1995 Tournament Of Towns

Part of Tournament Of Towns

Subcontests

(41)

1

TOT 482 1995 Autumn S A6 sum fo digtis in arithmetic progression

Does there exist an increasing arithmetic progression of (a)

11

10000

(c) infinitely many positive integers such that the sums of their digits in base

10

also form an increasing arithmetic progression?
(A Shapovalov)

1

TOT 480 1995 Autumn S A4 n spectators, n tickets

Along a track for cross-country skiing,

1000

seats are placed in a row and numbered in order from

1

1000

n

tickets were sold,

100 < n < 1000

, each with one of the numbers

1,2,..., 100

printed on it. Also for each number

1,2,..., 100

there exists at least one ticket with this number printed on it. Of course, there are tickets that have the same seat numbers. These

n

spectators arrive one at a time.
Each goes to the seat shown on his ticket and occupies it if it is still empty. If not, he just says “Oh” and moves to the seat with the next number. This is repeated until he finds an empty seat and occupies it, saying “Oh” once for each occupied seat passed over but not at any other time. Prove that all the spectators will be seated and that the total number of the exclamations “Oh” that have been made before all the spectators are seated does not depend on the order in which the n spectators arrive, although it does depend on the distribution of numbers on the tickets.
(A Shen)

1

TOT 479 1995 Autumn S A3 cutting a rectangle into right triangles

A rectangle with sides of lengths

a

b

a > b

) is cut into rightangled triangles so that any two of these triangles either have a common side, a common vertex or no common points. Moreover, any common side of two triangles is a leg of one of them and the hypotenuse of the other. Prove that

a > 2b

.
(A Shapovalov)

1

TOT 478 1995 Autumn S A2 n irrationals if their product -x_= odd integer

p

be the product of

n

x_1

x_2

...

x_n

. Prove that if

p - x_k

is an odd integer for

k = 1, 2,..., n

, then each of the numbers

x_1

x_2

...

x_n

is irrational.
(G Galperin)

1

treasure hunting, midpoint of orthocenters of 2 triangles, concyclic points

Version for Nordic Countries Six pine trees grow on the shore of a circular lake. It is known that a treasure is submerged at the mid-point

T

between the intersection points of the altitudes of two triangles, the vertices of one being at three of the

6

pines, and the vertices of the second one at the other three pines. At how many points

T

must one dive to find the treasure?
Version for Tropical Countries A captain finds his way to Treasure Island, which is circular in shape. He knows that there is treasure buried at the midpoint of the segment joining the orthocentres of triangles

ABC

DEF

A

B

C

D

E

F

are six palm trees on the shore of the island, not necessarily in cyclic order. He finds the trees all right, but does not know which tree is denoted by which letter. What is the maximum number of points at which the captain has to dig in order to recover the treasure?
(S Markelov)

1

locus P such that 4 angle bisectors create a parallelogram

If P is a point inside a convex quadrilateral

ABCD

, let the angle bisectors of

\angle APB

\angle BPC

\angle CPD

\angle DPA

AB

BC

CD

DA

K

L

M

N

respectively.
(a) Find a point

P

KLMN

is a parallelogram. (b) Find the locus of all such points

P

1

line l construction through C such that d(A,l) d(B,l) = max

Three different points

A

B

C

are placed in the plane. Construct a line

m

C

so that the product of the distances from

A

B

m

has the maximal value. Is

m

unique for every triple

A

B

C

?
(NB Vassiliev)

1

TOT 475 1995 Autumn S O3 12-digit number is perfect square

The first digit of a

6

-digit number is

5

. Is it true that it is always possible to write

6

more digits to the right of this number so that the resulting

12

-digit number is a perfect square?
(A Tolpygo)

1

TOT 474 1995 Autumn S O2 sum of any 3 of 4/5 is prime

Do there exist (a) four (b) five distinct positive integers such that the sum of any three of them is a prime number?
(V Senderov)

1

TOT 472 1995 Autumn J A6 game is played on a 1 x 1000 board

A game is played on a

1 \times 1000

board. There are n chips, all of which are initially in a box near the board. Two players move in turn. The first may choose

17

chips or less, from either on or off the board. She then puts them into unoccupied cells on the board so that there is no more than one chip in each of the cells. The second player may take off the board any number of chips occupying consecutive cells and put them back in the box. The first player wins if she can put all n chips on the board so that they occupy consecutive cells.
(a) Show that she can win if

n = 98

. (b) For what maximal value of

n

can she win?
(A Shapovalov)

1

TOT 471 1995 Autumn J A5 simple 7-gons

A simple polygon in the plane is a figure bounded by a closed nonself-intersecting broken line.
(a) Do there exist two congruent simple

