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Problems
Contests
International Contests
Tournament Of Towns
1985 Tournament Of Towns
1985 Tournament Of Towns
Part of
Tournament Of Towns
Subcontests
(27)
(106) 6
1
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TOT 106 1985 Autumn S6 <BEA=45^o prove <EHC=45^o
In triangle
A
B
C
,
A
H
ABC, AH
A
BC
,
A
H
is an altitude (
H
H
H
is on
B
C
BC
BC
) and
B
E
BE
BE
is a bisector (
E
E
E
is on
A
C
AC
A
C
) . We are given that angle
B
E
A
BEA
BE
A
equals
4
5
o
45^o
4
5
o
.Prove that angle
E
H
C
EHC
E
H
C
equals
4
5
o
45^o
4
5
o
. (I. Sharygin , Moscow)
(105) 5
1
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TOT 105 1985 Autumn S5 at least n- 1 acute angles <A_iOA_j
(a) The point
O
O
O
lies inside the convex polygon
A
1
A
2
A
3
.
.
.
A
n
A_1A_2A_3...A_n
A
1
A
2
A
3
...
A
n
. Consider all the angles
A
i
O
A
j
A_iOA_j
A
i
O
A
j
where
i
,
j
i, j
i
,
j
are distinct natural numbers from
1
1
1
to
n
n
n
. Prove that at least
n
−
1
n- 1
n
−
1
of these angles are not acute . (b) Same problem for a convex polyhedron with
n
n
n
vertices.(V. Boltyanskiy, Moscow)
(104) 1
1
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TOT 104 1985 Autumn S1 PA+PB+PC+PD <=L+D_1+D_2
We are given a convex quadrilateral and point
M
M
M
inside it . The perimeter of the quadrilateral has length
L
L
L
while the lengths of the diagonals are
D
1
D_1
D
1
and
D
2
D_2
D
2
. Prove that the sum of the distances from
M
M
M
to the vertices of the quadrilateral are not greater than
L
+
D
1
+
D
2
L + D_1 + D_2
L
+
D
1
+
D
2
.(V. Prasolov)
(103) 7
1
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TOT 103 1985 Autumn J7 S7 super-chess, 30 x 30 board, 20 different pieces
(a)The game of "super- chess" is played on a
30
×
30
30 \times 30
30
×
30
board and involves
20
20
20
different pieces. Each piece moves according to its own rules , but cannot move from any square to more than
20
20
20
other squares . A piece "captures" another piece which is on a square to which it has moved. A permitted move (e.g.
m
m
m
squares forward and
n
n
n
squares to the right) does not depend on the piece 's starting square . Prove that (i) A piece cannot cap ture a piece on a given square from more than
20
20
20
starting squares. (ii) It is possible to arrange all
20
20
20
pieces on the board in such a way that not one of them can capture any of the others in one move. (b) The game of "super-chess" is played on a
100
×
100
100 \times 100
100
×
100
board and involves
20
20
20
different pieces. Each piece moves according to its own rules , but cannot move from any square to more than
20
20
20
other squares. A piece "captures" another piece which is on a square to which it has moved. It is possible that a permitted move (e.g.
m
m
m
squares forward and
n
n
n
squares to the right) may vary, depending on the piece's position . Prove that one can arrange all
20
20
20
pieces on the board in such a way that not one of them can capture any of the others in one move.( A . K . Tolpygo, Kiev)PS. (a) for Juniors , (b) for Seniors
(102) 6
1
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TOT 102 1985 Autumn J6 x_{k+1} =x^2_k+x_k, sum 1/{x_i+1}
The numerical sequence
x
1
,
x
2
,
.
.
x_1 , x_2 ,..
x
1
,
x
2
,
..
satisfies
x
1
=
1
2
x_1 = \frac12
x
1
=
2
1
and
x
k
+
1
=
x
k
2
+
x
k
x_{k+1} =x^2_k+x_k
x
k
+
1
=
x
k
2
+
x
k
for all natural integers
k
k
k
. Find the integer part of the sum
1
x
1
+
1
+
1
x
2
+
1
+
.
.
.
+
1
x
100
+
1
\frac{1}{x_1+1}+\frac{1}{x_2+1}+...+\frac{1}{x_{100}+1}
x
1
+
1
1
+
x
2
+
1
1
+
...
