1997 Tournament Of Towns

Part of Tournament Of Towns

Subcontests

(42)

1

n congruent triangles by \\ parallel to sides of an equilateral

Lines parallel to the sides of an equilateral triangle are drawn so that they cut each of the sides into n equal segments and the triangle into n congruent triangles. Each of these n triangles is called a “cell”. Also lines parallel to each of the sides of the original triangle are drawn through each of the vertices of the original triangle. The cells between any two adjacent parallel lines form a “stripe”.
(a) If

n =10

, what is the maximum number of cells that can be chosen so that no two chosen cells belong to one stripe?
(b)The same question for

n = 9

.
(R Zhenodarov)

1

secret code in which every letter is replaced by a word with max 10 letters

Dima invented a secret code in which every letter is replaced by a word no longer than

10

letters. A code is called “good” if every encoded word can be decoded in only one way. Serjozha (with the help of a computer) checked that for Dima’s code, every possible word of at most

10000

letters can be decoded in only one way. Does it follow that Dima’s code is good? (Note that Dima and Serjozha are Russian, so they use the Cyrillic alphabet, which has

33

letters! A word is any sequence of letters.)
(D Piontkovskiy, S Shalunov)

1

several identical napkins on a table, hammering nails

(a) Several identical napkins, each in the shape of a regular hexagon, are put on a table (the napkins may overlap). Each napkin has one side which is parallel to a fixed line. Is it always possible to hammer a few nails into the table so that each napkin is nailed with exactly one nail?
(b) The same question for regular pentagons.
(A Kanel)

1

TOT 562 1997 Autumn S A3 \pi \sqrt1 \pm \sqrt2 \pm ... \pm \sqrt{100}

All expressions of the form

\pm \sqrt1 \pm \sqrt2 \pm ... \pm \sqrt{100}

(with every possible combination of signs) are multiplied together. Prove that the result is: (a) an integer; (b) the square of an integer.
(A Kanel)

1

broken line divides a polygon into 2 polygons

Which of the following statements are true?
(a) If a polygon can be divided into two congruent polygons by a broken line segment, it can be divided into two congruent polygons by a straight line segment.
(b) If a convex polygon can be divided into two congruent polygons by a broken line segment, it can be so divided by a straight line segment.
(c) If a convex polygon can be divided into two polygons by a broken line segment, one of which can be mapped onto the other by a combination of rotations and translations, it can be so divided by a straight line segment.
(S Markelov,)

1

is ABC isosceles if AP = AQ ?, angle bisectors wtice

M

N

are the midpoints of the sides

AB

AC

of a triangle ABC respectively.

P

Q

are points on the sides

AB

AC

respectively such that the bisector of the angle

ACB

also bisects the angle

MCP

, and the bisector of the angle

ABC

also bisects the angle

NBQ

AP = AQ

, does it follow that

ABC

is isosceles?
(V Senderov)

1

TOT 559 1997 Autumn S O4 max no of knight in 5x5 chessboard

The maximum possible number of knights are placed on a

5 \times 5

chessboard so that no two attack each other. Prove that there is only one possible placement.
(A Kanel)

1

TOT 558 1997 Autumn S O3 xy(x -y) + yz(y-z) + zx(z-x) = 6

Prove that the equation

xy(x -y) + yz(y-z) + zx(z-x) = 6

has infinitely many solutions in integers

x, y

z

.
(N Vassiliev)

1

TOT 557 1997 Autumn S O2 touchpoints of incircle and excircle trisect side

a

b

be two sides of a triangle. How should the third side

c

be chosen so that the points of contact of the incircle and the excircle with side

c

divide that side into three equal segments? (The excircle corresponding to the side

c

is the circle which is tangent to the side

c

and to the extensions of the sides

a

b

1

TOT 551 1997 Autumn J A1 x_{n+2} =x_n - 1/x_{n+1}

x_1,x_2, ...

is defined by the following equations:

x_1=19, \ \ x_2=97, \ \ x_{n+2} =x_n - \frac{1}{x_{n+1}}

n \ge 1

. Prove that there exists a positive integer

k

x_k=0

k

1

TOT 553 1997 Autumn J A3 checkers on a 1xn board

Initially there is a checker on every square of a

1\times n

board. The first move consists of moving a checker to an adjacent square thus creating a stack of two checkers. Then each time when making a move, one can choose a stack and move it in either direction as many squares on the board as there are checkers in the stack. If after the move the stack lands on a non-empty square, it is placed on top of the stack which is already there. Prove that it is possible to stack all the checkers on one square in

n - 1

moves.
(A Shapovalov)

1

TOT 555 1997 Autumn J A5 BW coloring of a cube

Each face of a cube is of the same size as each square of a chessboard. The cube is coloured black and white, placed on one of the squares of the chessboard and rolled so that each square of the chessboard is visited exactly once. Can this be done in such a way that the colour of the visited square and the colour of the bottom face of the cube are always the same?
(A Shapovalov)

