MathDB

1

TOT 052 1983 Autumn S5 kings and queens on a chessboard

A

of squares is given on a chessboard which is infinite in all directions. On each square of this chessboard which does not belong to

A

there is a king. On a command all kings may be moved in such a way that each king either remains on its square or is moved to an adjacent square, which may have been occupied by another king before the command. Each square may be occupied by at most one king. Does there exist such a number

k

and such a way of moving the kings that after

k

moves the kings will occupy all squares of the chessboard? Consider the following cases:
(a)

A

is the set of all squares, both of whose coordinates are multiples of

100

. (There is a horizontal line numbered by the integers from

-\infty

to

+\infty

A

is the set of all squares which are covered by

100

fixed arbitrary queens (i.e. each square covered by at least one queen).
Remark: If

A

consists of just one square, then

k = 1

and the required way is the following: all kings to the left of the square of

A

make one move to the right.

(051) 3

1

TOT 051 1983 Autumn S3 vectors OK + OM + OP = OI, perp. related

The centre

O

of the circumcircle of

\vartriangle ABC

lies inside the triangle. Perpendiculars are drawn rom

O

on the sides. When produced beyond the sides they meet the circumcircle at points

K, M

and

P

. Prove that

\overrightarrow{OK} + \overrightarrow{OM} + \overrightarrow{OP} = \overrightarrow{OI}

, where

I

is the centre of the inscribed circle of

\vartriangle ABC

.
(V Galperin, Moscow)

(050) 2

1

TOT 050 1983 Autumn S2 9-digits number from permutations of 1-9

Consider all nine-digit numbers, consisting of non-repeating digits from

1

to

9

in an arbitrary order. A pair of such numbers is called “conditional” if their sum is equal to

987654321

. (a) Prove that there exist at least two conditional pairs (noting that (

a,b

) and (

b,a

) is considered to be one pair). (b) Prove that the number of conditional pairs is odd.
(G Galperin, Moscow)

(049) 1

1

TOT 049 1983 Autumn S1 perimeter triangle double of a square

On sides

CB

and

CD

of square

ABCD

are chosen points

M

and

K

so that the perimeter of triangle

CMK

equals double the side of the square. Find angle

\angle MAK

.

(048) 5

1

TOT 048 1983 Autumn J5 N^2 pieces on NxN chessboard

N^2

pieces are placed on an

N \times N

chessboard. Is it possible to rearrange them in such a way that any two pieces which can capture each other (when considered to be knights) after the rearrangement are on adjacent squares (i.e. squares having at least one common boundary point)? Consider two cases: (a)

N = 3

. (b)

N = 8

(S Stefanov)

(047) 4

1

TOT 047 1983 Autumn J4 S4 monotone increasing a_{a_k} = 3k

a_1,a_2,a_3,...

is a monotone increasing sequence of natural numbers. It is known that for any

k, a_{a_k} = 3k

. a) Find

a_{100}

. b) Find

a_{1983}

.
(A Andjans, Riga)
PS. (a) for Juniors, (b) for Seniors

(046) 3

1

TOT 046 1983 Autumn J3 quadrilateral construction, sides and mid-diagonal given

Construct a quadrilateral given its side lengths and the length of the segment joining the midpoints of its diagonals.
(IZ Titovich)

(045) 2

1

TOT 045 1983 Autumn J2 find k, k = sum 2 relative prime \ne 1

Find all natural numbers

k

which can be represented as the sum of two relatively prime numbers not equal to

1

.

(044) 1

1

TOT 044 1983 Autumn J1 centroids form a square, starting w point in a square

Inside square

ABCD

consider a point

M

. Prove that the points of intersection of the medians of triangles

ABM, BCM, CDM

and

DAM

form a square.
(V Prasolov)

(041) O4

1

TOT 041 1983 Spring S-O4 K boys around a circle with sweets

There are

K

boys placed around a circle. Each of them has an even number of sweets. At a command each boy gives half of his sweets to the boy on his right. If, after that, any boy has an odd number of sweets, someone outside the circle gives him one more sweet to make the number even. This procedure can be repeated indefinitely. Prove that there will be a time at which all boys will have the same number of sweets.
(A Andjans, Riga)

(042) O5

1

TOT 042 1983 Spring S-O5 sum of lengths by projections in regular polygon

A point is chosen inside a regular

k

-gon in such a way that its orthogonal projections on to the sides all meet the respective sides at interior points. These points divide the sides into

2k

segments. Let these segments be enumerated consecutively by the numbers

1,2, 3, ... ,2k

. Prove that the sum of the lengths of the segments having even numbers equals the sum of the segments having odd numbers.
(A Andjans, Riga)

(043) A5

1

TOT 043 1983 Spring S-A5 coloring of k vertices in regular n-gon

k

vertices of a regular

n

-gon

P

are coloured. A colouring is called almost uniform if for every positive integer

m

the following condition is satisfied: If

M_1

is a set of m consecutive vertices of

P

and

M_2

is another such set then the number of coloured vertices of

M_1

differs from the number of coloured vertices of

M_2

at most by

1

. Prove that for all positive integers

k

and

n

(

k \le n

) an almost uniform colouring exists and that it is unique within a rotation.
(M Kontsevich, Moscow)

(040) O2

1

TOT 040 1983 Spring S-O2 S-A2 median vector sum proof, reverse

On sides

AB, BC

and

CA

of triangle

ABC

are located points

P, M

and

K

, respectively, so that

AM, BK

and

CP

intersect in one point and the sum of the vectors

\overrightarrow{AM}, \overrightarrow{BK}

and

\overrightarrow{CP}

equals

\overrightarrow{0}

. Prove that

K, M

and

P

are midpoints of the sides of triangle

ABC

on which they are located.