+
x
100
+
1
1
{A. Andjans, Riga)
(101) 5
1
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TOT 101 1985 Autumn J5 two throw coins.10-11 times, more heads
Two people throw coins. One throws his coin
10
10
10
times, the other throws his
11
11
11
times . What is the probability that the second coin fell showing "heads" a greater number of times than the first?(For those not familiar with Probability Theory this question could have been reformulated thus : Consider various arrangements of a
21
21
21
digit number in which each digit must be a "
1
1
1
" or a "
2
2
2
" . Among all such numbers what fraction of them will have more occurrences of the digit "
2
2
2
" among the last
11
11
11
digits than among the first
10
10
10
?)(S. Fomin , Leningrad)
(100) 4
1
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TOT 100 1985 Autumn J4 , 2 chessplayer, 40th move, 2h 30 min, time
Two chess players play each other at chess using clocks (when a player makes a move , the player stops his clock and starts the clock of his opponent) . It is known that when both players have just completed their
40
40
40
th move , both of their clocks read exactly
2
2
2
hr
30
30
30
min . Prove that there was a moment in the game when the clock of one player registered
1
1
1
min
51
51
51
sec less than that of the other . Furthermore , can one assert that the difference between the two clock readings was ever equal to
2
2
2
minutes?(S . Fomin , Leningrad)
(099) 3
1
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TOT 099 1985 Autumn J3 concentric circles, 3 chords-tangents, areas
A teacher gives each student in the class the following task in their exercise book . "Take two concentric circles of radius
1
1
1
and
10
10
10
. To the smaller circle produce three tangents whose intersections
A
,
B
A, B
A
,
B
and
C
C
C
lie in the larger circle . Measure the area
S
S
S
of triangle
A
B
C
ABC
A
BC
, and areas
S
1
,
S
2
S_1 , S_2
S
1
,
S
2
and
S
3
S_3
S
3
, the three sector-like regions with vertices at
A
,
B
A, B
A
,
B
and
C
C
C
(as in the diagram). Find the value of
S
1
+
S
2
+
S
3
−
S
S_1 +S_2 +S_3 -S
S
1
+
S
2
+
S
3
−
S
." Prove that each student would obtain the same result . https://1.bp.blogspot.com/-K3kHWWWgxgU/XWHRQ8WqqPI/AAAAAAAAKjE/0iO4-Yz6p9AcM2mklprX_M18xTyg9O81gCK4BGAYYCw/s200/TOT%2B1985%2BAutumn%2BJ3.png ( A . K . Tolpygo, Kiev)
(098) 2
1
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TOT 098 1985 Autumn J2 cat and mouse game, catch mouse strategy
In the game "cat and mouse" the cat chases the mouse in either labyrinth
A
,
B
A, B
A
,
B
or
C
C
C
. https://cdn.artofproblemsolving.com/attachments/4/5/429d106736946011f4607cf95956dcb0937c84.png The cat makes the first move starting at the point marked "
K
K
K
" , moving along a marked line to an adjacent point . The mouse then moves , under the same rules, starting from the point marked "
M
M
M
" . Then the cat moves again, and so on . If, at a point of time , the cat and mouse are at the same point the cat eats the mouse. Is there available to the cat a strategy which would enable it to catch the mouse , in cases
A
,
B
A, B
A
,
B
and
C
C
C
?(A. Sosinskiy, Moscow)
(097) 1
1
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TOT 097 1985 Autumn J1 8 football teams in tournament, no draws
Eight football teams participate in a tournament of one round (each team plays each other team once) . There are no draws. Prove that it is possible at the conclusion of the tournament to be able to find
4
4
4
teams , say
A
,
B
,
C
A, B, C
A
,
B
,
C
and
D
D
D
so that
A
A
A
defeated
B
,
C
B, C
B
,
C
and
D
,
B
D, B
D
,
B
defeated
C
C
C
and
D
D
D
, and
C
C
C
defeated
D
D
D
.
