MathDB

1

subsequences of permutations of 1-n with no divisors of n

n

, for which we can find a prime number less than

\sqrt{n}

that is not a divisor of

n

. The sequence

a_1, a_2,... ,a_n

is the numbers

1, 2,... ,n

arranged in some order. For this sequence, we will find the longest ascending subsequense

a_{i_1} < a_{i_2} < ... < a_{i_k}

, (

i_1 <...< i_k

) and the longest decreasing substring

a_{j_1} > ... > a_{j_l}

, (

j_1 < ... < j_l

) . Prove that at least one of these two subsequnsces

a_{i_1} , . . . , a_{i_k}

and

a_{j_1} > ... > a_{j_l}

contains a number that is not a divisor of

n

.

8

1

3n - 2 participants in chess festival

3n - 2

participants took part in the chess festival, some of them played one game of chess with each other. Prove that at least one of the following statements holds:
(A) One can find

n

chess players

A_1 , A_2 , . . . , A_n

suchthat Ai has played a game with

A_{i+1}

for all

i = 1, ...,n -1

.
(B) Seven chess players can be found in

B_1 , . . . , B_7

, who have not played with each other, except perhaps three pairs

(B_1, B_2)

,

(B_3, B_4)

and

(B_5, B_6)

, each of whom may or may not have played a game of chess.

7

1

commissions in the parliament of Nekurnekadzeme

In the parliament of Nekurnekadzeme, all activities take place in commissions, which consist of exactly three members. The constitution stipulates that any three commissions must have at least five members. We will call a family of commissions a clique if every two of them have exactly two members in common, but if any other commission is added to this family, this condition is no longer fulfilled. Prove that two different cliques cannot have more than one commission in common.

9

1

partition 1 to 2016 into 3 subsets, a+ b=c

The numbers from

1

to

2016

are divided into three (disjoint) subsets

A, B

and

C

, each one contains exactly

672

numbers. Prove that you can find three numbers, each from a different subset, such that the sum of two of them is equal to the third.
[hide=original wording]Skaitļi no 1 līdz 2016 ir sadalīti trīs (nešķeļošās) apakškopās A, B un C, katranotām satur tieši 672 skaitļus. Pierādīt, ka var atrast trīs tādus skaitļus, katru no citas apakškopas, ka divu no tiem summa ir vienāda ar trešo.

10

1

infinite 1x2 tiles cover infinite sheet ?

On an infinite sheet of tiles, an infinite number of

1 \times 2

tile rectangles are placed, their edges follow the lines of the tiles, and they do not touch each other, not even the corners. Is it true that the remaining checkered sheet can be completely covered with

1 \times 2

checkered rectangles?
[hide=original wording]Uz bezgalīgas rūtiņu lapas ir novietoti bezgaglīgi daudzi 1 x 2 rūtiņu taisnstūri, to malas iet pa rūtiņu līnijām, un tie nesaskaras cits ar citu pat ne ar stūriem. Vai tiesa, ka atlikušo rūtiņu lapu var pilnībā noklāt ar 1 x 2 rūtiņu tainstūriem?

18

1

diophantine 3x3: a^3=abc+2a+2c, b^3=abc-c , c^3=abc-a+b

Solve the system of equations in integers:

\begin{cases} a^3=abc+2a+2c \\ b^3=abc-c \\ c^3=abc-a+b \end{cases}

20

1

p^m=(a^6 + 21a^4b^2 + 35a^2b^4 + 7b^6) (b^6 + 21b^4a^2 + 35b^2a^4 + 7a^6)

For what pairs of natural numbers

(a, b)

is the expression

(a^6 + 21a^4b^2 + 35a^2b^4 + 7b^6) (b^6 + 21b^4a^2 + 35b^2a^4 + 7a^6)

the power of a prime number?

19

1

x^{2015} + y^{2015} = z^{2016}, diophantine

Prove that for equation

x^{2015} + y^{2015} = z^{2016}

there are infinitely many solutions where

x,y

and

z

are different natural numbers.