(039) O1

1

TOT 039 1983 Spring S-01 S-A1 500 segments from 1000 concyclic points

Numbers from

1

to

1000

are arranged around a circle. Prove that it is possible to form

500

non-intersecting line segments, each joining two such numbers, and so that in each case the difference between the numbers at each end (in absolute value) is not greater than

749

.
(AA Razborov, Moscow)

(038) A5

1

TOT 038 1983 Spring J-A5 4 in 5, are divisible by others or not, of 17

Prove that in any set of

17

distinct natural numbers one can either find five numbers so that four of them are divisible into the other or five numbers none of which is divisible into any other.
(An established theorem)

(037) A4

1

TOT 037 1983 Spring J-A4 S-A4 red + blue game in infinite square table

(a) An infinite sheet is divided into squares by two sets of parallel lines. Two players play the following game: the first player chooses a square and colours it red, the second player chooses a non-coloured square and colours it blue, the first player chooses a non-coloured square and colours it red, the second player chooses a non-coloured square and colours it blue, and so on. The goal of the first player is to colour four squares whose vertices form a square with sides parallel to the lines of the two parallel sets. The goal of the second player is to prevent him. Can the first player win?
(b) What is the answer to this question if the second player is permitted to colour two squares at once?
(DG Azov)
PS. (a) for Juniors, (a),(b) for Seniors

(036) O5

1

TOT 036 1983 Spring J-O5 billiard on triangular table

A version of billiards is played on a right triangular table, with a pocket in each of the three corners, and one of the acute angles being

30^o

. A ball is played from just in front of the pocket at the

30^o

. vertex toward the midpoint of the opposite side. Prove that if the ball is played hard enough, it will land in the pocket of the

60^o

vertex after

8

reflections.

(035) O4

1

TOT 035 1983 Spring J-O4 different permutations of same digits, sum of digits

The natural numbers

M

and

K

are represented by different permutations of the same digits. Prove that
(a) The sum of the digits of

2M

equals the sum of the digits of

2K

. (b) The sum of the digits of

M/2

equals the sum of the digits of

K/2

(

M, K

both even). (c) The sum of the digits of

5M

equals the sum of the digits of

5 K

.
(AD Lisitskiy)

(034) O3

1

TOT 034 1983 Spring J-O3 J-A3 S-O3 S-A3 N towns in Shvambrania

In Shvambrania there are

N

towns, every two of which are connected by a road. These roads do not intersect. If necessary, some of them pass over or under others via bridges. An evil magician establishes one-way rules along the roads in such a way that if someone goes out of a certain town he is unable to come back. Prove that
(a) It is possible to establish such rules. (b) There exists a town from which it is possible to reach any other town, and there exists a town from which it is not possible to go out. (c) There is one and only one route passing through all towns. (d) The magician can realise his intention in

N!

ways.
(LM Koganov, Moscow)
PS. (a),(b),(c) for Juniors, (a),(b),(d) for Seniors

(033) O2

1

TOT 034 1983 Spring J-O2 J-A2 regular 4k-gon is cut into parallelograms

(a) A regular

4k

-gon is cut into parallelograms. Prove that among these there are at least

k

rectangles.
(b) Find the total area of the rectangles in (a) if the lengths of the sides of the

4k

-gon equal

a

.
(VV Proizvolov, Moscow)

(032) O1

1

TOT 032 1983 Spring J-O1 J-A1 average speed of a pedestrian

A pedestrian walked for

3.5

hours. In every period of one hour’s duration he walked

5

kilometres. Is it true that his average speed was

5

kilometres per hour?
(NN Konstantinov, Moscow)

1983 Tournament Of Towns

Subcontests

TOT 052 1983 Autumn S5 kings and queens on a chessboard

TOT 051 1983 Autumn S3 vectors OK + OM + OP = OI, perp. related

TOT 050 1983 Autumn S2 9-digits number from permutations of 1-9

TOT 049 1983 Autumn S1 perimeter triangle double of a square

TOT 048 1983 Autumn J5 N^2 pieces on NxN chessboard

TOT 047 1983 Autumn J4 S4 monotone increasing a_{a_k} = 3k

TOT 046 1983 Autumn J3 quadrilateral construction, sides and mid-diagonal given

TOT 045 1983 Autumn J2 find k, k = sum 2 relative prime \ne 1

TOT 044 1983 Autumn J1 centroids form a square, starting w point in a square

TOT 041 1983 Spring S-O4 K boys around a circle with sweets

TOT 042 1983 Spring S-O5 sum of lengths by projections in regular polygon

TOT 043 1983 Spring S-A5 coloring of k vertices in regular n-gon

TOT 040 1983 Spring S-O2 S-A2 median vector sum proof, reverse

TOT 039 1983 Spring S-01 S-A1 500 segments from 1000 concyclic points

TOT 038 1983 Spring J-A5 4 in 5, are divisible by others or not, of 17

TOT 037 1983 Spring J-A4 S-A4 red + blue game in infinite square table

TOT 036 1983 Spring J-O5 billiard on triangular table

TOT 035 1983 Spring J-O4 different permutations of same digits, sum of digits

TOT 034 1983 Spring J-O3 J-A3 S-O3 S-A3 N towns in Shvambrania

TOT 034 1983 Spring J-O2 J-A2 regular 4k-gon is cut into parallelograms

TOT 032 1983 Spring J-O1 J-A1 average speed of a pedestrian