MathDB

1

TOT 031 1982 Autumn S5 closed selfintersecting curve, tunnel for Martian trains

The plan of a Martian underground is represented by a closed selfintersecting curve, with at most one self-intersection at each point. Prove that a tunnel for such a plan may be constructed in such a way that the train passes consecutively over and under the intersecting parts of the tunnel.

(030) 4

1

TOT 030 1982 Autumn S4 vector sum in a regular n-gon / tetrahedron

(a)

K_1,K_2,..., K_n

are the feet of the perpendiculars from an arbitrary point

M

inside a given regular

n

-gon to its sides (or sides produced). Prove that the sum

\overrightarrow{MK_1} + \overrightarrow{MK_2} + ... + \overrightarrow{MK_n}

equals

\frac{n}{2}\overrightarrow{MO}

, where

O

is the centre of the

n

-gon.
(b) Prove that the sum of the vectors whose origin is an arbitrary point

M

inside a given regular tetrahedron and whose endpoints are the feet of the perpendiculars from

M

to the faces of the tetrahedron equals

\frac43 \overrightarrow{MO}

, where

O

is the centre of the tetrahedron.
(VV Prasolov, Moscow)

(029) 3

1

TOT 029 1982 Autumn S3 60 symbols, either X or O on paper

60

symbols, each of which is either

X

or

O

, are written consecutively on a strip of paper. This strip must then be cut into pieces with each piece containing symbols symmetric about their centre, e.g.

O, XX, OXXXXX, XOX

, etc. (a) Prove that there is a way of cutting the strip so that there are no more than

24

such pieces. (b) Give an example of such an arrangement of the signs for which the number of pieces cannot be less than

15

. (c) Try to improve the result of (b).

(028) 2

1

TOT 028 1982 Autumn S2 polyhedron existence given list of edges

Does there exist a polyhedron (not necessarily convex) which could have the following complete list of edges?

AB, AC, BC, BD, CD, DE, EF, EG, FG, FH, GH, AH

.
http://1.bp.blogspot.com/-wTdNfQHG5RU/XVk1Bf4wpqI/AAAAAAAAKhA/8kc6u9KqOgg_p1CXim2LZ1ANFXFiWgnYACK4BGAYYCw/s1600/TOT%2B1982%2BAutum%2BS2.png

(027) 1

1

TOT 027 1982 Autumn S1 sum [ \sqrt[i]{n}] = sum [\log_i n]

Prove that for all natural numbers

n

greater than

1

:

[\sqrt{n}] + [\sqrt[3]{n}] +...+[ \sqrt[n]{n}] = [\log_2 n] + [\log_3 n] + ... + [\log_n n]

(VV Kisil)

(026) 4

1

TOT 026 1982 Autumn J4 20 points in circle, 20 equal arcs, 10 chords

(a)

10

points dividing a circle into

10

equal arcs are connected in pairs by

5

chords. Is it necessary that two of these chords are of equal length?
(b)

20

points dividing a circle into

20

equal arcs are connected in pairs by

10

chords. Prove that among these

10

chords there are two chords of equal length.
(VV Proizvolov, Moscow)

(025) 3

1

TOT 025 1982 Autumn J3 diophantine m!n! = k! , infinite solutions

Prove that the equation

m!n! = k!

has infinitely many solutions in which

m, n

and

k

are natural numbers greater than unity .

(024) 2

1

TOT 024 1982 Autumn J2 10 or 15 objects colored w 2 colours, in a line

A number of objects, each coloured in one of two given colours, are arranged in a line (there is at least one object having each of the given colours). It is known that each two objects, between which there are exactly

10

or

15

other objects, are of the same colour. What is the greatest possible number of such pieces?

(023) 1

1

TOT 023 1982 Autumn J1 36 cards game, take 4 each time

There are

36

cards in a deck arranged in the sequence spades, clubs, hearts, diamonds, spades, clubs, hearts, diamonds, etc. Somebody took part of this deck off the top, turned it upside down, and cut this part into the remaining part of the deck (i.e. inserted it between two consecutive cards). Then four cards were taken off the top, then another four, etc. Prove that in any of these sets of four cards, all the cards are of different suits.
(A Merkov, Moscow)

(022) 4

1

TOT 022 1982 Spring S4 polynomial takes values of form 2^M

A polynomial

P(x)

has unity as the coefficient of its highest power, and has the property that with natural number arguments, it can take all values of form

2^M

, where

M

is a natural number. Prove that the polynomial is of degree

1

.

(021) 2

1

TOT 021 1982 Spring S2 K^2 equal smaller squares, broken line, min

A square is subdivided into

K^2

equal smaller squares. We are given a broken line which passes through the centres of all the smaller squares (such a broken line may intersect itself). Find the minimum number of links in this broken line.
(A Andjans, Riga)

(020) 1

1

TOT 020 1982 Spring S1 sum {x_k}{x_{k-1}+x_{k+1}} >=2, sharp inequality

(a) Prove that for any positive numbers

x_1,x_2,...,x_k

(

k > 3

),

\frac{x_1}{x_k+x_2}+ \frac{x_2}{x_1+x_3}+...+\frac{x_k}{x_{k-1}+x_1}\ge 2

(b) Prove that for every

k

this inequality cannot be sharpened, i.e. prove that for every given

k

it is not possible to change the number

2

in the right hand side to a greater number in such a way that the inequality remains true for every choice of positive numbers

x_1,x_2,...,x_k

.
(A Prokopiev)

(019) 5

1

TOT 019 1982 Spring J5 S5 arithmetic progression in 1,1/2,1/3,1/4,...

