1996 Tournament Of Towns

Part of Tournament Of Towns

Subcontests

(41)

1

TOT 510 1996 Autumn J A3 sum (k^2-2}/k! < 3

\frac{2}{2!}+\frac{7}{3!}+\frac{14}{4!}+\frac{23}{5!}+...+\frac{k^2-2}{k!}+...+\frac{9998}{100!}<3

n! = 1 \times 2 \times ... \times n.

1

b - c/ (n-2)! < sum (n^3-a)/ n! < b

3-\frac{2}{(n-1)!} < \frac{2^2-2}{2!}+\frac{2^2-2}{3!}+...+\frac{n^2-2}{n!}<3

(b) Find some positive integers

a

b

c

such that for any

n > 2

b-\frac{c}{(n-2)!} < \frac{2^3-a}{2!}+\frac{3^3-a}{3!}+...+\frac{n^3-a}{n!}<b

(V Senderov, NB Vassiliev)

1

ports and roads on an island

A certain island has some ports along the coast and some towns inland. All roads on this island are one-way, and they do not meet except at a port or a town. Moreover, once you leave a certain port or town by road, there is no way you can return there by road. For any two ports

i

j

f_{ij}

denote the number of different routes along the roads between

i

j

.
(a) Suppose there are four ports on the island:

1, 2, 3

4

, in clockwise order. Show that

f_{14}f_{23} \ge f_{13}f_{24}

(b) Suppose there were six ports on the island:

1, 2, 3, 4, 5

6

, in clockwise order. Show that

f_{16}f_{25}f_{34} + f_{15}f_{24}f_{36} + f_{14}f_{26}f_{35}\ge f_{16}f_{24}f_{35}+ f_{15}f_{26}f_{34} + f_{14}f_{25}f_{36}

1

TOT 523 1996 Autumn S A6 mathlotto - senior version

The integers from

1

100

are written on a “mathlotto” ticket. When you buy a “mathlotto” ticket, you choose

10

100

numbers. Then 10 of the integers from

1

100

are drawn, and a winning ticket is one which does not contain any of them. Prove that
(a) if you buy

13

tickets, you can choose your numbers so that regardless of which numbers are drawn, you are guaranteed to have at least one winning ticket;
(b) if you buy only

12

tickets, it is possible for you not to have any winning tickets, regardless of how you choose your numbers.
(S Tokarev)

1

TOT 521 1996 Autumn S A4 f(f(x)) = x^2 - 1996

Prove that for any function

f(x)

, continuous or otherwise,

f(f(x)) = x^2 - 1996

cannot hold for all real numbers

x

.
(S Bogatiy, M Smurov,)

1

area chasing in convex hexagon

A', B', C', D', E'

F'

be the midpoints of the sides

AB

BC

CD

DE

EF

FA

of an arbitrary convex hexagon

ABCDEF

respectively. Express the area of

ABCDEF

in terms of the areas of the triangles

ABC

BCD'

CDS'

DEF'

EFA'

FAB'

.
(A Lopshi tz, NB Vassiliev)

1

TOT 518 1996 Autumn S A1 colour 4+4 verices of a cube in 2 colors

Can one paint four vertices of a cube red and the other four points black so that any plane passing through three points of the same colour contains a vertex of the other colour?
(Mebius, Sharygin)

1

TOT 517 1996 Autumn S O4 n +1 girls and n boys

For what integers

n > 1

can it happen that in a group of

n +1

n

boys, all the girls know a different number of boys while all the boys know the same number of girls?
(NB Vassiliev)

1

TOT 516 1996 Autumn S O3 reconstruct the axes , parabola y=x^2

y = x^2

is drawn in the coordinate plane and then the axes are erased so that the whole parabola stays on the picture but the origin is not shown on it. Reconstruct the axes with compass and ruler alone.
(A Egorov)

1

TOT 515 1996 Autumn S O2 paper circle in pices to a square?

Can a paper circle be cut into pieces and then rearranged into a square of the same area, if only a finite number of cuts is allowed and they must be along segments of straight lines or circular arcs?
(A Belov)

1

locus in cube equidistant from 3 edges

Consider three edges

a, b, c

of a cube such that no two of these edges lie in one plane. Find the locus of points inside the cube which are equidistant from

a

b

c

.
(V Proizvolov,)

1

total number of tiling triangles is even

(a) A square is cut into right triangles with legs of lengths

3

4

. Prove that the total number of the triangles is even.
(b) A rectangle is cut into right triangles with legs of lengths

1

2

. Prove that the total number of the triangles is even.
(A Shapovalov)

1

TOT 513 1996 Autumn J A6 36 numbers in mathlotto

The integers from

1

36

are written on a “mathlotto” ticket. When you buy a “mathlotto” ticket, you choose

6

36

6

of the integers from

1

36

are drawn, and a winning ticket is one which does not contain any of them. Prove that
(a) if you buy

9

tickets, you can choose your numbers so that regardless of which numbers are drawn, you are guaranteed to have at least one winning ticket;
(b) if you buy only

8

tickets, it is possible for you not to have any winning tickets, regardless of how you choose your numbers.
(S Tokarev)

1

TOT 512 1996 Autumn J A5 500 000 multiples of A not end in 6 same digits

Does there exist a

6

A

such that none of its

500 000

A

2A

3A

500 000A

6

identical digits?
(S Tokarev)

1

TOT 509 1996 Autumn J A2 p^2 + d is divisible by qr ...

