MathDB

1

TOT 163 1987 Autumn S6 policeman, bandit, infinite plane, grid

A certain town is represented as an infinite plane, which is divided by straight lines into squares. The lines are streets, while the squares are blocks. Along a certain street there stands a policeman on each

100

th intersection . Somewhere in the town there is a bandit , whose position and speed are unknown, but he can move only along the streets. The aim of the police is to see the bandit . Does there exist an algorithm available to the police to enable them to achieve their aim?
(A. Andjans, Riga)

(162) 6

1

TOT 162 1987 Autumn S6 bw equilateral triangles inside equilateral

An equilateral triangle is divided by lines, parallel to its sides, into equilateral triangles, all of the same size. One of the smaller triangles is black while the others are white. It is permitted to intersect simultaneously some small triangles with a line parallel to any side of the original triangle and to change the colour of each intersected small triangle from one colour to the other . Is it always possible to find a sequence of such operations so that the smaller triangles all become white?

(161) 5

1

TOT 161 1987 Autumn S5 colouring pairs of integers

Consider the set of all pairs of positive integers

(A , B)

in which

A < B

. Some of these pairs are to

be

designated as "black" , while the remainder are to be designated as "white" . Is it possible to designate these pairs in such a way that for any triple of positive integers of form

A, A + D, A + 2D

, in which

D > 0

, the associated pairs

(A, A + D )

,

(A , A + 2D)

and

(A + D, A + 2D)

would include at least one pair of each colour?

(160) 4

1

TOT 160 1987 Autumn S4 each segment equal to inradius

From point

M

in triangle

ABC

perpendiculars are dropped to each altitude. It can be shown that each of the line segments of altitudes, measured between the vertex and the foot of the perpendicular drawn to it, are of equal length. Prove that these lengths are each equal to the diameter of the circle inscribed in the triangle.

(159) 3

1

TOT 159 1987 Autumn S3 a^2 + 1 divisible by b , b^2 + 1 divisible by a .

Prove that there are infinitely many pairs of natural numbers

a

and

b

such that

a^2 + 1

is divisible by

b

and

b^2 + 1

is divisible by

a

.

(158) 2

1

TOT 158 1987 Autumn S2 boy in centre of square swimming pool

In the centre of a square swimming pool is a boy, while his teacher (who cannot swim) is standing at one corner of the pool. The teacher can run three times as fast as the boy can swim, but the boy can run faster than the teacher . Can the boy escape from the teacher?

(157) 1

1

TOT 157 1987 Autumn S1 area of quadrilateral by perpendiculars

From vertex

A

in square

ABCD

(of side length

1

) two lines are drawn , one intersecting side

BC

and the other intersecting side

CD

. The angle between these lines is

\theta

. From vertices

B

and

D

we construct perpendiculars to each of these lines . Find the area of the quadrilateral whose vertices are the four feet of these perpendiculars.

(156) 7

1

TOT 156 1987 Autumn J7 3 triangles (blue, green and red) , common P

Three triangles (blue, green and red) have a common interior point

M

. Prove that it is possible to choose one vertex from each triangle so that point

M

is located inside the triangle formed by these selected vertices.
(Imre Barani, Hungary)

(155) 6

1

TOT 155 1987 Autumn J6 2000 apples in several baskets

There are

2000

apples , contained in several baskets. One can remove baskets and /or remove apples from the baskets. Prove that it is possible to then have an equal number of apples in each of the remaining baskets, with the total number of apples being not less than

100

.
(A. Razborov)

(154) 5

1

TOT 154 1987 Autumn J5 A^2 + B^2 + C^2 \le 2(AB + BC + CA)

We are given three non-negative numbers

A , B

and

C

about which it is known that

A^4 + B^4 + C^4 \le 2(A^2B^2 + B^2C^2 + C^2A^2)

(a) Prove that each of

A, B

and

C

is not greater than the sum of the others. (b) Prove that

A^2 + B^2 + C^2 \le 2(AB + BC + CA)

. (c) Does the original inequality follow from the one in (b)?
(V.A. Senderov , Moscow)

(153) 4

1

TOT 153 1987 Autumn J4 lines bisecting areas, arvc and broken line

We are given a figure bounded by arc

AC

of a circle, and a broken line

ABC

, with the arc and broken line being on opposite sides of the chord

AC

. Construct a line passing through the mid-point of arc

AC

and dividing the area of the figure into two regions of equal area.

