MathDB

1

Infinite arithmetic progression of powers

Does there exist an infinite non-constant arithmetic progression, each term of which is of the form

a^b

, where

a

and

b

are positive integers with

b\ge 2

?

17

1

Sequence of binomials in periodic modulo 2002

Show that the sequence

\binom{2002}{2002},\binom{2003}{2002},\binom{2004}{2002},\ldots

considred modulo

2002

, is periodic.

15

1

Spider and fly on a cube

A spider and a fly are sitting on a cube. The fly wants to maximize the shortest path to the spider along the surface of the cube. Is it necessarily best for the fly to be at the point opposite to the spider? (“Opposite” means “symmetric with respect to the centre of the cube”.)

14

1

Angle LNM is a right angle if ∠ALB=∠MNB

Let

L,M

and

N

be points on sides

AC,AB

and

BC

of triangle

ABC

, respectively, such that

BL

is the bisector of angle

ABC

and segments

AN,BL

and

CM

have a common point. Prove that if

\angle ALB=\angle MNB

then

\angle LNM=90^{\circ}

.

13

1

FB is perpendicular to AC when FD is perpenduclar to BC

Let

ABC

be an acute triangle with

\angle BAC>\angle BCA

, and let

D

be a point on side

AC

such that

|AB|=|BD|

. Furthermore, let

F

be a point on the circumcircle of triangle

ABC

such that line

FD

is perpendicular to side

BC

and points

F,B

lie on different sides of line

AC

. Prove that line

FB

is perpendicular to side

AC

.

12

1

The four points in S are collinear

A set

S

of four distinct points is given in the plane. It is known that for any point

X\in S

the remaining points can be denoted by

Y,Z

and

W

so that

|XY|=|XZ|+|XW|

Prove that all four points lie on a line.

11

1

Drawn to more than 11 points

Let

n

be a positive integer. Consider

n

points in the plane such that no three of them are collinear and no two of the distances between them are equal. One by one, we connect each point to the two points nearest to it by line segments (if there are already other line segments drawn to this point, we do not erase these). Prove that there is no point from which line segments will be drawn to more than

11

points.

10

1

The smallest value of N for the 2nd player to win

Let

N

be a positive integer. Two persons play the following game. The first player writes a list of positive integers not greater than

25

, not necessarily different, such that their sum is at least

200

. The second player wins if he can select some of these numbers so that their sum

S

satisfies the condition

200-N\le S\le 200+N

. What is the smallest value of

N

for which the second player has a winning strategy?

9

1

Magicians' 4-card trick

Two magicians show the following trick. The first magician goes out of the room. The second magician takes a deck of

100

cards labelled by numbers

1,2,\ldots ,100

and asks three spectators to choose in turn one card each. The second magician sees what card each spectator has taken. Then he adds one more card from the rest of the deck. Spectators shuffle these

4

cards, call the first magician and give him these

4

cards. The first magician looks at the

4

cards and “guesses” what card was chosen by the first spectator, what card by the second and what card by the third. Prove that the magicians can perform this trick.

8

1

The set T of triangles

Let

P

be a set of

n\ge 3

points in the plane, no three of which are on a line. How many possibilities are there to choose a set

T

of

\binom{n-1}{2}

triangles, whose vertices are all in

P

, such that each triangle in

T

has a side that is not a side of any other triangle in

T

?

7

1

n convex quadrilaterals in the plane

We draw

n

convex quadrilaterals in the plane. They divide the plane into regions (one of the regions is infinite). Determine the maximal possible number of these regions.

6

1

Solitaire game on m x n chessboard

The following solitaire game is played on an

m\times n

rectangular board,

m,n\ge 2

, divided into unit squares. First, a rook is placed on some square. At each move, the rook can be moved an arbitrary number of squares horizontally or vertically, with the extra condition that each move has to be made in the

90^{\circ}

clockwise direction compared to the previous one (e.g. after a move to the left, the next one has to be done upwards, the next one to the right etc). For which values of

m

and

n

is it possible that the rook visits every square of the board exactly once and returns to the first square? (The rook is considered to visit only those squares it stops on, and not the ones it steps over.)

5

1

Rational equation

Find all pairs

(a,b)

of positive rational numbers such that

\sqrt{a}+\sqrt{b}=\sqrt{2+\sqrt{3}}.

4

1

Sum of x_1=1 inequality

Let

n

be a positive integer. Prove that

\sum_{i=1}^nx_i(1-x_i)^2\le\left(1-\frac{1}{n}\right)^2

for all nonnegative real numbers

x_1,x_2,\ldots ,x_n

such that

x_1+x_2+\ldots x_n=1

.

3

1

Find all sequences

0\le a_0\le a_1\le a_2\le \ldots

of real numbers such that

a_{m^2+n^2}=a_m^2+a_n^2

for all integers

m,n\ge 0

.

2

1

One of a,b,c,d is not greater than -1

Let

a,b,c,d

be real numbers such that

a+b+c+d=-2

ab+ac+ad+bc+bd+cd=0

Prove that at least one of the numbers

a,b,c,d

is not greater than

-1

.

1

Symmetric simultaneous equations

Solve the system of simultaneous equations

\begin{cases}a^3+3ab^2+3ac^2-6abc=1\\ b^3+3ba^2+3bc^2-6abc=1\\c^3+3ca^2+3cb^2-6abc=1\end{cases}

in real numbers.

18

1

Baltic way 2002

Find all integers

n>1

such that any prime divisor of

n^6-1

is a divisor of

(n^3-1)(n^2-1)

.

16

1

Find all non negative integers "m"

Find all nonnegative integers

m

such that

a_m=(2^{2m+1})^2+1

is divisible by at most two different primes.

19

1

Diophantine equation

Let

n

be a positive integer. Prove that the equation

x+y+\frac{1}{x}+\frac{1}{y}=3n

does not have solutions in positive rational numbers.

2002 Baltic Way

Subcontests

Infinite arithmetic progression of powers

Sequence of binomials in periodic modulo 2002

Spider and fly on a cube

Angle LNM is a right angle if ∠ALB=∠MNB

FB is perpendicular to AC when FD is perpenduclar to BC

The four points in S are collinear

Drawn to more than 11 points

The smallest value of N for the 2nd player to win

Magicians' 4-card trick

The set T of triangles

n convex quadrilaterals in the plane

Solitaire game on m x n chessboard

Rational equation

Sum of x_1=1 inequality

Find all sequences

One of a,b,c,d is not greater than -1

Symmetric simultaneous equations

Baltic way 2002

Find all non negative integers "m"

Diophantine equation

2002 Baltic Way

Subcontests

Infinite arithmetic progression of powers

Sequence of binomials in periodic modulo 2002

Spider and fly on a cube

Angle LNM is a right angle if ∠ALB=∠MNB

FB is perpendicular to AC when FD is perpenduclar to BC

The four points in S are collinear

Drawn to more than 11 points

The smallest value of N for the 2nd player to win

Magicians' 4-card trick

The set T of triangles

n convex quadrilaterals in the plane

Solitaire game on m x n chessboard

Rational equation

Sum of x_1=1 inequality

Find all sequences

One of a,b,c,d is not greater than -1

Symmetric simultaneous equations

Baltic way 2002

Find all non negative integers &quot;m&quot;

Diophantine equation

Find all non negative integers "m"