MathDB

1

z must have a factor congruent to 3 (mod 4)

x

and

y

be positive integers and assume that

z=\frac{4xy}{x+y}

is an odd integer. Prove that at least one divisor of

z

can be expressed in the form

4n-1

where

n

is a positive integer.

17

1

Find all a which generates squares in the sequence

A sequence

(x_n)_{n\ge 0}

is defined as follows:

x_0=a,x_1=2

and

x_n=2x_{n-1}x_{n-2}-x_{n-1}-x_{n-2}+1

for all

n>1

. Find all integers

a

such that

2x_{3n}-1

is a perfect square for all

n\ge 1

.

15

1

Prove that XY passes through H

Let the lines

e

and

f

be perpendicular and intersect each other at

H

. Let

A

and

B

lie on

e

and

C

and

D

lie on

f

, such that all five points

A,B,C,D

and

H

are distinct. Let the lines

b

and

d

pass through

B

and

D

respectively, perpendicularly to

AC

; let the lines

a

and

c

pass through

A

and

C

respectively, perpendicularly to

BD

. Let

a

and

b

intersect at

X

and

c

and

d

intersect at

Y

. Prove that

XY

passes through

H

.

14

1

Determine the angle PGQ

Let the medians of the triangle

ABC

meet at

G

. Let

D

and

E

be different points on the line

BC

such that

DC=CE=AB

, and let

P

and

Q

be points on the segments

BD

and

BE

, respectively, such that

2BP=PD

and

2BQ=QE

. Determine

\angle PGQ

.

12

1

AD and BC meet MN at P and Q

Let

ABCD

be a convex quadrilateral such that

BC=AD

. Let

M

and

N

be the midpoints of

AB

and

CD

, respectively. The lines

AD

and

BC

meet the line

MN

at

P

and

Q

, respectively. Prove that

CQ=DP

.

11

1

Line through circumcentres forms an isosceles triangle

Let the points

D

and

E

lie on the sides

BC

and

AC

, respectively, of the triangle

ABC

, satisfying

BD=AE

. The line joining the circumcentres of the triangles

ADC

and

BEC

meets the lines

AC

and

BC

at

K

and

L

, respectively. Prove that

KC=LC

.

16

1

p divides q - 1

Let

n

be a positive integer, let

p

be prime and let

q

be a divisor of

(n + 1)^p - n^p

. Show that

p

divides

q - 1

.

13

1

The smallest number of circles

What the smallest number of circles of radius

\sqrt{2}

that are needed to cover a rectangle

(a)

of size

6\times 3

?

(b)

of size

5\times 3

?

20

1

Find all positive integers

n=p_1p_2 \cdots p_k

which divide

(p_1+1)(p_2+1)\cdots (p_k+1)

where

p_1 p_2 \cdots p_k

is the factorization of

n

into prime factors (not necessarily all distinct).

19

1

2005 square number

Is it possible to find

2005

different positive square numbers such that their sum is also a square number ?

10

1

Divisors of 30030

Let

m = 30030

and let

M

be the set of its positive divisors which have exactly

2

prime factors. Determine the smallest positive integer

n

with the following property: for any choice of

n

numbers from

M

, there exist 3 numbers

a

,

b

,

c

among them satisfying

abc=m

.

9

1

A rectangle is divided into 200 x 3 unit squares

A rectangle is divided into

200\times 3

unit squares. Prove that the number of ways of splitting this rectangle into rectangles of size

1\times 2

is divisible by

3

.

8

1

25 x 25 grid of unit squares

Consider a

25 \times 25

grid of unit squares. Draw with a red pen contours of squares of any size on the grid. What is the minimal number of squares we must draw in order to colour all the lines of the grid?

7

1

A n x 6 - array

A rectangular array has

n

rows and

6

columns, where

n \geq 2

. In each cell there is written either

0

or

1

. All rows in the array are different from each other. For each two rows

(x_{1},x_{2},x_{3},x_{4},x_{5},x_{6})

and

(y_{1},y_{2},y_{3},y_{4},y_{5},y_{6})

, the row

(x_{1}y_{1},x_{2}y_{2},x_{3}y_{3},x_{4}y_{4},x_{5}y_{5},x_{6}y_{6})

can be found in the array as well. Prove that there is a column in which at least half of the entries are zeros.

6

1

Number of operations not greater than (2n/k)².

Let

N

and

K

be positive integers satisfying

1 \leq K \leq N

. A deck of

N

different playing cards is shuffled by repeating the operation of reversing the order of

K

topmost cards and moving these to the bottom of the deck. Prove that the deck will be back in its initial order after a number of operations not greater than

(2N/K)^2

.

4

1

Find three polynomials

Find three different polynomials

P(x)

with real coefficients such that

P\left(x^2 + 1\right) = P(x)^2 + 1

for all real

x

.

3

1

Strange sequence

Consider the sequence

\{a_k\}_{k \geq 1}

defined by

a_1 = 1

,

a_2 = \frac{1}{2}

and

a_{k + 2} = a_k + \frac{1}{2}a_{k + 1} + \frac{1}{4a_ka_{k + 1}}\ \textrm{for}\ k \geq 1.

Prove that

\frac{1}{a_1a_3} + \frac{1}{a_2a_4} + \frac{1}{a_3a_5} + \cdots + \frac{1}{a_{98}a_{100}} < 4.

2

1

Trigonometric inequality

Let

\alpha

,

\beta

and

\gamma

be three acute angles such that

\sin \alpha+\sin \beta+\sin \gamma = 1

. Show that

\tan^{2}\alpha+\tan^{2}\beta+\tan^{2}\gamma \geq \frac{3}{8}.

1

Can all terms of the sequence be distinct?

Let

a_0

be a positive integer. Define the sequence

\{a_n\}_{n \geq 0}

as follows: if

a_n = \sum_{i = 0}^jc_i10^i

where

c_i \in \{0,1,2,\cdots,9\}

, then

a_{n + 1} = c_0^{2005} + c_1^{2005} + \cdots + c_j^{2005}.

Is it possible to choose

a_0

such that all terms in the sequence are distinct?

5

1

inequality with abc=1

Let

a

,

b

,

c

be positive real numbers such that

abc=1

. Prove that

\frac a{a^{2}+2}+\frac b{b^{2}+2}+\frac c{c^{2}+2}\leq 1

2005 Baltic Way

Subcontests

z must have a factor congruent to 3 (mod 4)

Find all a which generates squares in the sequence

Prove that XY passes through H

Determine the angle PGQ

AD and BC meet MN at P and Q

Line through circumcentres forms an isosceles triangle

p divides q - 1

The smallest number of circles

Find all positive integers

2005 square number

Divisors of 30030

A rectangle is divided into 200 x 3 unit squares

25 x 25 grid of unit squares

A n x 6 - array

Number of operations not greater than (2n/k)².

Find three polynomials

Strange sequence

Trigonometric inequality

Can all terms of the sequence be distinct?

inequality with abc=1