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Problems
Contests
International Contests
Tournament Of Towns
1991 Tournament Of Towns
1991 Tournament Of Towns
Part of
Tournament Of Towns
Subcontests
(39)
(297) 4
1
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no of great circles not through 5 points on sphere
Five points are chosen on the sphere, no three of them lying on a great circle (a great circle is the intersection of the sphere with some plane passing through the sphere’s centre). Two great circles not containing any of the chosen points are called equivalent if one of them can be moved to the other without passing through any chosen points.(a) How many nonequivalent great circles not containing any chosen points can be drawn on the sphere? (b) Answer the same problem, but with
n
n
n
chosen points.
(319) 6
1
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TOT 319 1991 Autumn A S6 arithmetical progression eithout 9s
An arithmetical progression (whose difference is not equal to zero) consists of natural numbers without any nines in its decimal notation. (a) Prove that the number of its terms is less than
100
100
100
. (b) Give an example of such a progression with
72
72
72
terms. (c) Prove that the number of terms in any such progression does not exceed
72
72
72
.(V. Bugaenko and Tarasov, Moscow)
(318) 5
1
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rotating triangle about it's centroid at angles 120^o and 240^o
Let
M
M
M
be a centre of gravity (the intersection point of the medians) of a triangle
A
B
C
ABC
A
BC
. Under rotation by
120
120
120
degrees about the point
M
M
M
, the point
B
B
B
is taken to the point
P
P
P
; under rotation by
240
240
240
degrees about
M
M
M
, the point
C
C
C
is taken to the point
Q
Q
Q
. Prove that either
A
P
Q
APQ
A
PQ
is an equilateral triangle, or the points
A
,
P
,
Q
A, P, Q
A
,
P
,
Q
coincide. (Bykovsky, Khabarovsksk)
(317) 3
1
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TOT 317 1991 Autumn A S3 numbers in 9x9 table
Is it possible to put distinct positive integers less than
1991
1991
1991
in the cells of a
9
×
9
9\times 9
9
×
9
table so that the products of all the numbers in every column and every row are equal to each other?(N.B. Vasiliev, Moscow)
(316) 2
1
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TOT 316 1991 Autumn A S2 divide plane into polygons, rotation
Is it possible to divide the plane into polygons so that each polygon is transformed into itself under some rotation by
360
/
7
360/7
360/7
degrees about some point? All sides of these polygons must be greater than
1
1
1
cm. (A polygon is the part of a plane bounded by one non-self-intersect-ing closed broken line, not necessarily convex.)(A. Andjans, Riga)
(315) 1
1
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TOT 315 1991 Autumn A S1 area of cyclic ABCD =1/2 (AC)^2 sin A, BC=CD
In an inscribed quadrilateral
A
B
C
D
ABCD
A
BC
D
we have
B
C
=
C
D
BC = CD
BC
=
C
D
. Prove that the area of the quadrilateral is equal to
(
A
C
)
2
sin
A
2
\frac{(AC)^2 \sin A}{2}
2
(
A
C
)
2
s
i
n
A
(D. Fomin, Leningrad)
(314) 4
1
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TOT 314 1991 Autumn O S4 30 numbers on a circle a=|b-c|
Thirty numbers are placed on a circle. For every number
A
A
A
we have:
A
A
A
equals the absolute value of
(
B
−
C
)
(B- C)
(
B
−
C
)
, where
B
B
B
and
C
C
C
follow
A
A
A
clockwise. The total sum of the numbers equals
1
1
1
. Find all the numbers.(Folklore)
(313) 3
1
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TOT 313 1991 Autumn O S3 <C obtuse if AD/DC = AB/ BC
Point
D
D
D
lies on side
A
B
AB
A
B
of triangle
A
B
C
ABC
A
BC
, and
A
D
D
C
=
A
B
B
C
.
\frac{AD}{DC} = \frac{AB}{BC}.
D
C
A
D
=
BC
A
B
.
Prove that angle
C
C
C
is obtuse.(Sergey Berlov)
(312) 2
1
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TOT 312 1991 Autumn O S2 11 girls and n boys for mushrooms
11
11
11
girls and
n
n
n
boys went for mushrooms. They have found
n
2
+
9
n
−
2
n^2+9n -2
n
2
+
9
n
−
2
in total, and each child has found the same quantity. Which is greater: the number of girls or the number of boys?(A. Tolpygo, Kiev)
(311) 1
1
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TOT 311 1991 Autumn O S1 2 tangent circles, tangent to an angle
Two circles with centres
A
A
A
and
B
B
B
lie inside an angle. They touch each other and both sides of the angle. Prove that the circle with the diameter
A
B
AB
A
B
touches both sides of the angle.(V. Prasolov)
(310) 7
1
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TOT 310 1991 Autumn A J7 n children divide m identical pieces
n
n
n
children want to divide
m
m
m
identical pieces of chocolate into equal amounts, each piece being broken not more than once. (a) For what
n
n
n
is it possible, if
m
=
9
m = 9
m
=
9
? (b) For what
n
n
n
and
m
m
m
is it possible?(Y. Tschekanov, Moscow)
(309) 6
1
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sequence of diagonals in convex octagon
All internal angles of a convex octagon
A
B
C
D
E
F
G
H
ABCDEFGH
A
BC
D
EFG
H
are equal to each other and the edges are alternatively equal:
A
B
=
C
D
=
E
F
=
G
H
,
B
C
=
D
E
=
F
G
=
H
A
AB = CD = EF = GH,BC = DE = FG = HA
A
B
=
C
D
=
EF
=
G
H
,
BC
=
D
E
=
FG
=
H
A
(we call such an octagon semiregular). The diagonals
A
D
AD
A
D
,
B
E
BE
BE
,
C
F
CF
CF
,
D
G
DG
D
G
,
E
H
EH
E
H
,
F
A
FA
F
A
,
G
B
GB
GB
and
H
C
HC
H
C
divide the inside of the octagon into certain parts. Consider the part containing the centre of the octagon. If that part is an octagon, then this central octagon is semiregular (this is obvious). In this case we construct similar diagonals in the central octagon and so on. If, after several steps, the central figure is not an octagon, then the process stops. Prove that if the process never stops, then the initial octagon was regular. (A. Tolpygo, Kiev)
(308) 5
1
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TOT 308 1991 Autumn A J5 coloured cells in 9x9 square
A
9
×
9
9 \times 9
9
×
9
square is divided into
81
81
81
unit cells. Some of the cells are coloured. The distance between the centres of any two coloured cells is more than
2
2
2
. (a) Give an example of colouring with
17
17
17
coloured cells. (b) Prove that the numbers of coloured cells cannot exceed
17
17
17
.(S. Fomin, Leningrad)
(307) 4
1
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TOT 307 1991 Autumn A J4 a_{k+1}=3a_k^4+4a_k^3, NT
A sequence
a
n
a_n
a
n
is determined by the rules
a
0
=
9
a_0 = 9
a
0
=
9
and for any nonnegative
k
k
k
,
a
k
+
1
=
3
a
k
4
+
4
a
k
3
.