17

1

5 primes with p^3+q^3+r^3+s^3 =t^3

Can you find five prime numbers

p, q, r, s, t

such that

p^3+q^3+r^3+s^3 =t^3

?

16

1

max gcd (n^2 + 3, (n + 1)^2 + 3 )

What is the largest possible value of the expression

gcd \,\,\, (n^2 + 3, (n + 1)^2 + 3 )

for naturals

n

?
original wording]Kāda ir izteiksmes LKD (n2 + 3, (n + 1)2 + 3) lielākā iespējamā vērtība naturāliem n?

5

1

a + b + d <= 3c if a^2 + ab + b^2 = 3c^2 and a^3 + a^2b + ab^2 + b^3 = 4d^3

Given real positive numbers

a, b, c

and

d

, for which the equalities

a^2 + ab + b^2 = 3c^2

and

a^3 + a^2b + ab^2 + b^3 = 4d^3

are fulfilled. Prove that

a + b + d \le 3c.

4

1

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

Find all functions

f : R \to R

defined for real numbers, take real values and for all real

x

and

y

the equality holds:

f(2^x+2y) =2^yf(f(x))f(y).

3

1

roots of P (Q(x)) are -2015, ..., 2017 if P has degree 2016, Q is trinomial

Given a polynomial

P

of degree

2016

with real coefficients and a quadratic polynomial

Q

with real coefficients. Is it possible that the roots of the polynomial

P (Q(x))

are exactly all these numbers:

-2015, -2014, . . . , -2, -1, 1, 2, . . . , 2016, 2017?

2

1

0 <= mu + nv <= 2 \sqrt{X}

Given natural numbers

m, n

and

X

such that

X \ge m

and

X \ge n

. Prove that one can find two integers

u

and

v

such that

|u| + |v| > 0

,

|u| \le \sqrt{X}

,

|v| \le \sqrt{X}

and

0 \le mu + nv \le 2 \sqrt{X}.

1

delete 2 numbers from 1/2016 - 2/2016 and replace them with 3ab - 2a - 2b + 2

2016

numbers written on the board:

\frac{1}{2016}, \frac{2}{2016}, \frac{3}{2016}, ..., \frac{2016}{2016}

. In one move, it is allowed to choose any two numbers

a

and

b

written on the board, delete them, and write the number

3ab - 2a - 2b + 2

instead. Determine what number will remain written on the board after

2015

moves.

15

1

midpoints of AH,DK,EL,FM are concyclic if <BAC = 60^o, midpoints, altitudes

Let

ABC

be a triangle. Let its altitudes

AD

,

BE

and

CF

concur at

H

. Let

K, L

and

M

be the midpoints of

BC

,

CA

and

AB

, respectively. Prove that, if

\angle BAC = 60^o

, then the midpoints of the segments

AH

,

DK

,

EL

,

FM

are concyclic.

14

1

PC_|_AK wanted, 2 touchpoints of incircle related

Let

ABC

be a scalene triangle. Let

D

and

E

be the points where the incircle touches sides

BC

and

CA

, respectively. Let

K

be the common point of line

BC

and the bisector of the angle

\angle BAC

. Let

AD

intersect

EK

in

P

. Prove that

PC

is perpendicular to

AK

.

13

1

< BAC <=150^o if max (BX,CX) <= AX <= BC

Suppose that

A, B, C

, and

X

are any four distinct points in the plane with

\max \,(BX,CX) \le AX \le BC.

Prove that

\angle BAC \le 150^o

.

12

1

|PA_1|^2 +... + |PA_m|^2 =|PB_1|^2+...+|PA_n|^2, point P, m+n points

For what positive numbers

m

and

n

do there exist points

A_1, ..., Am

and

B_1 ..., B_n

in the plane such that, for any point

P

, the equation

|PA_1|^2 +... + |PA_m|^2 =|PB_1|^2+...+|PA_n|^2

holds true?