(152) 3

1

TOT 152 1987 Autumn J3 game, winmer chooses 1987

In a game two players alternately choose larger natural numbers. At each turn the difference between the new and the old number must be greater than zero but smaller than the old number. The original number is 2. The winner is considered to be the player who chooses the number

1987

. In a perfect game, which player wins?

(151) 2

1

TOT 151 1987 Autumn J2 locus inside a rhombus <AMB+CMD=180^p

Find the locus of points

M

inside the rhombus

ABCD

such that the sum of angles

AMB

and

CMD

equals

180^o

.

(150) 1

1

TOT 150 1987 Autumn J1 second last digit of each power of three is even

Prove that the second last digit of each power of three is even .
(V . I . Plachkos)

(149) 6

1

TOT 149 1987 Spring Main S6 game with 1 knight on an 8x8 chessboard

Two players play a game on an

8

by

8

chessboard according to the following rules. The first player places a knight on the board. Then each player in turn moves the knight , but cannot place it on a square where it has been before. The player who has no move loses. Who wins in an errorless game , the first player or the second one? (The knight moves are the normal ones. )
(V . Zudilin , year 12 student , Beltsy (Moldova))

(148) 5

1

TOT 148 1987 Spring Main S4 locus wanted, right,w equilateral related

Perpendiculars are drawn from an interior point

M

of the equilateral triangle

ABC

to its sides , intersecting them at points

D, E

and

F

. Find the locus of all points

M

such that

DEF

is a right triangle .
(J . Tabov , Sofia)

(147) 4

1

TOT 147 1987 Spring Main S4 \sqrt{2\sqrt{2}{\sqrt{3}\sqrt{4 ...}}} <3

For any natural

n

prove the inequality

\sqrt{2\sqrt{2}{\sqrt{3}\sqrt{4 ...\sqrt{n-1\sqrt{n}}}}} <3

(146) 3

1

TOT 146 1987 Spring Main S3 exchanges of apartments in a city

In a certain city only simple (pairwise) exchanges of apartments are allowed (if two families exchange fiats , they are not allowed to participate in another exchange on the same day). Prove that any compound exchange may be effected in two days. It is assumed that under any exchange (simple or comp ound) each family occupies one fiat before and after the exchange and the family cannot split up .
(A . Shnirelman , N .N . Konstantinov)

(145) 2

1

TOT 145 1987 Spring Main S2 disk of radius 1 is covered by 7 identical disks

Α disk of radius

1

is covered by seven identical disks. Prove that their radii are not less than

\frac12

.

(144) 1

1

TOT 144 1987 Spring Main S1 p(a) - p(b) = 1 , integer polynomial

Suppose

p(x)

is a polynomial with integer coefficients. It is known that

p(a) - p(b) = 1

(where

a

and

b

are integers). Prove that

a

and

b

differ by

1

.
(Folklore)

(143) 4

1

TOT 143 1987 Spring Train S4 a=b, sum of distances in a square on cheeboard

On a chessboard a square is chosen . The sum of the squares of distances from its centre to the centre of all black squares is designated by

a

and to the centre of all white squares by

b

. Prove that

a = b

.
(A. Andj ans, Riga)

(142) 2

1

TOT 142 1987 Spring Train S2 distance from plane in 3d space wanted

In

3

dimensional space we are given a parallelogram

ABCD

and plane

M

. The distances from vertices

A, B

and

C

to plane

M

are

a, b

and

c

respectively. Find the distance

d

from vertex

D

to the plane

M

.

(141) 1

1

TOT 141 1987 Spring Train S1 1986 as the sum of squares of 6 odd integers

Is it possible to represent the number

1986

as the sum of squares of

6

odd integers?

(140) 5

1

TOT 140 1987 Spring Main J5 certain number of cubes painted in 6 colours

A certain number of cubes are painted in six colours, each cube having six faces of different colours (the colours in different cubes may be arranged differently) . The cubes are placed on a table so as to form a rectangle. We are allowed to take out any column of cubes, rotate it (as a whole) along its long axis and replace it in the rectangle. A similar operation with rows is also allowed. Can we always make the rectangle monochromatic (i.e. such that the top faces of all the cubes are the same colour) by means of such operations?
( D. Fomin , Leningrad)