(096) 5
1
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TOT 096 1985 Spring S5 subset of projections of rectangles
A square is divided into rectangles. A "chain" is a subset
K
K
K
of the set of these rectangles such that there exists a side of the square which is covered by projections of rectangles of
K
K
K
and such that no point of this side is a projection of two inner points of two inner points of two different rectangles of
K
K
K
. (a) Prove that every two rectangles in such a division are members of a certain "chain". (b) Solve the similar problem for a cube, divided into rectangular parallelopipeds (in the definition of chain , replace "side" by"edge") . (A.I . Golberg, V.A. Gurevich)
(095) 4
1
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TOT 095 1985 Spring S4 2 convex sets F cover circle radius R
The convex set
F
F
F
does not cover a semi-circle of radius
R
R
R
. Is it possible that two sets, congruent to
F
F
F
, cover the circle of radius
R
R
R
? What if
F
F
F
is not convex?( N . B . Vasiliev , A. G . Samosvat)
(094) 2
1
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TOT 094 1985 Spring S2 radius of a circle rotates at a speed
The radius
O
M
OM
OM
of a circle rotates uniformly at a rate of
360
/
n
360/n
360/
n
degrees per second , where
n
n
n
is a positive integer . The initial radius is
O
M
0
OM_0
O
M
0
. After
1
1
1
second the radius is
O
M
1
OM_1
O
M
1
, after two more seconds (i.e. after three seconds altogether) the radius is
O
M
2
OM_2
O
M
2
, after
3
3
3
more seconds (after
6
6
6
seconds altogether) the radius is
O
M
3
OM_3
O
M
3
, ..., after
n
−
1
n - 1
n
−
1
more seconds its position is
O
M
n
−
1
OM_{n-1}
O
M
n
−
1
. For which values of
n
n
n
do the points
M
0
,
M
1
,
.
.
.
,
M
n
−
1
M_0, M_1 , ..., M_{n-1}
M
0
,
M
1
,
...
,
M
n
−
1
divide the circle into
n
n
n
equal arcs? (a) Is it true that the powers of
2
2
2
are such values? (b) Does there exist such a value which is not a power of
2
2
2
?(V. V. Proizvolov , Moscow)
(093) 1
1
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TOT 093 1985 Spring S1 area of a unit cube's projection on any plane
Prove that the area of a unit cube's projection on any plane equals the length of its projection on the perpendicular to this plane.
(092) T3
1
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TOT 092 1985 Spring Train S3 a+b+c>0 , bc+ca+ab>0, abc>0 => a,b,c>0
Three real numbers
a
,
b
a, b
a
,
b
and
c
c
c
are given . It is known that
a
+
b
+
c
>
0
,
b
c
+
c
a
+
a
b
>
0
a + b + c >0 , bc+ ca + ab > 0
a
+
b
+
c
>
0
,
b
c
+
c
a
+
ab
>
0
and
a
b
c
>
0
abc > 0
ab
c
>
0
. Prove that
a
>
0
,
b
>
0
a > 0 , b > 0
a
>
0
,
b
>
0
and
c
>
0
c > 0
c
>
0
.
(091) T2
1
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TOT 091 1985 Spring Train S2 largest subset with diff, of any 2 not prime
From the set of numbers
1
,
2
,
3
,
.
.
.
,
1985
1 , 2, 3, . . . , 1985
1
,
2
,
3
,
...
,
1985
choose the largest subset such that the difference between any two numbers in the subset is not a prime number (the prime numbers are
2
,
3
,
5
,
7
,
.
.
.
,
1
2, 3 , 5 , 7,... , 1
2
,
3
,
5
,
7
,
...
,
1
is not a prime number) .
(090) T1
1
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TOT 090 1985 Spring Train S1 AB=BC=1, <ABC=100^o, <CDA=130^o, BD=?
In quadrilateral ABCD it is given that
A
B
=
B
C
=
1
,
∠
A
B
C
=
10
0
o
AB = BC = 1, \angle ABC = 100^o
A
B
=
BC
=
1
,
∠
A
BC
=
10
0
o
, and
∠
C
D
A
=
13
0
o
\angle CDA = 130^o
∠
C
D
A
=
13
0
o
. Find the length of
B
D
BD
B
D
.
(089) 5
1
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TOT 089 1985 Spring J5 0-9 written in a 10x10 table, each 10 times
The digits
0
,
1
,
2
,
.
.
.
,
9
0, 1 , 2, ..., 9
0
,
1
,
2
,
...
