MathDB

1

maximal no of edges a plane intersects a polyhedron having 100 edges.

100

edges.
(a) Find the maximal possible number of its edges which can be intersected by a plane (not containing any vertices of the polyhedron) if the polyhedron is convex.
(b) Prove that for a non-convex polyhedron this number i. can be as great as

96

, ii. cannot be as great as

100

.
(A Andjans, Riga

(356) 5

1

1 of distances AM, BM, CM is equal to sum of 2 other distances.

The bisector of the angle

A

of triangle

ABC

intersects its circumscribed circle at the point

D

. Suppose

P

is the point symmetric to the incentre of the triangle with respect to the midpoint of the side

BC

, and

M

is the second intersection point of the line

PD

with the circumscribed circle. Prove that one of the distances

AM

,

BM

,

CM

is equal to the sum of two other distances.
(VO Gordon)

(355) 4

1

TOT 355 1992 Autumn A S4 permutations of mn elemens in mxn table

A table has

m

rows and

n

columns. The following permutations of its

mn

elements are permitted: an arbitrary permutation leaving each element in the same row (a“horizontal move”) and an arbitrary permutation leaving each element in the same column (a “vertical move”). Find the number

k

such that any permutation of

mn

elements can be obtained by

k

permitted moves but there exists a permutation that cannot be achieved in less than

k

moves.
(A Andjans, Riga0

(354) 3

1

TOT 354 1992 Autumn A S3 a(n + 1) = a(n) + [\sqrt{a(n)}]

Consider the sequence

a(n)

defined by the following conditions:

a(1) = 1\,\,\,\, a(n + 1) = a(n) + [\sqrt{a(n)}] \,\,\, , \,\,\,\, n = 1,2,3,...

How many perfect squares no greater in value than

1000 000

will be found among the first terms of the sequence? ( (Note:

[x]

means the integer part of

x

, that is the greatest integer not greater than

x

.)
(A Andjans)

(353) 2

1

TOT 353 1992 Autumn A S2 bw coloring of nxnxn coube

For which values of

n

is it possible to construct an

n

by

n

by

n

cube with

n^3

unit cubes, each of which is black or white, such that each cube shares a common face with exactly three cubes of the opposite colour?
(S Tokarev)

(352) 1

1

TOT 352 1992 Autumn A S1 sum of squares of any 2 consecutive terms is square.

Prove that there exists a sequence of

100

different integers such that the sum of the squares of any two consecutive terms is a perfect square.
(S Tokarev)

(351) 3

1

TOT 351 1992 Autumn O S3 total length of intervals increasing = ... decreasing

We are given a finite number of functions of the form

y = c2^{-|x-d|}

. In each case

c

and

d

are parameters with

c > 0

. The function

f(x)

is defined on the interval

[a, b]

as follows: For each

x

in

[a, b]

,

f(x)

is the maximum value taken by any of the given functions

y

(defined above) at that point

x

. It is known that

f(a) = f(b)

. Prove that the total length of the intervals in which the function

f

is increasing is equal to the total length of the intervals in which it is decreasing (that is, both are equal to

(b- a)/2

).
(NB Vasiliev)

(342) 4

1

<A > <C, <D > < B => BC> AD/2

(a) In triangle

ABC

, angle

A

is greater than angle

B

. Prove that the length of side

BC

is greater than half the length of side

AB

.
(b) In the convex quadrilateral

ABCD

, the angle at

A

is greater than the angle at

C

and the angle at

D

is greater than the angle at

B

. Prove that the length of side

BC

is greater than half of the length of side

AD

.
(F Nazarov)

(350) 2

1

spiral sequence of squares on infinite blackboard, centers lie on 2 lines

The following spiral sequence of squares is drawn on an infinite blackboard: The

1

st square

(1 \times 1)

has a common vertical side with the

2

nd square (also

1\times 1

) drawn on the right side of it; the 3rd square

(2 \times 2)

is drawn on the upper side of the

1

st and 2nd ones; the

4

th square

(3 \times 3)

is drawn on the left side of the

1

st and

3

rd ones; the

5

th square

(5 \times 5)

is drawn on the bottom side of the

4

th, 1st and

2

nd ones; the

6

th square

(8 \times 8)

is drawn on the right side, and so on. Each of the squares has a common side with the rectangle consisting of squares constructed earlier. Prove that the centres of all the squares except the

1

st lie on two straight lines.
(A Andjans, Riga)

(349) 1

1

TOT 349 1992 Autumn O S1 cover a cube nxn eith a tape

We are given a cube with edges of length

n

cm. At our disposal is a long piece of insulating tape of width

1

cm. It is required to stick this tape to the cube. The tape may freely cross an edge of the cube on to a different face but it must always be parallel to an edge of the cube. It may not overhang the edge of a face or cross over a vertex. How many pieces of the tape are necessary in order to completely cover the cube? (You may assume that

n

is an integer.)
(A Spivak)

(348) 6

1

TOT 348 1992 Autumn A J6 a(n + 1) = a(n) + [\sqrt{a(n)}]

Consider the sequence

a(n)

defined by the following conditions:

a(1) = 1\,\,\,\, a(n + 1) = a(n) + [\sqrt{a(n)}] \,\,\, , \,\,\,\, n = 1,2,3,...