7

-gons in the plane such that all the seven vertices of one of the

7

-gons are the vertices of the other one and yet these two

7

-gons have no common sides?
(b) Do there exist three such

7

-gons?
(V Proizvolov)

1

TOT 470 1995 Autumn J A4 n questions to find person Z among n persons

A journalist is looking for a person

Z

at a meeting of

n

persons. He has been told that

Z

knows all the other people at the meeting but none of them knows

Z

. The journalist may ask any person about any other person: “Do you know that person?” One person can be questioned many times. All answers are truthful.
(a) Can the journalist always find

Z

by asking less than

n

questions? (b) What is the minimal number of questions which are needed to find

Z

1

<PAQ = 90^o -/2 <BAC if AP = PK, AQ = QK, 3 angle bisectors

AK

BL

CM

be the angle bisectors of a triangle

ABC

K

BC

P

Q

be the points on the lines

BL

CM

respectively such that

AP = PK

AQ = QK

\angle PAQ = 90^o -\frac12 \angle B AC.

1

TOT 468 1995 Autumn J A2 product of the first 70 terms = sum of squares

The first five terms of a sequence are

1, 2, 3, 4

5

. From the sixth term on, each term is

1

less than the product of all the proceeding ones. Prove that the product of the first

70

terms is equal to the sum of their squares.
(LD Kurliandchik)

1

TOT 467 1995 Autumn J A1 - point equidistant from sides of triangle

Prove that inside any acute-angled triangle, there exists a point

P

such that the feet of the perpendiculars dropped from

P

to the sides of the triangle are the vertices of an equilateral triangle.
(NB Vassiliev)

1

TOT 465 1995 Autumn J O3 folding paper rectangle

A paper rectangle

ABCD

1

is folded along a straight line so that

C

A

. Prove that the area of the pentagon thus obtained is less than

3/4

1

MK = KN wanted, angle bisectors from A related

From the vertex

A

ABC

, three segments are drawn: the bisectors

AM

AN

of its interior and exterior angles and the tangent

AK

to the circumscribed circle of the triangle (the points

M

K

N

lie on the line

BC

MK = KN

1

TOT 464 1995 Autumn J O2 sum of 1000 pos. integers = lcm

100

positive integers such that their sum is equal to their least common multiple?
(S Tokarev)

1

TOT 463 1995 Autumn J O1 min questions to find if P lies inside a square

A square is placed in the plane and a point

P

is marked in this plane with invisible ink. A certain person can see this point through special glasses. One can draw a straight line and this person will say on which side of the line the point

P

P

lies on the line, the person says so. What is the minimal number of questions one needs to find out if

P

lies inside the square or not?
(Folklore)

1

TOT 460 1995 Spring S A5 divide line segment [0,1] into BW segments

(a) Divide the line segment

[0,1]

into smaller black and white segments so that, for any polynomial

p(x)

of degree no greater than two, the sum of increments of

p(x)

along all the black segments is equal to that along the white ones. (The increment of

p(x)

along the segment

[a, n]

p(b) - p(a)

.)
(b) Can this be done for all polynomials of degree no greater than

1995

1

TOT 462 1995 Spring S A7 common acquaintances in group of 50 people

Prove that in a group of

50

people there are always two who have an even number (possibly zero) of common acquaintances within the group.
(SI Tokarev)

1

TOT 461 1995 Spring S A6 nonconvex polyhedron existence

Does there exist a nonconvex polyhedron such that not one of its vertices is visible from a point

M

outside it? (The polyhedron is made out of an opaque material.)
(AY Belov, S Markelov)

1

TOT 459 1995 Spring S A4 lattice points and circle

Some points with integer coordinates in the plane are marked. It is known that no four of them lie on a circle. Show that there exists a circle of radius 1995 without any marked points inside.
(AV Shapovelov)

1

equal tangents from intersection of diagonals of trapezoid

The non-parallel sides of a trapezium serve as the diameters of two circles. Prove that all four tangents to the circles drawn from the point of intersection of the diagonals are equal (if this point lies outside the circles).
(S Markelov)

1

TOT 457 1995 Spring S A2 paint edges of prism, base in n-gon

For what values of

n

is it possible to paint the edges of a prism whose base is an

n

-gon so that there are edges of all three colours at each vertex and all the faces (including the upper and lower bases) have edges of all three colours?
(AV Shapovelov)

1

TOT 456 1995 Spring S A1 sphere passing through only 1 rational point

Does there exist a sphere passing through only one rational point? (A rational point is a point whose Cartesian coordinates are all rational numbers.)
(A Rubin)

1

TOT 452 1995 Spring S O1 sum 1/|x-a_i| < 40 if 0<a_i<=1

a, b, c

d

be points of the segment

[0,1]

of the real line (this means numbers

x

0 \le x \le 1

). Prove that there exists a point

x

on this segment such that

\frac{1}{|x-a|}+\frac{1}{|x-b|}+\frac{1}{|x-c|}+\frac{1}{|x-d|}< 40.

