1993 Tournament Of Towns

Part of Tournament Of Towns

Subcontests

(41)

1

TOT 3981993 Autumn A S6 x^4+ax^3+2x^2+bx+1=0

If it is known that the equation

x^4+ax^3+2x^2+bx+1=0

has a (real) root, prove the inequality

a^2+b^2 \ge 8.

1

4 frogs sit on vertices of a square, jumping

Four frogs sit on the vertices of a square, one on each vertex. They jump in arbitrary order but not simultaneously. Each frog jumps to the point symmetrical to its take-off position with respect to the centre of gravity of the three other frogs. Can one of them jump (at a given time) exactly on to one of the others (considering their locations as points)?
(A Andzans)

1

TOT 396 1993 Autumn A S4 convex 1993-gon is divided into convex 7-gons

1993

-gon is divided into convex

7

-gons. Prove that there are

3

neighbouring sides of the

1993

-gon belonging to one such

7

-gon. (A vertex of a

7

-gon may not be positioned on the interior of a side of the

1993

7

-gons either have no common points, exactly one common vertex or a complete common side.)
(A Kanel-Belov)

1

area of hexagon by vertices of 2 intersecting tirangles is fixed

Consider the hexagon which is formed by the vertices of two equilateral triangles (not necessarily equal) when the triangles intersect. Prove that the area of the hexagon is unchanged when one of the triangles is translated (without rotating) relative to the other in such a way that the hexagon continues to be defined.
(V Proizvolov)

1

TOT 394 1993 Autumn A S2 sequence 0-9 found in digit sequence

The decimal representation of all integers from

1

to an arbitrary integer

n

are written one after another as such:

123... 91011... 99100... (n).

Does there exist

n

such that each of the digits

0,1,2,...,9

appears the same number of times in the given sequence?
(A Andzans)

1

TOT 393 1993 Autumn A S1 max length in triangle by arc and 2 tangents

CA

CB

are drawn to a circle (

A

B

being the tangent points). Consider a “triangle” bounded by an arc

AB

(the smaller one) and segments

CA

CB

. Prove that the length of any segment inside the triangle is not greater than the length of

CA = CB

1

TOT 392 1993 Autumn O S4 unusual die

Peter wants to make an unusual die having different positive integers on each of its faces. For neighbouring faces the corresponding numbers should differ by at least two. Find the minimal sum of the six numbers.
(Folklore)

1

TOT 391 1993 Autumn O S3 1-25 in 5x5 table

Each of the numbers

1, 2, 3,... 25

is arranged in a

5

5

table. In each row they appear in increasing order (left to right). Find the maximal and minimal possible sum of the numbers in the third column.
(Folklore)

1

TOT 390 1993 Autumn O S2 <MCN =45^o , BC = BM, AC = AN, right triangle

M

N

are taken on the hypotenuse

AB

of a right triangle

ABC

BC = BM

AC = AN

. Prove that the angle

MCN

45

degrees.
(Folklore)

1

TOT 389 1993 Autumn O S1 x^2+y^3=z^2 NT

Consider the set of solutions of the equation

x^2+y^3=z^2.

in positive integers. Is it finite or infinite?
(Folklore)

1

TOT 388 1993 Autumn A J6 weight from 1-55 , even with lost weights

Construct a set of

k

integer weights that allows you to get any total integer weight from

1

55

grams even if some of the weights of the set are lost. Consider two versions: (a)

k = 10

, and any one of the weights may be lost; (b)

k = 12

, and any two of the weights may be lost.
(D Zvonkin)
(In both cases prove that the set found has the property required.)

1

TOT 387 1993 Autumn A J5 n +S(n) = p + S(p) = q + S(q), sum of digits

S(n)

denote the sum of digits of

n

(in decimal representation). Do there exist three different natural numbers

n

p

q

n +S(n) = p + S(p) = q + S(q)?

1

TOT 386 1993 Autumn A J4 diagonals of 1x1 in 8x8 table

1

1

square are arranged in an

8

8

table (one in each

1

1

square). Consider the union

W

64

diagonals drawn. The set

W

consists of several connected pieces (two points belong to the same piece if and only if W contains a path between them). Can the number of the pieces be greater than (a)

15

20

?
(NB Vassiliev)

1

midpoints of sides are vertices of square.

