MathDB

1

'Covering' the integer 1998

m

covers the number

1998

, if

1,9,9,8

appear in this order as digits of

m

. (For instance

1998

is covered by

2<span class='latex-bold'>1</span>59<span class='latex-bold'>9</span>36<span class='latex-bold'>98</span>

but not by

213326798

.) Let

k(n)

be the number of positive integers that cover

1998

and have exactly

n

digits (

n\ge 5

), all different from

0

. What is the remainder of

k(n)

on division by

8

?

19

1

Ping pong matches between two teams of 1000

Consider a ping-pong match between two teams, each consisting of

1000

players. Each player played against each player of the other team exactly once (there are no draws in ping-pong). Prove that there exist ten players, all from the same team, such that every member of the other team has lost his game against at least one of those ten players.

18

1

No two subsets of S with the same sum of elements

Determine all positive integers

n

for which there exists a set

S

with the following properties: (i)

S

consists of

n

positive integers, all smaller than

2^{n-1}

; (ii) for any two distinct subsets

A

and

B

of

S

, the sum of the elements of

A

is different from the sum of the elements of

B

.

17

1

Colouring the nk objects k different colours

Let

n

and

k

be positive integers. There are

nk

objects (of the same size) and

k

boxes, each of which can hold

n

objects. Each object is coloured in one of

k

different colours. Show that the objects can be packed in the boxes so that each box holds objects of at most two colours.

16

1

Covering a 13 x 13 chessboard without central square

Is it possible to cover a

13\times 13

chessboard with forty-two pieces of dimensions

4\times 1

such that only the central square of the chessboard remains uncovered?

15

1

E satisfies AE/ED=CD/DB

Given acute triangle

ABC

. Point

D

is the foot of the perpendicular from

A

to

BC

. Point

E

lies on the segment

AD

and satisfies the equation

\frac{AE}{ED}=\frac{CD}{DB}

Point

F

is the foot of the perpendicular from

D

to

BE

. Prove that

\angle AFC=90^{\circ}

.

14

1

Parallel of AC at B meets external bisector at D

Given triangle

ABC

with

AB<AC

. The line passing through

B

and parallel to

AC

meets the external angle bisector of

\angle BAC

at

D

. The line passing through

C

and parallel to

AB

meets this bisector at

E

. Point

F

lies on the side

AC

and satisfies the equality

FC=AB

. Prove that

DF=FE

.

12

1

The point D on the side BC for the right triangle ABC

In a triangle

ABC

,

\angle BAC =90^{\circ}

. Point

D

lies on the side

BC

and satisfies

\angle BDA=2\angle BAD

. Prove that

\frac{2}{AD}=\frac{1}{BD}+\frac{1}{CD}

10

1

Inscribed n-gon and n-1-gon

Let

n\ge 4

be an even integer. A regular

n

-gon and a regular

(n-1)

-gon are inscribed into the unit circle. For each vertex of the

n

-gon consider the distance from this vertex to the nearest vertex of the

(n-1)

-gon, measured along the circumference. Let

S

be the sum of these

n

distances. Prove that

S

depends only on

n

, and not on the relative position of the two polygons.

9

1

Trigonometry, arithmetic means and the greek alphabet.

Let the numbers

\alpha ,\beta

satisfy

0<\alpha <\beta <\frac{\pi}{2}

and let

\gamma

and

\delta

be the numbers defined by the conditions:

(\text{i})\ 0<\gamma<\frac{\pi}{2}

, and

\tan\gamma

is the arithmetic mean of

\tan\alpha

and

\tan\beta

;

(\text{ii})\ 0<\delta<\frac{\pi}{2}

, and

\frac{1}{\cos\delta}

is the arithmetic mean of

\frac{1}{\cos\alpha}

and

\frac{1}{\cos\beta}

.
Prove that

\gamma <\delta

.

8

1

Prove binomial sum for the polynomial P

Let

P_k(x)=1+x+x^2+\ldots +x^{k-1}

. Show that

\sum_{k=1}^n \binom{n}{k} P_k(x)=2^{n-1} P_n \left( \frac{x+1}{2} \right)

for every real number

x

and every positive integer

n

.

7

1

Find all functions

Let

\mathbb{R}

be the set of all real numbers. Find all functions

f:\mathbb{R}\rightarrow\mathbb{R}

satisfying for all

x,y\in\mathbb{R}

the equation

f(x)+f(y)=f(f(x)f(y))

.

6

1

The polynomial P is even if it is even for a,b

Let

P

be a polynomial of degree

6

and let

a,b

be real numbers such that

0<a<b

. Suppose that

P(a)=P(-a),P(b)=P(-b),P'(0)=0

. Prove that

P(x)=P(-x)

for all real

x

.

5

1

Multiple of 2^n has only digits a and b

Let

a

be an odd digit and

b

an even digit. Prove that for every positive integer

n

there exists a positive integer, divisible by

2^n

, whose decimal representation contains no digits other than

a

and

b

.

4

1

P(n) is a three-digit integer for n=1,2,..,1998

Let

P

be a polynomial with integer coefficients. Suppose that for

n=1,2,3,\ldots ,1998

the number

P(n)

is a three-digit positive integer. Prove that the polynomial

P

has no integer roots.

2

1

c is divisible by 5 in quasi-pythagorean triplet

A triple

(a,b,c)

of positive integers is called quasi-Pythagorean if there exists a triangle with lengths of the sides

a,b,c

and the angle opposite to the side

c

equal to

120^{\circ}

. Prove that if

(a,b,c)

is a quasi-Pythagorean triple then

c

has a prime divisor bigger than

5

.

1

Three conditions of the two variable function f

Find all functions

f

of two variables, whose arguments

x,y

and values

f(x,y)

are positive integers, satisfying the following conditions (for all positive integers

x

and

y

): \begin{align*} f(x,x)& =x,\\ f(x,y)& =f(y,x),\\ (x+y)f(x,y)& =yf(x,x+y).\end{align*}

11

1

9-th Baltic Way

If

a,b,c

be the lengths of the sides of a triangle. Let

R

denote its circumradius. Prove that

R\ge \frac{a^2+b^2}{2\sqrt{2a^2+2b^2-c^2}}

When does equality hold?

3

1

Solve 2x^2+5y^2=11(xy-11) in N

Find all positive integer solutions to

2x^2+5y^2=11(xy-11)

.

13

1

pentagon

In convex pentagon

ABCDE

, the sides

AE,BC

are parallel and

\angle ADE=\angle BDC

. The diagonals

AC

and

BE

intersect at

P

. Prove that

\angle EAD=\angle BDP

and

\angle CBD=\angle ADP

.

1998 Baltic Way

Subcontests

'Covering' the integer 1998

Ping pong matches between two teams of 1000

No two subsets of S with the same sum of elements

Colouring the nk objects k different colours

Covering a 13 x 13 chessboard without central square

E satisfies AE/ED=CD/DB

Parallel of AC at B meets external bisector at D

The point D on the side BC for the right triangle ABC

Inscribed n-gon and n-1-gon

Trigonometry, arithmetic means and the greek alphabet.

Prove binomial sum for the polynomial P

Find all functions

The polynomial P is even if it is even for a,b

Multiple of 2^n has only digits a and b

P(n) is a three-digit integer for n=1,2,..,1998

c is divisible by 5 in quasi-pythagorean triplet

Three conditions of the two variable function f

9-th Baltic Way

Solve 2x^2+5y^2=11(xy-11) in N

pentagon