1

100 triangles by lines // sides of equilateral triangle

Lines parallel to the sides of an equilateral triangle are drawn so that they cut each of the sides into

10

equal segments and the triangle into

100

congruent triangles. Each of these

100

triangles is called a “cell”. Also lines parallel to each of the sides of the original triangle are drawn through each of the vertices of the original triangle. The cells between any two adjacent parallel lines form a “stripe”. What is the maximum number of cells that can be chosen so that no two chosen cells belong to one stripe?
(R Zhenodarov)

1

TOT 554 1997 Autumn J A4 AD bisects <CAE, 2 intersecting circles

Two circles intersect at points

A

B

. A common tangent touches the first circle at point

C

and the second at point

D

\angle CBD > \angle CAD

CB

intersect the second circle again at point

E

AD

bisects the angle

\angle CAE

.
(P Kozhevnikov)

1

TOT 552 1997 Autumn J A2 line contruction

M

is the midpoint of the side

BC

ABC

. Construct a line

\ell

intersecting the triangle and parallel to

BC

such that the segment of

\ell

between the sides

AB

AC

is the hypotenuse of a right-angled triangle with

M

being its third vertex.
(Folklore)

1

TOT 550 1997 Autumn J O4 lines in a chessboard

We want to draw a number of straight lines such that for each square of a chessboard, at least one of the lines passes through an interior point of the square. What is the smallest number of lines needed for a (a)

3\times 3

4\times 4

chessboard? Use a picture to show that this many lines are enough, and prove that no smaller number would do.
(M Vyalyi)

1

TOT 549 1997 Autumn J O3 AK=DM+BK in square ABCD

ABCD

K

is a point on the side

BC

and the bisector of

\angle KAD

CD

M

. Prove that the length of segment

AK

is equal to the sum of the lengths of segments

DM

BK

1

TOT 547 1997 Autumn J O2 x^2 + y^2 - z^2 = 1997 NT

Prove that the equation

x^2 + y^2 - z^2 = 1997

has infinitely many solutions in integers

x

y

z

.
(N Vassiliev)

1

TOT 547 1997 Autumn J O1 a person on an escalator

On an escalator which is not moving, a person descends faster than he ascends. Is it faster for this person to descend and ascend once on an upward-moving escalator, or to descend and ascend once on a downward-moving escalator? (It is assumed that all the speeds mentioned here are constant, that the speed of the escalator is the same no matter if it is moving up or down and that the speed of the person relative to the escalator is always greater than the speed of the escalator.)
(Folklore)

1

TOT 545 1997 Spring S A6 F(x)G(x) = 1 +x + x^2 +...+ x^{n-1}

F(x)

G(x)

are polynomials with coefficients

0

1

F(x)G(x) = 1 +x + x^2 +...+ x^{n-1}

n > 1

, then one of them can be represented in the form

(1 +x + x^2 +...+ x^{k-1}) T(x)

k > 1

T(x)

is a polynomial with coefficients

0

1

.
(V Senderov, M Vialiy)

1

TOT 544 1997 Spring S A5 sum 1/(1+a+b) <1

\frac{1}{1+a+b}+\frac{1}{1+b+c}+\frac{1}{1+c+a} <1

a, b

c

are positive numbers such that

abc = 1

1

TOT 542 1997 Spring S A3 - 20 weights

20

weights such that any object of integer weight

m

1 \le m \le1997

, can be balanced by placing it on one pan of a balance and a subset of the weights on the other pan. What is the minimal value of largest of the

20

weights if the weights are (a) all integers; (b) not necessarily integers?
(M Rasin)

1

one can translate each strip // to itself so that together they cover circle

Several strips and a circle of radius

1

are drawn on the plane. The sum of the widths of the strips is

100

. Prove that one can translate each strip parallel to itself so that together they cover the circle.
(M Smurov )

1

support chord, convex polygon placed inside another convex polygon

A convex polygon

G

is placed inside a convex polygon

F

so that their boundaries have no common points. A segment

s

joining two points on the boundary of

F

is called a support chord for

G

if s contains a side or only a vertex of

G

. Prove that (a) there exists a support chord for

G

such that its midpoint lies on the boundary of

G

, (b) there exist at least two such chords.
(P Pushkar)

1

DE bisects <ADC, AD and BE angle bisectors

D

E

are points on the sides

BC

AC

ABC

AD

BE

are angle bisectors of the triangle

ABC

DE

\angle ADC

\angle A

1

TOT 540 1997 Spring S O5 red and green points

In a game, the first player paints a point on the plane red; the second player paints 10 uncoloured points on the plane green; then the first player paints an uncoloured point on the plane red; the second player paints 10 uncoloured points on the plane green; and so on. The first player wins if there are three red points which form an equilateral triangle. Can the second player prevent the first player from winning?
(A Kanel)

1

angle between planes ABC' and ACD; tetrahedron tetrahedron inscribed in sphere

All edges of a tetrahedron

ABCD

are equal. The tetrahedron

ABCD

is inscribed in a sphere.