Three angles of a non-convex, non-self-intersecting quadrilateral are equal to

45

degrees (i.e. the last equals

225

degrees). Prove that the midpoints of its sides are vertices of a square.
(V Proizvolov)

1

TOT 384 1993 Autumn A J2 equal sums of areas of pairs of quads

PQRS

is placed inside the square

ABCD

in such a way that the segments

AP

BQ

CR

DS

intersect neither each other nor the square

PQRS

. Prove that the sum of areas of quadrilaterals

ABQP

CDSR

is equal to the sum of the areas of quadrilaterals

BCRQ

DAPS

1

TOT 383 1993 Autumn A J1 rows of 10 integers

10

integers are written in a row. A second row of

10

integers is formed as follows: the integer written under each integer

A

of the first row is equal to the total number of integers that stand to the right side of

A

(in the first row) and are strictly greater than A. A third row is formed by the same way under the second one, and so on.
(a) Prove that after several steps a “zero row” (i.e. a row consisting entirely of zeros) appears. (b) What is the maximal possible number of non-zero rows (i.e. rows in which at least one entry is not zero)?
(S Tokarev)

1

TOT 382 1993 Autumn O J4 three players- tournament

Three players Alexander, Beverley and Catherine participate in a tournament (all of them play the same number of games with each other). Is it possible that Alexander gets more points than the others, Catherine gets less points than the others, but Alexander has a smaller number of wins than the others and Catherine has a greater number of wins than the others? (A win scores

1

point, a draw scores

\frac12

1

TOT 381 1993 Autumn O J3 A-> A+d NT

A natural number

A

is given. One may add to it one of its divisors

d

1 < d < A

). One may then repeat this operation with the new number

A + d

and so on. Prove that starting from

A = 4

one can get any composite number by these operations.
(M Vyalyi)

1

TOT 380 1993 Autumn O J2 collinear midpoints

A

B

C

of a triangle are connected with points

A'

B'

C'

lying in the opposite sides of the triangle (not at vertices). Can the midpoints of the segments

AA'

BB'

CC'

lie in a straight line?
(Folklore)

1

TOT 379 1993 Autumn O J1 numbers on sides and vertices of hexagon

We are given a hexagon with a number written on each of its sides and vertices. Each number on a vertex is equal to the sum of the two numbers on neighbouring sides. Assume all numbers of the sides and one vertex number were erased. Is it possible to find out the number that had been erased from a vertex?
(Folklore)

1

TOT 378 1993 Spring A S7 each plant characterized by 100 attributes

In a handbook of plants each plant is characterized by

100

attributes (each attribute may either be present in a plant or not). Two plants are called dissimilar if they differ by no less than

51

attributes.
(a) Prove that the handbook cannot describe more than

50

pair-wise dissimilar plants. (b) Can it describe

50

pairwise dissimilar plants?
(Dima Tereshin)

1

TOT 377 1993 Spring A S5 piecewise linear function f(f(x)) = -x

Does there exist a piecewise linear function

f

defined on the segment [

-1,1]

(including the ends) such that

f(f(x)) = -x

for all x? (A function is called piecewise linear if its graph is the union of a finite set of points and intervals; it may be discontinuous).

1

TOT 376 1993 Spring A S4 a_{n+1} \ne sum k_n *a_n

Positive integers are written on the blackboard one after another. The next integer

a_{n+1}

(to be written after

a_1

a_2

...

a_n

) is an arbitrary integer not representable as a sum of several previous integers taken one or more times (i.e.

a_{n+1}

is not of the form

k_1 *a_i + k_2 *a_2 + ... + k_n *a_n

k_1

k_2

..

k_n

are non-negative integers). Prove that the process of writing cannot be infinite.
(A Belov)

1

TOT 375 1993 Spring A S3 dividing an inheritance

A fixed number of people are dividing an inheritance among themselves. An heir will be called poor if he gets less than

\$99

and rich if he gets more than

\$10 000

(some heirs may be neither rich nor poor). The total inheritance and the number of heirs are such that the total income of the rich heirs will be no less than that of the poor ones no matter how the inheritance is divided. Prove that the total income of the rich heirs is no less than

100

times that of the poor ones.
(F Nazarov)

1

max sum CM + ON, square on side of a triangle

A square is constructed on the side

AB

ABC

(outside the triangle).

O

is the centre of the square.

M

N

are the midpoints of the sides

BC

AC

. The lengths of these sides are

a

b

respectively. Find the maximal possible value of the sum

CM + ON

(when the angle at

C

changes).
(IF Sharygin)

1

TOT 373 1993 Spring A S1 sum of perimeters >1993, square grid

Inside a square with sides of length

1

unit several non-overlapping smaller squares with sides parallel to the sides of the large square are placed (the small squares may differ in size). Draw a diagonal of the large square and consider all of the small squares intersecting it. Can the sum of their perimeters be greater than

1993

?
(AN Vblmogorov)

1

TOT 371 1993 Spring O S3 like pascal triangle with fractions

Each number in the second, third, and further rows of the following triangle: https://cdn.artofproblemsolving.com/attachments/1/5/589d9266749477b0f56f0f503d4f18a6e5d695.png is equal to the difference of two neighbouring numbers standing above it. Find the last number (at the bottom of the triangle).
(GW Leibnitz,)