Prove that the sequence contains an infinite number of perfect squares. (Note:

[x]

means the integer part of

x

, that is the greatest integer not greater than

x

.)
(A Andjans)

(347) 5

1

fixed point fro line MN, <CAM=<CAN

An angle with vertex

O

and a point

A

inside it are placed on a plane. Points

M

and

N

are chosen on different sides of the angle so that the angles

CAM

and

CAN

are equal. Prove that the straight line

MN

always passes through a fixed point (or is always parallel to a fixed line).
(S Tokarev)

(346) 4

1

combo geo, with reflections on broken line ABCD

On the plane is give a broken line

ABCD

in which

AB = BC = CD = 1

, and

AD

is not equal to

1

. The positions of

B

and

C

are fixed but

A

and

D

change their positions in turn according to the following rule (preserving the distance rules given): the point

A

is reflected with respect to the line

BD

, then

D

is reflected with respect to the line

AC

(in which

A

occupies its new position), then

A

is reflected with respect to the line

BD

(

D

occupying its new position),

D

is reflected with respect to the line

AC

, and so on. Prove that after several steps

A

and

D

coincide with their initial positions.
(M Kontzewich)

(345) 3

1

TOT 345 1992 Autumn A J3 (P-Q)(x), P(x), (P+Q)(x) re squares of polynomials

Do there exist two polynomials

P(x)

and

Q(x)

with integer coefficients such that

(P-Q)(x), \,\,\,\, P(x) \,\,\,\, and \,\,\,\,(P+Q)(x)

are squares of polynomials (and

Q

is not equal to

cP

, where

c

is a real number)?
(V Prasolov)

(344) 2

1

TOT 344 1992 Autumn A J2 a squarea and 1993$ equilaterals

On the plane a square is given, and

1993

equilateral triangles are inscribed in this square. All vertices of any of these triangles lie on the border of the square. Prove that one can find a point on the plane belonging to the borders of no less than

499

of these triangles.
(N Sendrakyan)

(343) 1

1

TOT 343 1992 Autumn A J1 numbers in nxn table

Numbers in an

n

by

n

table may be changed by adding

1

to each number on an arbitrary closed non-selfintersecting “rook path” (a broken line with segments parallel to the borders of the table). Originally

1

’s stand on one of the diagonals, and $0S’s in the other cells of the table. Can one get (after several transformations) a table in which all numbers are equal to each other? (A “rook path” contains all cells through which it passes.)
(AA Egorov)

(341) 3

1

TOT 341 1992 Autumn O J3 odd sum of its digits

Prove that for any positive integer

M

there exists an integer divisible by

M

such that the sum of its digits (in its decimal representation) is odd.
(D Fomin, St Petersburg)

(340) 2

1

TOT 340 1992 Autumn O J2 circumcenters are vertices of #

On each side of a parallelogram an arbitrary point is chosen. Each pair of chosen points on neighbouring sides (i.e. sides with a common vertex) are connected by a line segment. Prove that the centres of the circumscribed circles of the four triangles so created are themselves vertices of a parallelogram.
(ED Kulanin)

(339) 1

1

TOT 339 1992 Autumn O J1 101 chess players in tournaments

There are

101

chess players who participated in several tournaments. There was no tournament in which all of them participated. Each pair of these

101

players met exactly once during these tournaments. Prove that one of them participated in no less than

11

tournaments. (Assume that each pair of participants in each tournament plays each other once in that tournament).
(A Andjans, Riga)

(338) 6

1

TOT 338 1992 Spring A S6 no of decompositions in product of integers

For natural numbers

n

and

b

, let

V(n, b)

denote the number of decompositions of

n

into the product of integers each of which is greater than

b

: for example

36 = 6\times 6 = 4\times 9 = 3\times 3\times 4 = 3\times 12,

i.e.