(139) 4

1

TOT 139 1987 Spring Main J4 altitudes bisector passes through circumcenter

Angle

A

of the acute-angled triangle

ABC

equals

60^o

. Prove that the bisector of one of the angles formed by the altitudes drawn from

B

and

C

, passes through the circumcircle 's centre.
(V . Pogrebnyak , year 12 student , Vinnitsa,)

(138) 3

1

TOT 138 1987 Spring Main J3 9 pawns in a 3x3 square on 8x8 chessboard

Nine pawns forming a

3

by

3

square are placed in the lower left hand corner of an

8

by

8

chessboard. Any pawn may jump over another one standing next to it into a free square, i .e. may be reflected symmetrically with respect to a neighb our's centre (jumps may be horizontal , vertical or diagonal) . It is required to rearrange the nine pawns in another corner of the chessboard (in another

3

by

3

square) by means of such jumps. Can the pawns be thus re-arranged in the (a) upper left hand corner? (b) upper right hand corner?
(J . E . Briskin)

(137) 2

1

TOT 137 1987 Spring Main J2 tangentials from cuts of an octagon

Quadrilaterals may be obtained from an octagon by cutting along its diagonals (in

8

different ways) . Can it happen that among these

8

quadrilaterals (a) four (b ) five possess an inscribed circle?
(P. M . Sedrakyan , Yerevan)

(136) 1

1

TOT 136 1987 Spring Main J1 machine gives 5 pennies for each nickel

A machine gives out five pennies for each nickel inserted into it and five nickels for each penny. Can Peter , who starts out with one penny, use the machine several times in such a way as to end up with an equal number of nickels and pennies?
(F. Nazarov, Leningrad Olympiad, 1987)

(135) 4

1

TOT 135 1987 Spring Train J4 tiles of right triangle with perp. sides 1,2

We are given tiles in the form of right angled triangles having perpendicular sides of length

1

cm and

2

cm. Is it possible to form a square from

20

such tiles?
( S . Fomin , Leningrad)

(134) 3

1

TOT 134 1987 Spring Train J3 two bottles with water and salt, mix

We are given two three-litre bottles, one containing

1

litre of water and the other containing

1

litre of

2\%

salt solution . One can pour liquids from one bottle to the other and then mix them to obtain solutions of different concentration . Can one obtain a

1 . 5\%

solution of salt in the bottle which originally contained water?
(S . Fomin, Leningrad),

(133) 2

1

TOT 133 1987 Spring Train J2 trangle of feet of altitudes. parallel sides

In an acute angled triangle the feet of the altitudes are joined to form a new triangle. In this new triangle it is known that two sides are parallel to sides of the original triangle . Prove that the third side is also parallel to one of the sides of the original triangle .

(132) 1

1

TOT 132 1987 Spring Train J1 3(1+a^2+a^4) \ge (1+a+a^2)^2

Prove that for all values of

a

,

3(1+a^2+a^4) \ge (1+a+a^2)^2

.

1987 Tournament Of Towns

Subcontests

TOT 163 1987 Autumn S6 policeman, bandit, infinite plane, grid

TOT 162 1987 Autumn S6 bw equilateral triangles inside equilateral

TOT 161 1987 Autumn S5 colouring pairs of integers

TOT 160 1987 Autumn S4 each segment equal to inradius

TOT 159 1987 Autumn S3 a^2 + 1 divisible by b , b^2 + 1 divisible by a .

TOT 158 1987 Autumn S2 boy in centre of square swimming pool

TOT 157 1987 Autumn S1 area of quadrilateral by perpendiculars

TOT 156 1987 Autumn J7 3 triangles (blue, green and red) , common P

TOT 155 1987 Autumn J6 2000 apples in several baskets

TOT 154 1987 Autumn J5 A^2 + B^2 + C^2 \le 2(AB + BC + CA)