1

TOT 372 1993 Spring O S4 three piles of stones

Three piles of stones are given. One may add to, or remove from one of the piles in one operation the number of stones in the other two piles. For example

[12,3,5]

[12,20,5]

17 = 12 + 5

stones to pile 2 or

[4,3,5]

8 = 3 + 5

stones from pile

1

. Is it possible starting from the piles with

1993

199

19

stones to get one empty heap after several operations?
(MN Gusarov)

1

CM = CN if BM = DN for cyclic ABCD

ABCD

is inscribed in a circle,

M

is the intersection point of the lines

AB

CD

N

is the intersection point of the lines

BC

AD

. It is known that

BM = DN

CM = CN

1

TOT 369 1993 Spring O S1 2^n, delete first digit, again power of 2

Find all integers of the form

2^n

n

is a natural number) such that after deleting the first digit of its decimal representation we again get a power of

2

1

TOT 367 1993 Spring A J6 boat can move along a river

The width of a long winding river is not greater than

1

km. This means by definition that from any point of each bank of the river one can reach the other bank swimming

1

km or less. Is it true that a boat can move along the river so that its distances from both banks are never greater than (a)

0.7

0.8

km?
(Grigory Kondakov, Moscow)
You may assume that the banks consist of segments and arcs of circles.

1

red and blue points on a line

Two coloured points are marked on a line, with the blue one at the left and the red one at the right. You may add to the line two neighbouring points of the same color (both red or both blue) or delete two such points (neighbouring means that there is no coloured point between these two). Prove that after several such transformation you cannot again get only two points on the line in which the red one is at the left and the blue one is at the right.
(A Belov)

1

TOT 366 1993 Spring A J5 cut triangle 140-20-20

A paper triangle with the angles

20^o

20^o

140^o

is cut into two triangles by the bisector of one of its angles. Then one of these triangles is cut into two by its bisector, and so on. Prove that it is impossible to get a triangle similar to the initial one.
(AI Galochkin)

1

TOT 365 1993 Spring A J4 25 students in Peter’s class

25

students in Peter’s class (not counting him). Peter has observed that all

25

have different numbers of friends in this class. How many friends does Peter have in this class? (Give all possible answers.)
(S Toparev)

1

TOT 364 1993 Spring A J3 operation x(z) = (x* y)+ z

An operation denoted by

*

defines, for each pair of numbers

(x, y)

x*y

so that for all

x, y

z

x*x = 0 \,\,\,\,\, (1)

x*(*z) = (x* y)+ z \,\,\,\,\, (2)

+

denoting ordinary addition of numbers). Find

1993* 1932

1

concyclic wanted, excenter related

O

be the centre of the circle touching the side

AC

ABC

and the continuations of the sides

BA

BC

D

is the centre of the circle passing through the points

A

B

O

. Prove that the points

A

B

C

D

lie on a circle.
(YF Akurlich)

1

TOT 362 1993 Spring A J1 sum of integers, product of integers

One of two wizards, named Steven, was told the sum of three positive integers and the other, named Peter, their product. “If I knew”, said Steven, “that your number is greater than mine, I could find the integers”. “But my number is less than yours,” replied Peter, “and the integers are

X

Y

Z

”. Find these numbers.
(L Borisov)

1

TOT 360 1993 Spring O J3 (a -b) perfect square if ab/(a-b)=c

Positive integers

a

b

c

are positive integers with greatest common divisor equal to

1

(i.e. they have no common divisors greater than

1

\frac{ab}{a-b}=c

a -b

is a perfect square.
(SL Berlov)

1

TOT 361 1993 Spring O J4 ant crawls on a cube

An ant crawls along the edges of a cube turning only at its vertices. It has visited one of the vertices

25

times. Is it possible that it has visited each of the other

7

vertices exactly

20

times?
(S Tokarev)

1

TOT 359 1993 Spring O J2 cats and dogs , two houses with 2 flats each

Each of two houses

A

B

is divided into two flats. Several cats and dogs live there. It is known that the fraction of cats in the first flat of

A

(the ratio between the number of cats and the total number of animals in the flat) is greater than the fraction of cats in the first flat of

B

, and the fraction of cats in the second flat of

A

is greater than the fraction of cats in the second flat of

B

. Is it true that the fraction of cats in house

A

is greater than the fraction of cats in house

B

?
(AK Kovaldji)

1

TOT 358 1993 Spring O J1 distance between 2 orthocenters

M

be a point on the side

AB

ABC

AB = c

\angle CMA=\phi

are given. Find the distance between the orthocentres (intersection points of altitudes) of the triangles

AMC

BMC

.
(IF Sharygin)