CC'

DD'

are diameters. Find the angle between the planes

ABC

ACD'

.
(A Zaslavskiy)

1

TOT 538 1997 Spring S O3 difference of areas of circle parts

A circle centred at

(a, b)

contains the origin

(0,0)

S^+

the total area of the parts of the circle in the first and third quadrants, and by

S^-

the total area of the parts of the circle in the second and the fourth quadrants. Compute

S^+ -S^-

1

TOT 537 1997 Spring S O2 a=b if ab | (a^2 + b^2)

a

b

be positive integers. If

a^2 + b^2

is divisible by

ab

a = b

1

TOT 536 1997 Spring S O1 cube is cut into 99 smaller cubes

A cube is cut into 99 smaller cubes, exactly 98 of which are unit cubes. Find the volume of the original cube.
(V Proizvolov)

1

TOT 535 1997 Spring J A7 ten weights , power of 2

You are given a balance and one copy of each of ten weights of

1, 2, 4, 8, 16, 32, 64, 128, 256

512

grams. An object weighing

M

M

is a positive integer, is put on one of the pans and may be balanced in different ways by placing various combinations of the given weights on either pan of the balance. (a) Prove that no object may be balanced in more than

89

ways. (b) Find a value of

M

such that an object weighing

M

grams can be balanced in

89

ways.
(A Shapovalov, A Kulakov)

1

TOT 533 1997 Spring J A5 1997^{17} not sum of perfect squares

Prove that the number
(a)

97^{97}

1997^{17}

cannot be equal to a sum of cubes of several consecutive integers.
(AA Egorov)

1

[ABC] = 1/2 [AC'BA'C B'$] convex , <A+<B+<C =<A' +<B'+<C'

AC' BA'C B'

is a convex hexagon such that

AB' = AC'

BC' = BA'

CA' = CB'

\angle A +\angle B + \angle C = \angle A' + \angle B' + \angle C'

. Prove that the area of the triangle

ABC

is half the area of the hexagon.
(V Proizvolov)

1

angle chasing inside an isosceles triangle

P

be a point inside the triangle

ABC

AB = BC

\angle ABC = 80^o

\angle PAC = 40^o

\angle ACP = 30^o

\angle BPC

1

TOT 531 1997 Spring J A3 - 2n players in chess tournament,

In a chess tournament, each of

2n

players plays every other player once in each of two rounds. A win is worth

1

point, a draw is worth

\frac12

point and a loss is worth nothing. Prove that if for every player, the total score in the first round differs from that in the second round by at least n points, then this difference is exactly n points for every player.
(B Frenkin)

1

TOT 530 1997 Spring J A2 - 25 pieces of cheese of different weights.

25

pieces of cheese of different weights. Is it always possible to cut one of the pieces into two parts and put the

26

pieces in two packets so that

\bullet

each packet contains

13

\bullet

the total weights of the two packets are equal;

\bullet

the two parts of the piece which has been cut are in different packets?
(VL Dolnikov)

1

TOT 529 1997 Spring J A1, angles <A =1/3 (<B +<C)

One side of a triangle is equal to one third of the sum of the other two. Prove that the angle opposite the first side is the smallest angle of the triangle.
(AK Tolpygo)

1

isosceles triangle wanted, parallelogram related

E

is the midpoint of the side

AD

of a parallelogram

ABCD

F

is the foot of the perpendicular from the vertex

B

CE

ABF

is an isosceles triangle.
(MA Bolchkevich)

1

TOT 527 1997 Spring J O4 square cut into 25 smaller squares,

A square is cut into 25 smaller squares, exactly 24 of which are unit squares. Find the area of the original square.
(V Proizvolov)

1

TOT 526 1997 Spring J O3 difference of sum of areas of circle pieces

The vertical diameter of a circle is moved a centimetres to the right, and the horizontal diameter of this circle is moved

b

centimetres up. These two lines divide the circle into four pieces. Consider the sum of the areas of the largest and the smallest pieces, and the sum of the areas of the other two pieces. Find the difference between these two sums.
(G Galperin, NB Vassiliev)

1

TOT 525 1997 Spring J O2 equilateral triangle billiard

Baron Munchausen plays billiards on a table with the shape of an equilateral triangle. He claims to have shot a ball from one of the sides of this table so that it passed through a certain point three times in three different directions and then returned to the original point on the side. Can that be true, assuming that the usual law of reflection holds?
(Μ Evdokimov)

1

TOT 524 1997 Spring J O1 sum of digits divisible by 5

How many integers from

1

1997

have the sum of their digits divisible by

5

?
(AI Galochkin)