MathDB

1

TOT 131 1986 Autumn S7 21 points on a circle, angle of arc <=120

21

points. Prove that among the arcs which join any two of these points, at least

100

of them must subtend an angle at the centre of the circle not exceeding

120^o

.
( A . F . Sidorenko)

(130) 6

1

TOT 130 1986 Autumn S6 numbers 1-32 twice in a 8x8 chessboard

Squares of an

8 \times 8

chessboard are each allocated a number between

1

and

32

, with each number being used twice. Prove that it is possible to choose

32

such squares, each allocated a different number, so that there is at least one such square on each row or column .
(A . Andjans, Riga

(129) 4

1

TOT 129 1986 Autumn S4 1987 /( 1 986 !! + 1985 !! )

We define

N !!

to be

N(N - 2)(N -4)...5 \cdot 3 \cdot 1

if

N

is odd and

N(N -2)(N -4)... 6\cdot 4\cdot 2

if

N

is even . For example,

8 !! = 8 \cdot 6\cdot 4\cdot 2

, and

9 !! = 9v 7 \cdot 5\cdot 3 \cdot 1

. Prove that

1986 !! + 1985 !!

i s divisible by

1987

.
(V.V . Proizvolov , Moscow)

(128) 3

1

TOT 128 1986 Autumn S3 100 triangles, not 1 can be covered by other 99

Does there exist a set of

100

triangles in which not one of the triangles can be covered by the other

99

?

(127) 2

1

TOT 127 1986 Autumn S2 N-1 infinite ar. progressions with diff 2,3,4,..,Ν

Does there exist a number

N

so that there are

N - 1

infinite arithmetic progressions with differences

2 , 3 , 4 ,..., N

, and every natural number belongs to at least one of these progressions?

(126) 1

1

TOT 126 1986 Autumn S1 tangential trapezoid, area wanted

We are given trapezoid

ABCD

and point

M

on the intersection of its diagonals. The parallel sides are

AD

and

BC

and it is known that

AB

is perpendicular to

AD

and that the trapezoid can have an inscribed circle. If the radius of this inscribed circle is

R

find the area of triangle

DCM

.

(125) 7

1

TOT 125 1986 Autumn J7 blue and red on chesboard, new moves for queen

Each square of a chessboard is painted either blue or red . Prove that the squares of one colour possess the property that the chess queen can perform a tour of all of them. The rules are that the queen may visit the squares of this colour not necessarily only once each , and may not be placed on squares of the other colour, although she may pass over them ; the queen moves along any horizontal , vertical or diagonal file over any distance.
(A . K . Tolpugo , Kiev)

(124) 6

1

TOT 124 1986 Autumn J6 28 teams in football tournament, 75 %

In a football tournament of one round (each team plays each other once,

2

points for win ,

1

point for draw and

0

points for loss),

28

teams compete. During the tournament more than

75\%

of the matches finished in a draw . Prove that there were two teams who finished with the same number of points.
(M . Vora, gymnasium student , Hungary)

(123) 5

1

TOT 123 1986 Autumn J5 locus of orthocenters of triangles inscribed in circle

Find the locus of the orthocentres (i.e. the point where three altitudes meet) of the triangles inscribed in a given circle .
(A. Andjans, Riga)

(122) 4

1

TOT 122 1986 Autumn J4 sum of products of reciprocals of {1,2,...,N}

Consider subsets of the set

1 , 2,..., N

. For each such subset we can compute the product of the reciprocals of each member. Find the sum of all such products.

(121) 3

1

TOT 121 1986 Autumn J3 game with 6 x 10 pieces of chocolate

A game has two players. In the game there is a rectangular chocolate bar, with

60

pieces, arranged in a

6 \times 1 0

formation , which can be broken only along the lines dividing the pieces. The first player breaks the bar along one line, discarding one section . The second player then breaks the remaining section, discarding one section. The first player repeats this process with the remaining section , and so on. The game is won by the player who leaves a single piece. In a perfect game which player wins?
{S. Fomin , Leningrad)

(120) 2

1

TOT 120 1986 Autumn J2 square and circle interesect in 8 points, perimeter sum

Square

ABCD

and circle

O

intersect in eight points, forming four curvilinear triangles,

AEF , BGH , CIJ

and

DKL

(

EF , GH, IJ

and

KL

are arcs of the circle) . Prove that (a) The sum of lengths of

EF

and

IJ

equals the sum of the lengths of

GH

and

KL

. (b) The sum of the perimeters of curvilinear triangles

AEF

and

CIJ

equals the sum of the perimeters of the curvilinear triangles

BGH

and

DKL

.
( V . V . Proizvolov , Moscow)

(119) 1

1

TOT 119 1986 Autumn J1 find 2-digit numbers , one is twice as big as the other

We are given two two-digit numbers ,

x

and

y

. It is known that

x

is twice as big as

y

. One of the digits of

y

is the sum, while the other digit of

y

is the difference, of the digits of

x

. Find the values of

x

and

y

, proving that there are no others.