V(36,2) = 5

. Prove that

V(n, b) < n/b

for all

n

and

b

.
(N.B. Vasiliev, Moscow)

(337) 5

1

TOT 337 1992 Spring A S5 100 silver and 101 gold coins

100

silver coins ordered by weight and

101

gold coins also ordered by weight are given. All coins have different weights. You are given a balance to compare weights of any two coins. How can you find the “middle” coin (that occupies the

101

-st place in weight among all

201

coins) using the minimal number of weighings? Find this number and prove that a smaller number of weighings would be insufficient.
(A. Andjans, Riga)

(336) 4

1

set of 27 triangles can be divided into two parts of the same total area

Three triangles

A_1A_2A_3

,

B_1B_2B_3

,

C_1C_2C_3

are given such that their centres of gravity (intersection points of their medians) lie on a straight line, but no three of the

9

vertices of the triangles lie on a straight line. Consider the set of

27

triangles

A_iB_jC_k

(where

i

,

j

,

k

take the values

1

,

2

,

3

independently). Prove that this set of triangles can be divided into two parts of the same total area.
(A. Andjans, Riga)

(335) 3

1

TOT 335 1992 Spring A S3 numbers 1/(i + j - 1) in nxn table

The numbers

\frac{1}{i+j-1} \,\,\,\,\,\,\, (i = 1,2,...,n; j = 1,2,...,n)

are written in an

n

by

n

table: the number

1/(i + j - 1)

stands at the intersection of the

i

-th row and

j

-th column. Chose any

n

squares of the table so that no two of them stand in the same row and no two of them stand in the same column. Prove that the sum of the numbers in these

n

squares is not less than

1

.
(Sergey Ivanov, St Petersburg)

(334) 2

1

TOT 334 1992 Spring A S2 sum of areas of 51 triangles >= 3S

Let

a

and

S

be the length of the side and the area of regular triangle inscribed in a circle of radius

1

. A closed broken line

A_1A_2...A_{51}A_1

consisting of

51

segments of the same length

a

is placed inside the circle. Prove that the sum of areas of the

51

triangles between the neighboring segments

A_1A_2A_3, A_2A_3A_4,..., A_{49}A_{50}A_{51}, A_{50}A_{51}A_1, A_{51}A_1A_2

is not less than

3S

.
(A. Berzinsh, Riga)

(331) 3

1

TOT 331 1992 Spring O S3 sum of vectors not zero

Let

O

be the centre of a regular

n

-gon whose vertices are labelled

A_1

,

...

,

A_n

. Let

a_1>a_2>...>a_n>0

. Prove that the vector

a_1\overrightarrow{OA_1}+a_2\overrightarrow{OA_2}+...+a_n\overrightarrow{OA_n}

is not equal to the zero vector.
(D. Fomin, Alexey Kirichenko)

(332) 4

1

TOT 332 1992 Spring O S4 10 numbers on a circle

10

numbers are placed on a circle. Their sum is equal to

100

. A sum of any three neighbouring numbers is no less than

29

. Find the minimal number

A

such that for any such set of 10 numbers none of them is greater than

A

. Prove that this value for

A

is really minimal.
(A. Tolpygo, Kiev)

(330) 2

1

TOT 330 1992 Spring O S2 midpoint wanted by angle bisectors

Sides of a triangle are equal to

3

,

4

and

5

. Each side is extended until it intersects the bisector of the external angle to the angle opposite to it. Three such points are obtained in all. Prove that one of the three points we get is the midpoint of the segment joining the other two points.
(V. Prasolov)

(333) 1

1

TOT 333 1992 Spring A S1 not perfect square

Prove that the product of all integers from

2^{1917} +1

up to

2^{1991} -1

is not the square of an integer.
(V. Senderov, Moscow)

(326) 3

1

TOT 326 1992 Spring A J3 compare nested radicals

Let

n, m, k

be natural numbers, with

m > n

. Which of the numbers is greater:

\sqrt{n+\sqrt{m+\sqrt{n+...}}}\,\,\, or \,\,\,\, \sqrt{m+\sqrt{n+\sqrt{m+...}}}\,\, ?

Note: Each of the expressions contains

k

square root signs;

n, m

alternate within each expression.
(N. Kurlandchik)

(328) 5

1

TOT 328 1992 Spring A J5 7 weighings for 101 coins

50

silver coins ordered by weight and

51

gold coins also ordered by weight are given. All coins have different weights. You are given a balance to compare weights of any two coins. How can you find the “middle” coin (that occupying the

51

st place in weight among all

101

coins) using

7

weighings?
(A. Andjans)

(329) 6

1

n + 1 pawns on n sectors of a circle

A circle is divided into

n

sectors. Pawns stand on some of the sectors; the total number of pawns equals

n + 1

. This configuration is changed as follows. Any two of the pawns standing on the same sector move simultaneously to the neighbouring sectors in different directions. Prove that after several such transformations a configuration in which no less than half of the sectors are occupied by pawns, will inevitably appear.
(D. Fomin, St Petersburg)