TOT 153 1987 Autumn J4 lines bisecting areas, arvc and broken line

TOT 152 1987 Autumn J3 game, winmer chooses 1987

TOT 151 1987 Autumn J2 locus inside a rhombus <AMB+CMD=180^p

TOT 150 1987 Autumn J1 second last digit of each power of three is even

TOT 149 1987 Spring Main S6 game with 1 knight on an 8x8 chessboard

TOT 148 1987 Spring Main S4 locus wanted, right,w equilateral related

TOT 147 1987 Spring Main S4 \sqrt{2\sqrt{2}{\sqrt{3}\sqrt{4 ...}}} <3

TOT 146 1987 Spring Main S3 exchanges of apartments in a city

TOT 145 1987 Spring Main S2 disk of radius 1 is covered by 7 identical disks

TOT 144 1987 Spring Main S1 p(a) - p(b) = 1 , integer polynomial

TOT 143 1987 Spring Train S4 a=b, sum of distances in a square on cheeboard

TOT 142 1987 Spring Train S2 distance from plane in 3d space wanted

TOT 141 1987 Spring Train S1 1986 as the sum of squares of 6 odd integers

TOT 140 1987 Spring Main J5 certain number of cubes painted in 6 colours

TOT 139 1987 Spring Main J4 altitudes bisector passes through circumcenter

TOT 138 1987 Spring Main J3 9 pawns in a 3x3 square on 8x8 chessboard

TOT 137 1987 Spring Main J2 tangentials from cuts of an octagon

TOT 136 1987 Spring Main J1 machine gives 5 pennies for each nickel

TOT 135 1987 Spring Train J4 tiles of right triangle with perp. sides 1,2

TOT 134 1987 Spring Train J3 two bottles with water and salt, mix

TOT 133 1987 Spring Train J2 trangle of feet of altitudes. parallel sides

TOT 132 1987 Spring Train J1 3(1+a^2+a^4) \ge (1+a+a^2)^2

1987 Tournament Of Towns

Subcontests

TOT 163 1987 Autumn S6 policeman, bandit, infinite plane, grid

TOT 162 1987 Autumn S6 bw equilateral triangles inside equilateral

TOT 161 1987 Autumn S5 colouring pairs of integers

TOT 160 1987 Autumn S4 each segment equal to inradius

TOT 159 1987 Autumn S3 a^2 + 1 divisible by b , b^2 + 1 divisible by a .

TOT 158 1987 Autumn S2 boy in centre of square swimming pool

TOT 157 1987 Autumn S1 area of quadrilateral by perpendiculars

TOT 156 1987 Autumn J7 3 triangles (blue, green and red) , common P

TOT 155 1987 Autumn J6 2000 apples in several baskets

TOT 154 1987 Autumn J5 A^2 + B^2 + C^2 \le 2(AB + BC + CA)

TOT 153 1987 Autumn J4 lines bisecting areas, arvc and broken line

TOT 152 1987 Autumn J3 game, winmer chooses 1987

TOT 151 1987 Autumn J2 locus inside a rhombus &lt;AMB+CMD=180^p

TOT 150 1987 Autumn J1 second last digit of each power of three is even

TOT 149 1987 Spring Main S6 game with 1 knight on an 8x8 chessboard

TOT 148 1987 Spring Main S4 locus wanted, right,w equilateral related

TOT 147 1987 Spring Main S4 \sqrt{2\sqrt{2}{\sqrt{3}\sqrt{4 ...}}} &lt;3

TOT 146 1987 Spring Main S3 exchanges of apartments in a city

TOT 145 1987 Spring Main S2 disk of radius 1 is covered by 7 identical disks

TOT 144 1987 Spring Main S1 p(a) - p(b) = 1 , integer polynomial

TOT 143 1987 Spring Train S4 a=b, sum of distances in a square on cheeboard

TOT 142 1987 Spring Train S2 distance from plane in 3d space wanted

TOT 141 1987 Spring Train S1 1986 as the sum of squares of 6 odd integers

TOT 140 1987 Spring Main J5 certain number of cubes painted in 6 colours

TOT 139 1987 Spring Main J4 altitudes bisector passes through circumcenter

TOT 138 1987 Spring Main J3 9 pawns in a 3x3 square on 8x8 chessboard

TOT 137 1987 Spring Main J2 tangentials from cuts of an octagon

TOT 136 1987 Spring Main J1 machine gives 5 pennies for each nickel

TOT 135 1987 Spring Train J4 tiles of right triangle with perp. sides 1,2

TOT 134 1987 Spring Train J3 two bottles with water and salt, mix

TOT 133 1987 Spring Train J2 trangle of feet of altitudes. parallel sides

TOT 132 1987 Spring Train J1 3(1+a^2+a^4) \ge (1+a+a^2)^2

TOT 151 1987 Autumn J2 locus inside a rhombus <AMB+CMD=180^p

TOT 147 1987 Spring Main S4 \sqrt{2\sqrt{2}{\sqrt{3}\sqrt{4 ...}}} <3