MathDB

1

Does there exist an angle

\alpha\in(0,\pi/2)

such that

\sin\alpha

,

\cos\alpha

,

\tan\alpha

and

\cot\alpha

, taken in some order, are consecutive terms of an arithmetic progression?

5

1

Romeo, Juliet and a game

Suppose that Romeo and Juliet each have a regular tetrahedron to the vertices of which some positive real numbers are assigned. They associate each edge of their tetrahedra with the product of the two numbers assigned to its end points. Then they write on each face of their tetrahedra the sum of the three numbers associated to its three edges. The four numbers written on the faces of Romeo's tetrahedron turn out to coincide with the four numbers written on Juliet's tetrahedron. Does it follow that the four numbers assigned to the vertices of Romeo's tetrahedron are identical to the four numbers assigned to the vertices of Juliet's tetrahedron?

6

1

Set of positive integers

Find all finite sets of positive integers with at least two elements such that for any two numbers

a

,

b

(

a > b

) belonging to the set, the number \frac {b^2}{a \minus{} b} belongs to the set, too.

7

1

Integer Equation

How many pairs

(m,n)

of positive integers with

m < n

fulfill the equation \frac {3}{2008} \equal{} \frac 1m \plus{} \frac 1n?

8

1

The greatest element of A

Consider a set

A

of positive integers such that the least element of

A

equals

1001

and the product of all elements of

A

is a perfect square. What is the least possible value of the greatest element of

A

?

10

1

Sum of digits...

For a positive integer

n

, let

S(n)

denote the sum of its digits. Find the largest possible value of the expression

\frac {S(n)}{S(16n)}

.

11

1

A contains a perfect square

Consider a subset

A

of

84

elements of the set

\{1,\,2,\,\dots,\,169\}

such that no two elements in the set add up to

169

. Show that

A

contains a perfect square.

14

1

Building a cube

Is it possible to build a

4\times 4\times4

cube from blocks of the following shape consisting of

4

unit cubes?

12

1

Making a present

In a school class with

3n

children, any two children make a common present to exactly one other child. Prove that for all odd

n

it is possible that the following holds: For any three children

A

,

B

and

C

in the class, if

A

and

B

make a present to

C

then

A

and

C

make a present to

B

.

13

1

Number of participating countries...

For an upcoming international mathematics contest, the participating countries were asked to choose from nine combinatorics problems. Given how hard it usually is to agree, nobody was surprised that the following happened: i) Every country voted for exactly three problems. ii) Any two countries voted for different sets of problems. iii) Given any three countries, there was a problem none of them voted for. Find the maximal possible number of participating countries.

15

1

Problem with dominoes

Some

1\times 2

dominoes, each covering two adjacent unit squares, are placed on a board of size

n\times n

such that no two of them touch (not even at a corner). Given that the total area covered by the dominoes is

2008

, find the least possible value of

n

.

20

1

Prove ABC is an isosceles triangle

Let

M

be a point on

BC

and

N

be a point on

AB

such that

AM

and

CN

are angle bisectors of the triangle

ABC

. Given that \frac {\angle BNM}{\angle MNC} \equal{} \frac {\angle BMN}{\angle NMA}, prove that the triangle

ABC

is isosceles.

19

1

Combinatorical Geometry

In a circle of diameter

1

, some chords are drawn. The sum of their lengths is greater than

19

. Prove that there is a diameter intersecting at least

7

chords.

18

1

Lines pass through a common point

Let

AB

be a diameter of a circle

S

, and let

L

be the tangent at

A

. Furthermore, let

c

be a fixed, positive real, and consider all pairs of points

X

and

Y

lying on

L

, on opposite sides of

A

, such that |AX|\cdot |AY| \equal{} c. The lines

BX

and

BY

intersect

S

at points

P

and

Q

, respectively. Show that all the lines

PQ

pass through a common point.

17

1

Inequality in a cyclic quadrilateral

Assume that

a

,

b

,

c

and

d

are the sides of a quadrilateral inscribed in a given circle. Prove that the product (ab \plus{} cd)(ac \plus{} bd)(ad \plus{} bc) acquires its maximum when the quadrilateral is a square.

9

1

hard equation

Suppose that the positive integers

a

and

b

satisfy the equation a^b\minus{}b^a\equal{}1008 Prove that

a

and

b

are congruent modulo 1008.

4

1

nice polyminal

The polynomial

P

has integer coefficients and

P(x)=5

for five different integers

x

. Show that there is no integer

x

such that

-6\le P(x)\le 4

or

6\le P(x)\le 16

.

16

1

Parallelogram

Let

ABCD

be a parallelogram. The circle with diameter

AC

intersects the line

BD

at points

P

and

Q

. The perpendicular to the line

AC

passing through the point

C

intersects the lines

AB

and

AD

at points

X

and

Y

, respectively. Prove that the points

P,Q,X

and

Y

lie on the same circle.

2

1

Inequality with condition

Prove that if the real numbers

a,b

and

c

satisfy

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

then

\frac{a^2}{2+b+c^2}+\frac{b^2}{2+c+a^2}+\frac{c^2}{2+a+b^2}\ge\frac{(a+b+c)^2}{12}

When does the inequality hold?

1

Polynomials

Determine all polynomials

p(x)

with real coefficients such that

p((x+1)^3)=(p(x)+1)^3

and

p(0)=0

.

2008 Baltic Way

Subcontests

Does there exist an angle

Romeo, Juliet and a game

Set of positive integers

Integer Equation

The greatest element of A

Sum of digits...

A contains a perfect square

Building a cube

Making a present

Number of participating countries...

Problem with dominoes

Prove ABC is an isosceles triangle

Combinatorical Geometry

Lines pass through a common point

Inequality in a cyclic quadrilateral

hard equation

nice polyminal

Parallelogram

Inequality with condition

Polynomials