(327) 4

1

3 lines concurrent with a circle

Let

P

be a point on the circumcircle of triangle

ABC

. Construct an arbitrary triangle

A_1B_1C_1

whose sides

A_1B_1

,

B_1C_1

and

C_1A_1

are parallel to the segments

PC

,

PA

and

PB

respectively and draw lines through the vertices

A_1

,

B_1

and

C_1

and parallel to the sides

BC

,

CA

and

AB

respectively. Prove that these three lines have a common point lying on the circumcircle of triangle

A_1B_1C_1

.
(V. Prasolov)

(325) 2

1

TOT 325 1992 Spring Α J2 ADE is right iff c:b:a =3 : 4 : 5

Consider a right triangle

ABC

, where

A

is the right angle, and

AC > AB

. Points

E

on

AC

and

D

on

BC

are chosen so that

AB = AE = BD

. Prove that the triangle

ADE

is right if and only if the ratio

AB : AC : BC

of sides of the triangle

ABC

is

3 : 4 : 5

.
(A. Parovan)

(324) 1

1

TOT 324 1992 Spring A J1 crowded collections of numbers

A collection of

n > 2

numbers is called crowded if each of them is less than their sum divided by

n - 1

. Let

\{a, b, c, ,...\}

be a crowded collection of

n

numbers whose sum equals

S

. Prove that:
(a) each of the numbers is positive,
(b) we always have

a + b > c

,
(c) we always have

a + b \ge \frac{S}{n-1}

. (Regina Schleifer)

(323) 4

1

TOT 323 1992 Spring O J4 circle is divided into 7 arcs.

A circle is divided into

7

arcs. The sum of the angles subtending any two neighbouring arcs is no more than

103^o

. Find the maximal number

A

such that any of the

7

arcs is subtended by no less than

A^o

. Prove that this value

A

is really maximal.
(A. Tolpygo, Kiev)

(322) 3

1

TOT 322 1992 Spring O J3 coins in a one-layer box

A numismatist Fred has some coins. A diameter of any coin is no more than

10

cm. All the coins are contained in a one-layer box of dimensions

30

cm by

70

cm. He is presented with a new coin. Its diameter is

25

cm. Prove that it is possible to put all the coins in a one-layer box of dimensions

55

cm by

55

cm.
(Fedja Nazarov, St Petersburg)

(321) 2

1

TOT 321 1992 Spring O J2 isosceles trapezoid wanteed

In trapezoid

ABCD

the sides

BC

and

AD

are parallel,

AC = BC + AD

, and the angle between the diagonals is equal to

60^o

. Prove that

AB = CD

.
(Stanislav Smirnov, St Petersburg)

(320) 1

1

TOT 320 1992 Spring O J1 10 different products for sale

At the beginning of a month a shop has

10

different products for sale, each with equal prices. Every day the price of each product is either doubled or trebled. By the beginning of the following month all the prices have become different. Prove that the ratio (the maximal price) /(the minimal price) is greater than

27

.
(D. Fomin and Stanislav Smirnov, St Petersburg)

1992 Tournament Of Towns

Subcontests

maximal no of edges a plane intersects a polyhedron having 100 edges.

1 of distances AM, BM, CM is equal to sum of 2 other distances.

TOT 355 1992 Autumn A S4 permutations of mn elemens in mxn table

TOT 354 1992 Autumn A S3 a(n + 1) = a(n) + [\sqrt{a(n)}]

TOT 353 1992 Autumn A S2 bw coloring of nxnxn coube

TOT 352 1992 Autumn A S1 sum of squares of any 2 consecutive terms is square.

TOT 351 1992 Autumn O S3 total length of intervals increasing = ... decreasing

<A > <C, <D > < B => BC> AD/2

spiral sequence of squares on infinite blackboard, centers lie on 2 lines

TOT 349 1992 Autumn O S1 cover a cube nxn eith a tape

TOT 348 1992 Autumn A J6 a(n + 1) = a(n) + [\sqrt{a(n)}]