(LD Kurliandchik)

1

4 grasshoppers sit at the vertices of a square

Four grasshoppers sit at the vertices of a square. Every second, one of them jumps over one of the others to the symmetrical point on the other side (if

X

Y

X'

X

Y

X'

lie on a straight line and

XY = YX'

). Prove that after several jumps no three grasshoppers can be:
(a) on a line parallel to a side of the square, (b) on a straight line.
(AK Kovaldzhy)

1

CO _|_ DE wanted, 2 circles related

ABC

is inscribed in a circle with center

O

q

be the circle passing through

A

O

B

CA

CB

q

D

E

(different from

A

B

). Prove that the lines

CO

DE

are perpendicular to each other.
(S Markelov)

1

TOT 455 1995 Spring S O4 a0... 09 not a perfext square

\overline{a0... 09}

a > 0

is a digit and there is at least one zero) is not a perfect square.
(VA Senderov)

1

cuting 4 equal right triangles

Four equal right-angled triangles are given. We are allowed to cut any triangle into two new ones along the altitude dropped on to the hypotenuse. This operation may be repeated with any of the triangles from the new set. Prove that after any number of such operations there will be congruent triangles among those obtained.
(AV Shapovalov)

1

TOT 451 1995 Spring A J7 80 tin cans of provisions

A team of geologists on a field expedition have taken with them

80

tin cans of provisions. The

80

cans have different weights, which are known (there is a list). After a while the names of the contents of the cans have become illegible. The cook knows what is in each can and claims that he can prove it without opening any can and only using the list and a balance which indicates the difference of weight of the objects placed on its two pans. Show that in order to do so, (a) four weight measurements will be enough, (b) three will not
(AK Tolpygo)

1

TOT 450 1995 Spring A J6 6 parallelepipeds in space

Can it happen that

6

parallelepipeds, no two of which have common points, are placed in space so that there is a point outside of them from which no vertex of a parallelepiped is visible? (The parallelepipeds are not transparent.)
(V Proizvolov)

1

TOT 448 1995 Spring A J4 is a+b+c+d prime if ab = cd ?

a + b + c + d

a, b, c

d

are positive integers and

ab = cd

1

TOT 447 1995 Spring A J3 ocus of P interior of equilateral ABC ,AP = BP

Given the equilateral triangle

ABC

, find the locus of all points

P

such that the segments of the lines

AP

BP

lying inside the triangle are equal.

1

TOT 446 1995 Spring A J2 regular 10-gon

10

ABCDEFGHIJ

1

a straight line cuts off a triangle

PAQ

PA +AQ = 1

. Find the sum of angles under which the segment

PQ

is seen from the points

B

C

D

E

F

G

H

I

J

.
(V Proizvolov)

1

TOT 445 1995 Spring A J1 a/b+b/c+c/a and a/c+c/b+b/a are integers

a

b

c

are integers and the sums

\frac{a}{b}+\frac{b}{c}+\frac{c}{a} \,\,\,\, and \,\,\,\, \frac{a}{c}+\frac{c}{b}+\frac{b}{a}

are also integers, then we have

|a| = |v| = |c|

1

TOT 444 1995 Spring O J4 40...09 not a perfect square

Prove that the number

\overline{40...0}9

(with at least one zero) is not a perfect square.
(VA Senderov)

1

angles of convex octagon wanted, squares inscribed and circumscribed in circle

L

is the circle inscribed in the square

T_1

T_2

is the square inscribed in

L

, so that vertices of

T_1

lie on the straight lines containing the sides of

T_2

. Find the angles of the convex octagon whose vertices are at the tangency points of

L

with the sides of

T_1

and at the vertices of

T_2

1

three grasshoppers on a line

Three grasshoppers

A

B

C

are placed on a line. Grasshopper

B

sits at the midpoint between

A

C

. Every second, one of the grasshoppers jumps over one of the others to the symmetrical point on the other side (if

X

Y

X'

XY

YX'

). After several jumps it so happened that they returned to the three initial points (but maybe in different order). Prove that in this case

B

returns to his initial middle position.
(AK Kovaldzhy)

1

TOT 441 1995 Spring O J1 sonia has stamps worthy of 5S= dollars

10

15

20

cent stamps with total face value of

\$5

30

stamps altogether. Prove that she has more

20

cent stamps than

10