11

1

cut square of side \sqrt{2015} into no more than 5 pieces, integer rectangle

Is it possible to cut a square with side

\sqrt{2015}

into no more than five pieces so that these pieces can be rearranged into a rectangle with sides of integer length? (The cuts should be made using straight lines, and flipping of the pieces is disallowed.)

2016 Latvia Baltic Way TST

Subcontests

subsequences of permutations of 1-n with no divisors of n

3n - 2 participants in chess festival

commissions in the parliament of Nekurnekadzeme

partition 1 to 2016 into 3 subsets, a+ b=c

infinite 1x2 tiles cover infinite sheet ?

diophantine 3x3: a^3=abc+2a+2c, b^3=abc-c , c^3=abc-a+b

p^m=(a^6 + 21a^4b^2 + 35a^2b^4 + 7b^6) (b^6 + 21b^4a^2 + 35b^2a^4 + 7a^6)

x^{2015} + y^{2015} = z^{2016}, diophantine

5 primes with p^3+q^3+r^3+s^3 =t^3

max gcd (n^2 + 3, (n + 1)^2 + 3 )

a + b + d <= 3c if a^2 + ab + b^2 = 3c^2 and a^3 + a^2b + ab^2 + b^3 = 4d^3

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

roots of P (Q(x)) are -2015, ..., 2017 if P has degree 2016, Q is trinomial

0 <= mu + nv <= 2 \sqrt{X}

delete 2 numbers from 1/2016 - 2/2016 and replace them with 3ab - 2a - 2b + 2

midpoints of AH,DK,EL,FM are concyclic if <BAC = 60^o, midpoints, altitudes

PC_|_AK wanted, 2 touchpoints of incircle related

< BAC <=150^o if max (BX,CX) <= AX <= BC

|PA_1|^2 +... + |PA_m|^2 =|PB_1|^2+...+|PA_n|^2, point P, m+n points

cut square of side \sqrt{2015} into no more than 5 pieces, integer rectangle

2016 Latvia Baltic Way TST

Subcontests

subsequences of permutations of 1-n with no divisors of n

3n - 2 participants in chess festival

commissions in the parliament of Nekurnekadzeme

partition 1 to 2016 into 3 subsets, a+ b=c

infinite 1x2 tiles cover infinite sheet ?

diophantine 3x3: a^3=abc+2a+2c, b^3=abc-c , c^3=abc-a+b

p^m=(a^6 + 21a^4b^2 + 35a^2b^4 + 7b^6) (b^6 + 21b^4a^2 + 35b^2a^4 + 7a^6)

x^{2015} + y^{2015} = z^{2016}, diophantine

5 primes with p^3+q^3+r^3+s^3 =t^3

max gcd (n^2 + 3, (n + 1)^2 + 3 )

a + b + d &lt;= 3c if a^2 + ab + b^2 = 3c^2 and a^3 + a^2b + ab^2 + b^3 = 4d^3

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

roots of P (Q(x)) are -2015, ..., 2017 if P has degree 2016, Q is trinomial

0 &lt;= mu + nv &lt;= 2 \sqrt{X}

delete 2 numbers from 1/2016 - 2/2016 and replace them with 3ab - 2a - 2b + 2

midpoints of AH,DK,EL,FM are concyclic if &lt;BAC = 60^o, midpoints, altitudes

PC_|_AK wanted, 2 touchpoints of incircle related

&lt; BAC &lt;=150^o if max (BX,CX) &lt;= AX &lt;= BC

|PA_1|^2 +... + |PA_m|^2 =|PB_1|^2+...+|PA_n|^2, point P, m+n points

cut square of side \sqrt{2015} into no more than 5 pieces, integer rectangle

a + b + d <= 3c if a^2 + ab + b^2 = 3c^2 and a^3 + a^2b + ab^2 + b^3 = 4d^3

0 <= mu + nv <= 2 \sqrt{X}

midpoints of AH,DK,EL,FM are concyclic if <BAC = 60^o, midpoints, altitudes

< BAC <=150^o if max (BX,CX) <= AX <= BC