(118) 6

1

TOT 118 1986 Spring S6 a^2_1-a^2_2+ ...+a^2_{2n- l} \ge (a_1- ...+a_{2n- l})^2

Given the nonincreasing sequence of non-negative numbers in which

a_1 \ge a_2 \ge a_3 \ge ... \ge a_{2n-1}\ge 0

.
Prove that

a^2_1 -a^2_2 + a^2_3 - ... + a^2_{2n- l} \ge (a_1 - a_2 + a_3 - ... + a_{2n- l} )^2

.
( L . Kurlyandchik , Leningrad )

(117) 5

1

TOT 117 1986 Spring S5 circumcenter lies on other circumcircle, inside #

The bisector of angle

BAD

in the parallelogram

ABCD

intersects the lines

BC

and

CD

at the points

K

and

L

respectively. It is known that

ABCD

is not a rhombus. Prove that the centre of the circle passing through the points

C, K

and

L

lies on the circle passing through the points

B, C

and

D

.

(116) 4

1

TOT 116 1986 Spring S4 F(x+1)F(x)+F(x+1)+1=0, F not continuous

The function

F

, defined on the entire real line, satisfies the following relation (for all

x

) :

F(x +1 )F(x) + F(x + 1 ) + 1 = 0

. Prove that

F

is not continuous.
(A.I. Plotkin, Leningrad)

(115) 3

1

TOT 115 1986 Spring S3 edges of tetrahedron coincide with vectors, 0 sum?

Vectors coincide with the edges of an arbitrary tetrahedron (possibly non-regular). Is it possible for the sum of these six vectors to equal the zero vector?
(Problem from Leningrad)

(114) 1

1

TOT 114 1986 Spring S1 max of k^2/1.001^k when k is natural

For which natural number

k

does

\frac{k^2}{1.001^k}

attain its maximum value?

(113) 7

1

TOT 113 1986 Spring J7 30 pupils from the same class exchange visits

Thirty pupils from the same class decided to exchange visits. Any pupil may make several visits during one evening, but must stay home if he is receiving guests that evening. Prove that in order that each pupil visit each of his classmates (a) four evenings are not enough (b) five evenings are not enough (c) ten evenings are enough (d) even seven evenings are enough

(112) 6

1

TOT 112 1986 Spring J6 Sisyphian Labour, 1001 steps, 500 rocks

( "Sisyphian Labour" ) There are

1001

steps going up a hill , with rocks on some of them {no more than 1 rock on each step ) . Sisyphus may pick up any rock and raise it one or more steps up to the nearest empty step . Then his opponent Aid rolls a rock (with an empty step directly below it) down one step . There are

500

rocks, originally located on the first

500

steps. Sisyphus and Aid move rocks in turn , Sisyphus making the first move . His goal is to place a rock on the top step. Can Aid stop him?
( S . Yeliseyev)

(111) 5

1

TOT 111 1986 Spring J5 20 football teams in a tournament

20

football teams take part in a tournament . On the first day all the teams play one match . On the second day all the teams play a further match . Prove that after the second day it is possible to select

10

teams, so that no two of them have yet played each other.
( S . A . Genkin)

(110) 4

1

TOT 110 1986 Spring J4 intersection of circumcircles lies on square's diagonal

We are given the square

ABCD

. On sides

AB

and

CD

we are given points

K

and

L

respectively, and on segment

KL

we are given point

M

. Prove that the second intersection point (i.e. the one other than

M

) of the intersection points of circles circumscribed around triangles

AKM

and

MLC

lies on the diagonal

AC

.
(V . N . Dubrovskiy)

(109) 3

1

TOT 109 1986 Spring J3 strees of a town arranged in 3 directions, equilateral

The streets of a town are arranged in three directions , dividing the town into blocks which are equilateral triangles of equal area. Traffic , when reaching an intersection, may only go straight ahead, or turn left or right through

120^0

, as shown in the diagram. https://cdn.artofproblemsolving.com/attachments/3/6/a100a5c39bf15116582bc0bceb76fcbae28af9.png No turns are permitted except the ones at intersections . One car departs for a certain nearby intersection and when it reaches it a second car starts moving toward it. From then on both cars continue travelling at the same speed (but do not necessarily take the same turns). Is it possible that there will be a time when they will encounter each other somewhere?
( N . N . Konstantinov , Moscow )

(108) 2

1

TOT 108 1986 Spring J2 each digit of decimal repr. of N divides N

A natural number

N

is written in its decimal representation . It is known that for each digit in this representation , this digit divides exactly into the number

N

(the digit

0

is not encountered). What is the maximum number of different digits which there can be in such a representation of

N

?
(S . Fomin, Leningrad)

(107) 1

1

TOT 107 1986 Spring J1 3 out of 4 regions have equal areas

Through vertices

A

and

B

of triangle

ABC

are constructed two lines which divide the triangle into four regions (three triangles and one quadrilateral). It is known that three of them have equal area. Prove that one of these three regions is the quadrilateral .
(G . Galperin , A . Savin, Moscow)