Consider the sequence

1, \frac12, \frac13, \frac14 ,...

Does there exist an arithmetic progression composed of terms of this sequence (a) of length

5

, (b) of length greater than

5

(if so, what possible length)?
(G Galperin, Moscow)

(018) 4

1

TOT 018 1982 Spring J4 air routes between 101 towns

In a certain country there are more than

101

towns. The capital of this country is connected by direct air routes with

100

towns, and each town, except for the capital, is connected by direct air routes with

10

towns (if

A

is connected with

B, B

is connected with

A

). It is known that from any town it is possible to travel by air to any other town (possibly via other towns). Prove that it is possible to close down half of the air routes connected with the capital, and preserve the capability of travelling from any town to any other town within the country.
(IS Rubanov)

(017) 3

1

TOT 017 1982 Spring J3 S3 100 members with a_k > k

a) Prove that in an infinite sequence

{a_k}

of integers, pairwise distinct and each member greater than

1

, one can find

100

members for which

a_k > k

.
b) Prove that in an infinite sequence

{a_k}

of integers, pairwise distinct and each member greater than

1

there are infinitely many such numbers

a_k

such that

a_k > k

.
(A Andjans, Riga)
PS. (a) for juniors (b) for seniors

(016) 2

1

TOT 016 1982 Spring J2 lengths sides, diagonals ABCD <1, in circle rad=0,9

The lengths of all sides and both diagonals of a quadrilateral are less than

1

metre. Prove that it may be placed in a circle of radius

0.9

metres.

(015) 1

1

TOT 015 1982 Spring J1 30 / n, n has 30 divisors

Find all natural numbers which are divisible by

30

and which have exactly

30

different divisors.
(M Levin)

1982 Tournament Of Towns

Subcontests

TOT 031 1982 Autumn S5 closed selfintersecting curve, tunnel for Martian trains

TOT 030 1982 Autumn S4 vector sum in a regular n-gon / tetrahedron

TOT 029 1982 Autumn S3 60 symbols, either X or O on paper

TOT 028 1982 Autumn S2 polyhedron existence given list of edges

TOT 027 1982 Autumn S1 sum [ \sqrt[i]{n}] = sum [\log_i n]

TOT 026 1982 Autumn J4 20 points in circle, 20 equal arcs, 10 chords

TOT 025 1982 Autumn J3 diophantine m!n! = k! , infinite solutions

TOT 024 1982 Autumn J2 10 or 15 objects colored w 2 colours, in a line

TOT 023 1982 Autumn J1 36 cards game, take 4 each time

TOT 022 1982 Spring S4 polynomial takes values of form 2^M

TOT 021 1982 Spring S2 K^2 equal smaller squares, broken line, min

TOT 020 1982 Spring S1 sum {x_k}{x_{k-1}+x_{k+1}} >=2, sharp inequality

TOT 019 1982 Spring J5 S5 arithmetic progression in 1,1/2,1/3,1/4,...

TOT 018 1982 Spring J4 air routes between 101 towns

TOT 017 1982 Spring J3 S3 100 members with a_k > k

TOT 016 1982 Spring J2 lengths sides, diagonals ABCD <1, in circle rad=0,9

TOT 015 1982 Spring J1 30 / n, n has 30 divisors

1982 Tournament Of Towns

Subcontests

TOT 031 1982 Autumn S5 closed selfintersecting curve, tunnel for Martian trains

TOT 030 1982 Autumn S4 vector sum in a regular n-gon / tetrahedron

TOT 029 1982 Autumn S3 60 symbols, either X or O on paper

TOT 028 1982 Autumn S2 polyhedron existence given list of edges

TOT 027 1982 Autumn S1 sum [ \sqrt[i]{n}] = sum [\log_i n]

TOT 026 1982 Autumn J4 20 points in circle, 20 equal arcs, 10 chords

TOT 025 1982 Autumn J3 diophantine m!n! = k! , infinite solutions

TOT 024 1982 Autumn J2 10 or 15 objects colored w 2 colours, in a line

TOT 023 1982 Autumn J1 36 cards game, take 4 each time

TOT 022 1982 Spring S4 polynomial takes values of form 2^M

TOT 021 1982 Spring S2 K^2 equal smaller squares, broken line, min

TOT 020 1982 Spring S1 sum {x_k}{x_{k-1}+x_{k+1}} &gt;=2, sharp inequality

TOT 019 1982 Spring J5 S5 arithmetic progression in 1,1/2,1/3,1/4,...

TOT 018 1982 Spring J4 air routes between 101 towns

TOT 017 1982 Spring J3 S3 100 members with a_k &gt; k

TOT 016 1982 Spring J2 lengths sides, diagonals ABCD &lt;1, in circle rad=0,9

TOT 015 1982 Spring J1 30 / n, n has 30 divisors

TOT 020 1982 Spring S1 sum {x_k}{x_{k-1}+x_{k+1}} >=2, sharp inequality

TOT 017 1982 Spring J3 S3 100 members with a_k > k

TOT 016 1982 Spring J2 lengths sides, diagonals ABCD <1, in circle rad=0,9