MathDB

1

Exactly two balanced numbers among n, n+1, n+2 and n+3

n\ge 1

is called balanced if it has an even number of distinct prime divisors. Prove that there exist infinitely many positive integers

n

such that there are exactly two balanced numbers among

n,n+1,n+2

and

n+3

.

19

1

Arithmetic sequence of p terms with product equal to a cube

Let

p\neq 3

be a prime number. Show that there is a non-constant arithmetic sequence of positive integers

x_1,x_2,\ldots ,x_p

such that the product of the terms of the sequence is a cube.

18

1

Good old difference of two squares

Determine all pairs

(p,q)

of primes for which both

p^2+q^3

and

q^2+p^3

are perfect squares.

17

1

d divides any integer obtained by rearranging the digits

Determine all positive integers

d

such that whenever

d

divides a positive integer

n

,

d

will also divide any integer obtained by rearranging the digits of

n

.

16

1

Maximum exponent of prime dividing x_{2011}-1

Let

a

be any integer. Define the sequence

x_0,x_1,\ldots

by

x_0=a

,

x_1=3

, and for all

n>1

x_n=2x_{n-1}-4x_{n-2}+3.

Determine the largest integer

k_a

for which there exists a prime

p

such that

p^{k_a}

divides

x_{2011}-1

.

15

1

Angle EBA is equal to Angle DCB

Let

ABCD

be a convex quadrilateral such that

\angle ADB=\angle BDC

. Suppose that a point

E

on the side

AD

satisfies the equality

AE\cdot ED + BE^2=CD\cdot AE.

Show that

\angle EBA=\angle DCB

.

14

1

CH is parallel to AB

The incircle of a triangle

ABC

touches the sides

BC,CA,AB

at

D,E,F

, respectively. Let

G

be a point on the incircle such that

FG

is a diameter. The lines

EG

and

FD

intersect at

H

. Prove that

CH\parallel AB

.

13

1

Constructing triangles holding many similarities

Let

E

be an interior point of the convex quadrilateral

ABCD

. Construct triangles

\triangle ABF,\triangle BCG,\triangle CDH

and

\triangle DAI

on the outside of the quadrilateral such that the similarities

\triangle ABF\sim\triangle DCE,\triangle BCG\sim \triangle ADE,\triangle CDH\sim\triangle BAE

and

\triangle DAI\sim\triangle CBE

hold. Let

P,Q,R

and

S

be the projections of

E

on the lines

AB,BC,CD

and

DA

, respectively. Prove that if the quadrilateral

PQRS

is cyclic, then

EF\cdot CD=EG\cdot DA=EH\cdot AB=EI\cdot BC.

12

1

Determine the angle BPA

Let

P

be a point inside a square

ABCD

such that

PA:PB:PC

is

1:2:3

. Determine the angle

\angle BPA

.

11

1

Length of RS is independent on P

Let

AB

and

CD

be two diameters of the circle

C

. For an arbitrary point

P

on

C

, let

R

and

S

be the feet of the perpendiculars from

P

to

AB

and

CD

, respectively. Show that the length of

RS

is independent of the choice of

P

.

10

1

Players wanting to obtain first non-positIve integer to win

Two persons play the following game with integers. The initial number is

2011^{2011}

. The players move in turns. Each move consists of subtraction of an integer between

1

and

2010

inclusive, or division by

2011

, rounding down to the closest integer when necessary. The player who first obtains a non-positive integer wins. Which player has a winning strategy?

9

1

Which m x n grids have a valid colouring?

Given a rectangular grid, split into

m\times n

squares, a colouring of the squares in two colours (black and white) is called valid if it satisfies the following conditions:
[*]All squares touching the border of the grid are coloured black. [*]No four squares forming a

2\times 2

square are coloured in the same colour. [*]No four squares forming a

2\times 2

square are coloured in such a way that only diagonally touching squares have the same colour. Which grid sizes

m\times n

(with

m,n\ge 3

) have a valid colouring?

8

1

Student from one school knows all students from next school

In Greifswald there are three schools called

A,B

and

C

, each of which is attended by at least one student. Among any three students, one from

A

, one from

B

and one from

C

, there are two knowing each other and two not knowing each other. Prove that at least one of the following holds:
[*]Some student from

A

knows all students from

B

. [*]Some student from

B

knows all students from

C

. [*] Some student from

C

knows all students from

A

.

7

1

Largest possible size of subset S of T

Let

T

denote the

15

-element set

\{10a+b:a,b\in\mathbb{Z},1\le a<b\le 6\}

. Let

S

be a subset of

T

in which all six digits

1,2,\ldots ,6

appear and in which no three elements together use all these six digits. Determine the largest possible size of

S

.

6

1

Number of lines through origin and precisely one other point

Let

n

be a positive integer. Prove that the number of lines which go through the origin and precisely one other point with integer coordinates

(x,y),0\le x,y\le n

, is at least

\frac{n^2}{4}

.

5

1

Determine f(0) if given f(f(x))

Let

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

be a function such that

f(f(x))=x^2-x+1

for all real numbers

x

. Determine

f(0)

.

4

1

Inequality for a+b+c+d=4

Let

a,b,c,d

be non-negative reals such that

a+b+c+d=4

. Prove the inequality

\frac{a}{a^3+8}+\frac{b}{b^3+8}+\frac{c}{c^3+8}+\frac{d}{d^3+8}\le\frac{4}{9}

3

1

Does sequence become periodic?

A sequence

a_1,a_2,a_3,\ldots

of non-negative integers is such that

a_{n+1}

is the last digit of

a_n^n+a_{n-1}

for all

n>2

. Is it always true that for some

n_0

the sequence

a_{n_0},a_{n_0+1},a_{n_0+2},\ldots

is periodic?

2

1

Prove that f is bounded if equation holds

Let

f:\mathbb{Z}\to\mathbb{Z}

be a function such that for all integers

x

and

y

, the following holds:

f(f(x)-y)=f(y)-f(f(x)).

Show that

f

is bounded.

1

Prove that all x_i are equal

The real numbers

x_1,\ldots ,x_{2011}

satisfy

x_1+x_2=2x_1',\ x_2+x_3=2x_2', \ \ldots, \ x_{2011}+x_1=2x_{2011}'

where

x_1',x_2',\ldots,x_{2011}'

is a permutation of

x_1,x_2,\ldots,x_{2011}

. Prove that

x_1=x_2=\ldots =x_{2011}

.

2011 Baltic Way

Subcontests

Exactly two balanced numbers among n, n+1, n+2 and n+3

Arithmetic sequence of p terms with product equal to a cube

Good old difference of two squares

d divides any integer obtained by rearranging the digits

Maximum exponent of prime dividing x_{2011}-1

Angle EBA is equal to Angle DCB

CH is parallel to AB

Constructing triangles holding many similarities

Determine the angle BPA

Length of RS is independent on P

Players wanting to obtain first non-positIve integer to win

Which m x n grids have a valid colouring?

Student from one school knows all students from next school

Largest possible size of subset S of T

Number of lines through origin and precisely one other point

Determine f(0) if given f(f(x))

Inequality for a+b+c+d=4

Does sequence become periodic?

Prove that f is bounded if equation holds

Prove that all x_i are equal