

BC

. Let

Q

and

R

be reflections of point

P

over sides

AB, AC

. It is given that points

Q,H,R

are collinear. Prove that

\triangle ABC

is right angled.

11

1

Brocard's theorem FTW

Circle centred at point

O

intersects sides

AC, AB

of triangle

\triangle ABC

at points

B_1

and

C_1

respectively and passes through points

B,C

. It is known that lines

AO, CC_1, BB_1

are concurrent. Prove that

\triangle ABC

is isosceles.

9

1

Existence of circle

Given

\triangle ABC

, whose all sides have different length. Point

P

is chosen on altitude

AD

. Lines

BP

and

CP

intersect lines

AC, AB

respectively and point

X, Y

.It is given that

AX=AY

. Prove that there is circle, whose centre lies on

BC

and is tangent to sides

AC

and

AB

at points

X,Y

.

12

1

Rhombus and 2 circles

There are rhombus

ABCD

and circle

\Gamma_B

, which is centred at

B

and has radius

BC

, and circle

\Gamma_C

, which is centred at

C

and has radius

BC

. Circles

\Gamma_B

and

\Gamma_C

intersect at point

E

. The line

ED

intersects

\Gamma_B

at point

F

. Find all possible values of

\angle AFB

.

14

1

Infinitely many lattice points

Prove that there are infinitely many different triangles in coordinate plane satisfying: 1) their vertices are lattice points 2) their side lengths are consecutive integers Remark: Triangles that can be obtained by rotation or translation or shifting the coordinate system are considered as equal triangles

16

1

Sequence with gcd and lcm

Given sequence

\{a_n\}

satisfying:

a_{n+1} = \frac{ lcm(a_n,a_{n-1})}{\gcd(a_n, a_{n-1})}

It is given that

a_{209} =209

and

a_{361} = 361

. Find all possible values of

a_{2020}

.

15

1

Prime number and perfect cube

Let

p

be a prime. Prove that

p^2+p+1

is never a perfect cube.

13

1

Modified well - known NT problem

It is given that

n

and

\sqrt{12n^2+1}

are both positive integers. Prove that:

\sqrt{ \frac{\sqrt{12n^2+1}+1}{2}}

is also positive integer.

4

1

Cubic poly with irrational roots

Given cubic polynomial with integer coefficients and three irrational roots. Show that none of these roots can be root of quadratic equation with integer coefficients.

2

1

Asymmetric FE over reals

Determine all functions

f:\mathbb R\to\mathbb R

that satisfy equation:

f(x^3+y^3) =f(x^3) + 3x^2f(x)f(y) + 3f(x)f(y)^2 + y^6f(y)

for all reals

x,y

3

1

Similar to harmonic series, but with 2020

Prove that for all positive integers

n

the following inequality holds:

\frac{1}{1^2 +2020}+\frac{1}{2^2+2020} + \ldots + \frac{1}{n^2+2020} < \frac{1}{22}

1

Symmetric inequality FTW

Prove that for positive reals

a,b,c

satisfying

a+b+c=3

the following inequality holds:

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

2020 Latvia Baltic Way TST

Subcontests

Magic combi with numbered cards

Old and classical combi

Blink and you miss the solution

Combinatorial game with number choosing

Orthocenter and reflections

Brocard's theorem FTW

Existence of circle

Rhombus and 2 circles

Infinitely many lattice points

Sequence with gcd and lcm

Prime number and perfect cube

Modified well - known NT problem

Cubic poly with irrational roots

Asymmetric FE over reals

Similar to harmonic series, but with 2020

Symmetric inequality FTW