Do there exist three different prime numbers

p

q

r

p^2 + d

is divisible by

qr

q^2 + d

is divisible by

rp

r^2 + d

is divisible by

pq

d = 10

d = 11

1

sum of lengths of blue segments equals that of red ones, circle, rhombus

A circle cuts each side of a rhombus twice thus dividing each side into three segments. Let us go around the perimeter of the rhombus clockwise beginning at a vertex and paint these segments successively in red, white and blue. Prove that the sum of lengths of the blue segments equals that of the red ones.
(V Proizvolov)

1

TOT 506 1996 Autumn J O3 .... 10 girls and 9 boys

(a) Can it happen that in a group of

10

9

boys, ball the girls know a different number of boys while all the boys know the same number of girls?
(b) What if there are

11

10

boys?
(NB Vassiliev)

1

TOT 505 1996 Autumn J O2 tile an equilateral triangle with trapezoids

For what positive integers

n

is it possible to tile an equilateral triangle of side

n

with trapezoids each of which has sides

1, 1, 1, 2

?
(NB Vassiliev)

1

TOT 504 1996 Autumn J O1 10 consecutives with sum of squares

10

consecutive positive integers such that the sum of their squares is equal to the sum of squares of the next

9

integers?
(Inspired by a diagram in an old text book)

1

TOT 508 1996 Autumn J A1 4 points red and 4 points black

Can one paint four points in the plane red and another four points black so that any three points of the same colour are vertices of a parallelogram whose fourth vertex is a point of the other colour?
(NB Vassiliev)

1

TOT 503 1996 Spring S A6 -1, +1 on cells of 2^n x n table

2^n

2^n \times n

table were filled with all different

n

-tuples of numbers

+1

-1

. Then some of the numbers were replaced by Os. Prove that one can choose a (non-empty) set of rows such that:
(a) the sum of all the numbers in all the chosen rows is

0

;
(b) the sum of all the chosen rows equals the zero row, that is, the sum of numbers in each column of the chosen rows equals

0

.
(G Kondakov, V Chernorutskii)

1

TOT 502 1996 Spring S A5 n-1 , n, n + 1 and sum of squares

Prove that there exist an infinite number of triples

n-1

n

n + 1

n

can be represented as the sum of two squares of natural numbers but neither of

n-1

n+1

can;
(b) each of these three numbers can be represented as the sum of two squares.
(V Senderov)

1

strict laws in country of Militaria about to military service

There are two very strict laws in the country of Militaria.
(i) Anyone who is shorter than

80\%

(or more) of his “neighbours” (i.e. men living at distance less then

r

from him) is freed from the military service.
(ii) Anyone who is taller than

80\%

(or more) of his “neighbours” (i.e. men living at distance less then

R

from him) is allowed to serve in the police.
A nice thing is that each man

X

may choose his own (possibly different) positive numbers

r = r(X)

R = R(X)

. Can it happen that

90\%

(or more) of the men in Militaria are free from the army and, at the same time,

90\%

(or more) of the men in Militaria are allowed to serve in the police? (The places of living of the men are fixed points in the plane.)
(N Konstantinov)

1

grasshopper jumping in [0,1]x[0,1]

0\le x\le 1

0\le y\le 1

is drawn in the plane

Oxy

. A grasshopper sitting at a point

M

with noninteger coordinates outside this square jumps to a new point which is symmetrical to

M

with respect to the leftmost (from the grasshopper’s point of view) vertex of the square. Prove that no matter how many times the grasshopper jumps, it will never reach the distance more than

10 d

from the center

C

of the square, where

d

is the distance between the initial position

M

C

1

TOT 499 1996 Spring S A1 8 distances in a cube

Does there exist a cube in space such that the perpendiculars dropped from its eight vertices to a given plane are of length

0, 1, 2, 3, 4, 5, 6

7

?
(V Proizvolov)

1

3 squares on sides of a triangle exterally, difference of areas

ABMN

BCKL

ACPQ

are constructed outside triangle

ABC

. The difference between the areas of

AB MN

BCKL

d

. Find the difference between the areas of the squares with sides

NQ

PK

respectively, if

\angle ABC

is
(a) a right angle; (b) not necessarily a right angle.
(A Gerko)

1

TOT 497 1996 Spring S O4 tile space with reg. tetrahedra and octahedra

Is it possible to tile space using a combination of regular tetrahedra and regular octahedra?
(A Belov)

1

TOT 496 1996 Spring S O3 product of 99 factorials a perfect sqaure

Consider the factorials of the first

100

positive integers, namely,

1!, 2!

