MathDB

1

The sum 1996k+1997k has no carries

1996

to

1997

, we first add the unit digits

6

and

7

. Obtaining

13

, we write down

3

and “carry”

1

to the next column. Thus we make a carry. Continuing, we see that we are to make three carries in total.
Does there exist a positive integer

k

such that adding

1996\cdot k

to

1997\cdot k

no carry arises during the whole calculation?

1

f(x)f(y)=f(x-y) for reals x,y

Determine all functions

f

from the real numbers to the real numbers, different from the zero function, such that

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

for all real numbers

x

and

y

.

2

1

1/2(a_1+a_m) is also a member of sequence

Given a sequence

a_1,a_2,a_3,\ldots

of positive integers in which every positive integer occurs exactly once. Prove that there exist integers

\ell

and

m,\ 1<\ell <m

, such that

a_1+a_m=2a_{\ell}

.

3

1

Determine x_1997 in recursive sequence

Let

x_1=1

and

x_{n+1} =x_n+\left\lfloor \frac{x_n}{n}\right\rfloor +2

, for

n=1,2,3,\ldots

where

x

denotes the largest integer not greater than

x

. Determine

x_{1997}

.

4

1

Arithmetic mean a of x_i

Prove that the arithmetic mean

a

of

x_1,\ldots ,x_n

satisfies

(x_1-a)^2+\ldots +(x_n-a)^2\le \frac{1}{2}(|x_1-a|+\ldots +|x_n-a|)^2

5

1

Sequence is eventually periodic

In a sequence

u_0,u_1,\ldots

of positive integers,

u_0

is arbitrary, and for any non-negative integer

n

,

u_{n+1}=\begin{cases}\frac{1}{2}u_n & \text{for even }u_n \\ a+u_n & \text{for odd }u_n \end{cases}

where

a

is a fixed odd positive integer. Prove that the sequence is periodic from a certain step.

6

1

a^3,b^2,c suggestive diophantine equation

Find all triples

(a,b,c)

of non-negative integers satisfying

a\ge b\ge c

and

1\cdot a^3+9\cdot b^2+9\cdot c+7=1997

7

1

Both a and a+1997 are roots of P, Q(P(x))=1 has no solutions

Let

P

and

Q

be polynomials with integer coefficients. Suppose that the integers

a

and

a+1997

are roots of

P

, and that

Q(1998)=2000

. Prove that the equation

Q(P(x))=1

has no integer solutions.

9

1

Gandalf visiting worlds numbered n, 2n or 3n+1

The worlds in the Worlds’ Sphere are numbered

1,2,3,\ldots

and connected so that for any integer

n\ge 1

, Gandalf the Wizard can move in both directions between any worlds with numbers

n,2n

and

3n+1

. Starting his travel from an arbitrary world, can Gandalf reach every other world?

10

1

13|S(n) for some n among 79 consecutive integers

Prove that in every sequence of

79

consecutive positive integers written in the decimal system, there is a positive integer whose sum of digits is divisible by

13

.

11

1

Infinite sum of the angles made by the points A_i,B_i

On two parallel lines, the distinct points

A_1,A_2,A_3,\ldots

respectively

B_1,B_2,B_3,\ldots

are marked in such a way that

|A_iA_{i+1}|=1

and

|B_iB_{i+1}|=2

for

i=1,2,\ldots

. Provided that

A_1A_2B_1=\alpha

, find the infinite sum

\angle A_1B_1A_2+\angle A_2B_2A_3+\angle A_3B_3A_4+\ldots

12

1

Midpoint of AB is also the midpoint of YZ

Two circles

\mathcal{C}_1

and

\mathcal{C}_2

intersect in

P

and

Q

. A line through

P

intersects

\mathcal{C}_1

and

\mathcal{C}_2

again at

A

and

B

, respectively, and

X

is the midpoint of

AB

. The line through

Q

and

X

intersects

C_1

and

C_2

again at

Y

and

Z

, respectively. Prove that

X

is the midpoint of

YZ

.