fixed point fro line MN, <CAM=<CAN

combo geo, with reflections on broken line ABCD

TOT 345 1992 Autumn A J3 (P-Q)(x), P(x), (P+Q)(x) re squares of polynomials

TOT 344 1992 Autumn A J2 a squarea and 1993$ equilaterals

TOT 343 1992 Autumn A J1 numbers in nxn table

TOT 341 1992 Autumn O J3 odd sum of its digits

TOT 340 1992 Autumn O J2 circumcenters are vertices of #

TOT 339 1992 Autumn O J1 101 chess players in tournaments

TOT 338 1992 Spring A S6 no of decompositions in product of integers

TOT 337 1992 Spring A S5 100 silver and 101 gold coins

set of 27 triangles can be divided into two parts of the same total area

TOT 335 1992 Spring A S3 numbers 1/(i + j - 1) in nxn table

TOT 334 1992 Spring A S2 sum of areas of 51 triangles >= 3S

TOT 331 1992 Spring O S3 sum of vectors not zero

TOT 332 1992 Spring O S4 10 numbers on a circle

TOT 330 1992 Spring O S2 midpoint wanted by angle bisectors

TOT 333 1992 Spring A S1 not perfect square

TOT 326 1992 Spring A J3 compare nested radicals

TOT 328 1992 Spring A J5 7 weighings for 101 coins

n + 1 pawns on n sectors of a circle

3 lines concurrent with a circle

TOT 325 1992 Spring Α J2 ADE is right iff c:b:a =3 : 4 : 5

TOT 324 1992 Spring A J1 crowded collections of numbers

TOT 323 1992 Spring O J4 circle is divided into 7 arcs.

TOT 322 1992 Spring O J3 coins in a one-layer box

TOT 321 1992 Spring O J2 isosceles trapezoid wanteed

TOT 320 1992 Spring O J1 10 different products for sale

1992 Tournament Of Towns

Subcontests

maximal no of edges a plane intersects a polyhedron having 100 edges.

1 of distances AM, BM, CM is equal to sum of 2 other distances.

TOT 355 1992 Autumn A S4 permutations of mn elemens in mxn table

TOT 354 1992 Autumn A S3 a(n + 1) = a(n) + [\sqrt{a(n)}]

TOT 353 1992 Autumn A S2 bw coloring of nxnxn coube

TOT 352 1992 Autumn A S1 sum of squares of any 2 consecutive terms is square.

TOT 351 1992 Autumn O S3 total length of intervals increasing = ... decreasing

&lt;A &gt; &lt;C, &lt;D &gt; &lt; B =&gt; BC&gt; AD/2

spiral sequence of squares on infinite blackboard, centers lie on 2 lines

TOT 349 1992 Autumn O S1 cover a cube nxn eith a tape

TOT 348 1992 Autumn A J6 a(n + 1) = a(n) + [\sqrt{a(n)}]

fixed point fro line MN, &lt;CAM=&lt;CAN

combo geo, with reflections on broken line ABCD

TOT 345 1992 Autumn A J3 (P-Q)(x), P(x), (P+Q)(x) re squares of polynomials

TOT 344 1992 Autumn A J2 a squarea and 1993$ equilaterals

TOT 343 1992 Autumn A J1 numbers in nxn table

TOT 341 1992 Autumn O J3 odd sum of its digits

TOT 340 1992 Autumn O J2 circumcenters are vertices of #

TOT 339 1992 Autumn O J1 101 chess players in tournaments

TOT 338 1992 Spring A S6 no of decompositions in product of integers

TOT 337 1992 Spring A S5 100 silver and 101 gold coins

set of 27 triangles can be divided into two parts of the same total area

TOT 335 1992 Spring A S3 numbers 1/(i + j - 1) in nxn table

TOT 334 1992 Spring A S2 sum of areas of 51 triangles &gt;= 3S

TOT 331 1992 Spring O S3 sum of vectors not zero

TOT 332 1992 Spring O S4 10 numbers on a circle

TOT 330 1992 Spring O S2 midpoint wanted by angle bisectors

TOT 333 1992 Spring A S1 not perfect square

TOT 326 1992 Spring A J3 compare nested radicals

TOT 328 1992 Spring A J5 7 weighings for 101 coins

n + 1 pawns on n sectors of a circle

3 lines concurrent with a circle

TOT 325 1992 Spring &Alpha; J2 ADE is right iff c:b:a =3 : 4 : 5

TOT 324 1992 Spring A J1 crowded collections of numbers

TOT 323 1992 Spring O J4 circle is divided into 7 arcs.

TOT 322 1992 Spring O J3 coins in a one-layer box

TOT 321 1992 Spring O J2 isosceles trapezoid wanteed

TOT 320 1992 Spring O J1 10 different products for sale

<A > <C, <D > < B => BC> AD/2

fixed point fro line MN, <CAM=<CAN

TOT 334 1992 Spring A S2 sum of areas of 51 triangles >= 3S

TOT 325 1992 Spring Α J2 ADE is right iff c:b:a =3 : 4 : 5