...

100!

. Is it possible to delete one of them so that the product of the remaining ones is a perfect square?
(S Tokarev)

1

TOT 495 1996 Spring S O2 9 digits sum

1,2,3,..., 9

are written down in some order so that they form a

9

-digit number. Consider all triples of consecutive digits and find the sum of these seven

3

-digit numbers. What is the largest possible value of this sum?
(A Galochkin)

1

TOT 494 1996 Spring S O1 bettere precident

People are asked “Do you think that the new president will be better than the most recent one?” Suppose

a

people say “better”,

b

say “the same” and

c

“worse”. Sociologists then calculate two measures of “social optimism”:

m =a + \frac{b}{2}

n = a - c

. Suppose exactly

100

people respond to this survey and it turns out that

m = 40

n

1

sum of n - 1 angles is 30^o, n equal parts of side of equilateral

In an equilateral triangle

ABC

D

be a point on the side

AB

AD = AB /n

. Prove that the sum of

n - 1

\angle DP_lA

\angle DP_2A

...

\angle DP_nA

P_1

P_2

...

P_{n-1}

are the points dividing the side

BC

n

equal parts, is equal to

30

n = 3

n

is an arbitrary integer,

n > 2

.
(V Proizvolov)

1

TOT 492 1996 Spring J A5 8 students 8 problems

Eight students were asked to solve

8

problems (the same set of problems for each of the students).
(a) Each problem was solved by

5

students. Prove that one canfind two students so that each of the problems was solved by at least one of them.
(b) If each problem was solved by

4

students, then it is possible that no such pair of students exists. Prove this.
(S Tokarev)

1

TOT 491 1996 Spring J A4 rook at a m x n board

A rook stands at a corner of an

m \times n

squared board. Two players move the rook in turn (vertically or horizontally through any numbers of squares). As the rook moves, it paints the squares that it visits (stopping or passing through). The rook is not allowed to pass through or stop at the painted squares. The player who cannot move, loses. Who has a guaranteed win: the first player (who starts the game) or the other, and how should he/she play?
(B Begun)

1

TOT 490 1996 Spring J A3 sequence of 1996 reals

Prove that from any sequence of

1996

a_1

a_2

...

a_{1996}

one can choose one or several numbers standing successively one after another so that their sum differs from an integer by less than

0.001

1

ext. common tangent to 2 non-intersecting circles

An exterior common tangent to two non-intersecting circles with centers and

O_2

touches them at the points

A_1

A_2

respectively. The segment

O_1O_2

intersects the circles at the points

B_1

B_2

C

is the point where the straight lines

A_1B_1

A_2B_2

D

is the point on the line

A_1A_2

CD

is perpendicular to

B_1B_2

A_1D = DA_2

1

TOT 488 1996 Spring J A1 a^2 + b^2 - ab = c^2

a, b

c

are positive numbers such that

a^2 + b^2 - ab = c^2,

(a - c)(b - c) < 0.

1

TOT 487 1996 Spring J O5 2player game on 10x10 chessboard

A game is played between two players on a

10 \times 10

checkerboard. They move alternately, the first player marking

X

s on vacant cells and the second

O

100

cells have been marked, they calculate two numbers

C

Z

C

is the total number of five consecutive

X

s in a row, a column or a diagonal, so that

6

X

s contribute a count of

2

C

7

X

3

, and so on. Similarly,

Z

is the total number of five consecutive Os. The first player wins if

C > Z

C < Z

C = Z

. Does the first player have a strategy which guarantees (a) a draw or a win (b) a win regardless of the counter-strategy of the second player?
(A Belov)

1

max no of self- intersections of hexagon

All vertices of a hexagon, whose sides may intersect at points other than the vertices, lie on a circle. (a) Draw a hexagon such that it has the largest possible number of points of self-intersection. (b) Prove that this number is indeed maximum.
(NB Vassiliev)

1

TOT 485 1996 Spring J O3 <PCQ , tangents of incircle of right triangle

The two tangents to the incircle of a right-angled triangle

ABC

that are perpendicular to the hypotenuse

AB

intersect it at the points

P

Q

\angle PCQ

.
(M Evdokimov,)

1

TOT 484 1996 Spring J O2 n- 1996, n, n + 1996 are perfect squares

Does there exist an integer n such that all three numbers
(a)

n - 96

n

n + 96

n - 1996

n

n + 1996

are positive prime numbers?
(V Senderov)

1

TOT 483 1996 1996 Spring J O1 min angle =1/5 largest angle

In an acute-angled triangle, each angle is an integral number of degrees, and the smallest angle is one-fifth of the largest one. Find these angles.
(G Galperin)