1986 Tournament Of Towns

Subcontests

TOT 131 1986 Autumn S7 21 points on a circle, angle of arc <=120

TOT 130 1986 Autumn S6 numbers 1-32 twice in a 8x8 chessboard

TOT 129 1986 Autumn S4 1987 /( 1 986 !! + 1985 !! )

TOT 128 1986 Autumn S3 100 triangles, not 1 can be covered by other 99

TOT 127 1986 Autumn S2 N-1 infinite ar. progressions with diff 2,3,4,..,Ν

TOT 126 1986 Autumn S1 tangential trapezoid, area wanted

TOT 125 1986 Autumn J7 blue and red on chesboard, new moves for queen

TOT 124 1986 Autumn J6 28 teams in football tournament, 75 %

TOT 123 1986 Autumn J5 locus of orthocenters of triangles inscribed in circle

TOT 122 1986 Autumn J4 sum of products of reciprocals of {1,2,...,N}

TOT 121 1986 Autumn J3 game with 6 x 10 pieces of chocolate

TOT 120 1986 Autumn J2 square and circle interesect in 8 points, perimeter sum

TOT 119 1986 Autumn J1 find 2-digit numbers , one is twice as big as the other

TOT 118 1986 Spring S6 a^2_1-a^2_2+ ...+a^2_{2n- l} \ge (a_1- ...+a_{2n- l})^2

TOT 117 1986 Spring S5 circumcenter lies on other circumcircle, inside #

TOT 116 1986 Spring S4 F(x+1)F(x)+F(x+1)+1=0, F not continuous

TOT 115 1986 Spring S3 edges of tetrahedron coincide with vectors, 0 sum?

TOT 114 1986 Spring S1 max of k^2/1.001^k when k is natural

TOT 113 1986 Spring J7 30 pupils from the same class exchange visits

TOT 112 1986 Spring J6 Sisyphian Labour, 1001 steps, 500 rocks

TOT 111 1986 Spring J5 20 football teams in a tournament

TOT 110 1986 Spring J4 intersection of circumcircles lies on square's diagonal

TOT 109 1986 Spring J3 strees of a town arranged in 3 directions, equilateral

TOT 108 1986 Spring J2 each digit of decimal repr. of N divides N

TOT 107 1986 Spring J1 3 out of 4 regions have equal areas

1986 Tournament Of Towns

Subcontests

TOT 131 1986 Autumn S7 21 points on a circle, angle of arc &lt;=120

TOT 130 1986 Autumn S6 numbers 1-32 twice in a 8x8 chessboard

TOT 129 1986 Autumn S4 1987 /( 1 986 !! + 1985 !! )

TOT 128 1986 Autumn S3 100 triangles, not 1 can be covered by other 99

TOT 127 1986 Autumn S2 N-1 infinite ar. progressions with diff 2,3,4,..,&Nu;

TOT 126 1986 Autumn S1 tangential trapezoid, area wanted

TOT 125 1986 Autumn J7 blue and red on chesboard, new moves for queen

TOT 124 1986 Autumn J6 28 teams in football tournament, 75 %

TOT 123 1986 Autumn J5 locus of orthocenters of triangles inscribed in circle

TOT 122 1986 Autumn J4 sum of products of reciprocals of {1,2,...,N}

TOT 121 1986 Autumn J3 game with 6 x 10 pieces of chocolate

TOT 120 1986 Autumn J2 square and circle interesect in 8 points, perimeter sum

TOT 119 1986 Autumn J1 find 2-digit numbers , one is twice as big as the other

TOT 118 1986 Spring S6 a^2_1-a^2_2+ ...+a^2_{2n- l} \ge (a_1- ...+a_{2n- l})^2

TOT 117 1986 Spring S5 circumcenter lies on other circumcircle, inside #

TOT 116 1986 Spring S4 F(x+1)F(x)+F(x+1)+1=0, F not continuous

TOT 115 1986 Spring S3 edges of tetrahedron coincide with vectors, 0 sum?

TOT 114 1986 Spring S1 max of k^2/1.001^k when k is natural

TOT 113 1986 Spring J7 30 pupils from the same class exchange visits

TOT 112 1986 Spring J6 Sisyphian Labour, 1001 steps, 500 rocks

TOT 111 1986 Spring J5 20 football teams in a tournament

TOT 110 1986 Spring J4 intersection of circumcircles lies on square&#039;s diagonal

TOT 109 1986 Spring J3 strees of a town arranged in 3 directions, equilateral

TOT 108 1986 Spring J2 each digit of decimal repr. of N divides N

TOT 107 1986 Spring J1 3 out of 4 regions have equal areas

TOT 131 1986 Autumn S7 21 points on a circle, angle of arc <=120

TOT 127 1986 Autumn S2 N-1 infinite ar. progressions with diff 2,3,4,..,Ν

TOT 110 1986 Spring J4 intersection of circumcircles lies on square's diagonal