,
9
are written in a
10
x
10
10 x 10
10
x
10
table , each number appearing
10
10
10
times . (a) Is it possible to write them in such a way that in any row or column there would be not more than
4
4
4
different digits? (b) Prove that there must be a row or column containing more than
3
3
3
different digits .{ L . D . Kurlyandchik , Leningrad)
(088) 4
1
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TOT 088 1985 Spring J4 square divided into 5 rectangles, 4/5 equal areas
A square is divided into
5
5
5
rectangles in such a way that its
4
4
4
vertices belong to
4
4
4
of the rectangles , whose areas are equal , and the fifth rectangle has no points in common with the side of the square (see diagram) . Prove that the fifth rectangle is a square. https://3.bp.blogspot.com/-TQc1v_NODek/XWHHgmONboI/AAAAAAAAKi4/XES55OJS5jY9QpNmoURp4y80EkanNzmMwCK4BGAYYCw/s1600/TOT%2B1985%2BSpring%2BJ4.png
(087) 3
1
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TOT 087 1985 Spring J3 class of 32 pupils is organised into 33 clubs
A certain class of
32
32
32
pupils is organised into
33
33
33
clubs , so that each club contains
3
3
3
pupils and no two clubs have the same composition. Prove that there are two clubs which have exactly one common member.
(086) 2
1
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TOT 086 1985 Spring J2 fractional part equation {A} +{1/A}=1
The integer part
I
(
A
)
I (A)
I
(
A
)
of a number
A
A
A
is the greatest integer which is not greater than
A
A
A
, while the fractional part
F
(
A
)
F(A)
F
(
A
)
is defined as
A
−
I
(
A
)
A - I(A)
A
−
I
(
A
)
. (a) Give an example of a positive number
A
A
A
such that
F
(
A
)
+
F
(
1
/
A
)
=
1
F(A) + F( 1/A) = 1
F
(
A
)
+
F
(
1/
A
)
=
1
. (b) Can such an
A
A
A
be a rational number?(I. Varge, Romania)
(085) 1
1
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TOT 085 1985 Spring J1 in triangle c >= (a + b) \sin (\gamma/2)
a
,
b
a, b
a
,
b
and
c
c
c
are sides of a triangle, and
γ
\gamma
γ
is its angle opposite
c
c
c
. Prove that
c
≥
(
a
+
b
)
sin
γ
2
c \ge (a + b) \sin \frac{\gamma}{2}
c
≥
(
a
+
b
)
sin
2
γ
(V. Prasolov )
(084) T5
1
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TOT 084 1985 Spring Train J5 123456 exists in sequence ? (sum of digits)
Every member of a given sequence, beginning with the second , is equal to the sum of the preceding one and the sum of its digits . The first member equals
1
1
1
. Is there, among the members of this sequence, a number equal to
123456
123456
123456
?(S. Fomin , Leningrad)
(083) T4
1
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TOT 083 1985 Spring Train J4 3 grasshoppers jump, every second
Three grasshoppers are on a straight line. Every second one grasshopper jumps. It jumps across one (but not across two) of the other grasshoppers . Prove that after
1985
1985
1985
seconds the grasshoppers cannot be in the initial position .(Leningrad Mathematical Olympiad 1985)
(082) T3
1
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TOT 082 1985 Spring Train J3 (x+y)^3=z, (y+z)^3=x, (z+x)^3=y
Find all real solutions of the system of equations
{
(
x
+
y
)
3
=
z
(
y
+
z
)
3
=
x
(
z
+
x
)
3
=
y
\begin{cases} (x + y) ^3 = z \\ (y + z) ^3 = x \\ ( z+ x) ^3 = y \end{cases}
⎩
⎨
⎧
(
x
+
y
)
3
=
z
(
y
+
z
)
3
=
x
(
z
+
x
)
3
=
y
(Based on an idea by A . Aho , J. Hop croft , J. Ullman )
(081) T2
1
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TOT 081 1985 Spring Train J2 68 coins, 100 weighings , heaviest - lightest
There are
68
68
68
coins , each coin having a different weight than that of each other . Show how to find the heaviest and lightest coin in
100
100
100
weighings on a balance beam.(S. Fomin, Leningrad)
(080) T1
1
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TOT 080 1985 Spring Train J1 median, bisector, altitude concurrent, equilateral
A median , a bisector and an altitude of a certain triangle intersect at an inner point
O
O
O
. The segment of the bisector from the vertex to
O
O
O
is equal to the segment of the altitude from the vertex to
O
O
O
. Prove that the triangle is equilateral .