13

1

Line joining circumcentres is perpendicular to FC

Five distinct points

A,B,C,D

and

E

lie on a line with

|AB|=|BC|=|CD|=|DE|

. The point

F

lies outside the line. Let

G

be the circumcentre of the triangle

ADF

and

H

the circumcentre of the triangle

BEF

. Show that the lines

GH

and

FC

are perpendicular.

14

1

2b^2=a^2+c^2 in triangle ABC inequality

In the triangle

ABC

,

AC^2

is the arithmetic mean of

BC^2

and

AB^2

. Show that

\cot^2B\ge \cot A\cdot\cot C

.

15

1

MN passes through incentre of ABC

In the acute triangle

ABC

, the bisectors of

A,B

and

C

intersect the circumcircle again at

A_1,B_1

and

C_1

, respectively. Let

M

be the point of intersection of

AB

and

B_1C_1

, and let

N

be the point of intersection of

BC

and

A_1B_1

. Prove that

MN

passes through the incentre of

\triangle ABC

.

16

1

Running out of squares on 5 x 5 chessboard

On a

5\times 5

chessboard, two players play the following game: The first player places a knight on some square. Then the players alternately move the knight according to the rules of chess, starting with the second player. It is not allowed to move the knight to a square that was visited previously. The player who cannot move loses. Which of the two players has a winning strategy?

17

1

Dividing rectangle into n and n+76 equal squares

A rectangle can be divided into

n

equal squares. The same rectangle can also be divided into

n+76

equal squares. Find

n

.

18

1

Expressing n as a+b from the infinite sets A and B

a) Prove the existence of two infinite sets

A

and

B

, not necessarily disjoint, of non-negative integers such that each non-negative integer

n

is uniquely representable in the form

n=a+b

with

a\in A,b\in B

. b) Prove that for each such pair

(A,B)

, either

A

or

B

contains only multiples of some integer

k>1

.

19

1

Campaign-making animals (?) and paths

In a forest each of

n

animals (

n\ge 3

) lives in its own cave, and there is exactly one separate path between any two of these caves. Before the election for King of the Forest some of the animals make an election campaign. Each campaign-making animal visits each of the other caves exactly once, uses only the paths for moving from cave to cave, never turns from one path to another between the caves and returns to its own cave in the end of its campaign. It is also known that no path between two caves is used by more than one campaign-making animal.
a) Prove that for any prime

n

, the maximum possible number of campaign-making animals is

\frac{n-1}{2}

.
b) Find the maximum number of campaign-making animals for

n=9

.

20

1

Twelve cards with each side either black or white

Twelve cards lie in a row. The cards are of three kinds: with both sides white, both sides black, or with a white and a black side. Initially, nine of the twelve cards have a black side up. The cards

1-6

are turned, and subsequently four of the twelve cards have a black side up. Now cards

4-9

are turned, and six cards have a black side up. Finally, the cards

1-3

and

10-12

are turned, after which five cards have a black side up. How many cards of each kind were there?

1997 Baltic Way

Subcontests

The sum 1996k+1997k has no carries

f(x)f(y)=f(x-y) for reals x,y

1/2(a_1+a_m) is also a member of sequence

Determine x_1997 in recursive sequence

Arithmetic mean a of x_i

Sequence is eventually periodic

a^3,b^2,c suggestive diophantine equation

Both a and a+1997 are roots of P, Q(P(x))=1 has no solutions

Gandalf visiting worlds numbered n, 2n or 3n+1

13|S(n) for some n among 79 consecutive integers

Infinite sum of the angles made by the points A_i,B_i

Midpoint of AB is also the midpoint of YZ

Line joining circumcentres is perpendicular to FC

2b^2=a^2+c^2 in triangle ABC inequality

MN passes through incentre of ABC

Running out of squares on 5 x 5 chessboard

Dividing rectangle into n and n+76 equal squares

Expressing n as a+b from the infinite sets A and B

Campaign-making animals (?) and paths

Twelve cards with each side either black or white