a_{k+1}=3a_k^4+4a_k^3.
a
k
+
1
=
3
a
k
4
+
4
a
k
3
.
Prove that
a
10
a_{10}
a
10
contains more than
1000
1000
1000
nines in decimal notation.(Yao)
(306) 3
1
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TOT 306 1991 Autumn A J3 numbers in a 4x4 table
Is it possible to put pairwise distinct positive integers less than
100
100
100
in the cells of a
4
×
4
4 \times 4
4
×
4
table so that the products of all the numbers in every column and every row are equal to each other?(N.B. Vasiliev, Moscow))
(305) 2
1
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TOT 305 1991 Autumn A J2 famous 80-80-20 problem with AD =BC
In
△
A
B
C
\vartriangle ABC
△
A
BC
,
A
B
=
A
C
AB = AC
A
B
=
A
C
and
∠
B
A
C
=
2
0
o
\angle BAC = 20^o
∠
B
A
C
=
2
0
o
. A point
D
D
D
lies on the side
A
B
AB
A
B
and
A
D
=
B
C
AD = BC
A
D
=
BC
. Find
∠
B
C
D
\angle BCD
∠
BC
D
.(LF. Sharygin, Moscow)
(304) 1
1
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TOT 304 1991 Autumn A J1 32 knights and their servants
32
32
32
knights live in a kingdom. Some of them are servants of others. A servant may have only one master and any master is more wealthy than any of his servants. A knight having not less than
4
4
4
servants is called a baron. What is the maximum number of barons? (The kingdom is ruled by the law: “My servant’s servant is not my servant”. (A. Tolpygo, Kiev)
(302) 3
1
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TOT 302 1991 Autumn O J3 nested fractions sum
Prove that
1
2
+
1
3
+
1
4
+
1
.
.
.
+
1
9991
+
1
1
+
1
1
+
1
3
+
1
4
+
1
.
.
.
+
1
9991
=
1
\dfrac{1}{2+\dfrac{1}{3+\dfrac{1}{4+\dfrac{1}{...+\dfrac{1}{9991}}}}}+\dfrac{1}{1+\dfrac{1}{1+\dfrac{1}{3+\dfrac{1}{4+\dfrac{1}{...+\dfrac{1}{9991}}}}}}=1
2
+
3
+
4
+
...
+
9991
1
1
1
1
1
+
1
+
1
+
3
+
4
+
...
+
9991
1
1
1
1
1
1
=
1
This means
1
/
(
2
+
(
1
/
(
3
+
(
1
/
(
4
+
(
.
.
.
+
1
/
1991
)
)
)
)
)
)
+
1
/
(
1
+
(
1
/
(
1
+
(
1
/
(
3
+
(
1
/
(
4
+
(
.
.
.
+
1
/
1991...
)
)
)
)
)
)
)
)
=
1.
)
1/(2+ (1/(3+ (1/(4+(...+1/1991)))))) +1/(1 + (1/(1 + (1/(3 + (1/(4 + (...+ 1/1991...)))))))) = 1.)
1/
(
2
+
(
1/
(
3
+
(
1/
(
4
+
(
...
+
1/1991
))))))
+
1/
(
1
+
(
1/
(
1
+
(
1/
(
3
+
(
1/
(
4
+
(
...
+
1/1991...
))))))))
=
1.
)
(G. Galperin, Moscow-Tel Aviv)
(303) 4
1
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TOT 303 1991 Autumn O J4 6 numbers on a circle a=|b-c|
Six numbers are placed on a circle. For every number
A
A
A
we have:
A
A
A
equals the absolute value of
(
B
−
C
)
(B- C)
(
B
−
C
)
, where
B
B
B
and
C
C
C
follow
A
A
A
clockwise. The total sum of the numbers equals
1
1
1
. Find all the numbers. (Folklore)
(301) 2
1
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TOT 301 1991 Autumn O J2 flying rook on 4x4 chessboard
The “flying rook” moves as the usual chess rook but can’t move to a neighbouring square in one move. Is it possible for the flying rook on a
4
×
4
4 \times 4
4
×
4
chess-board to visit every square once and return to the initial square in
16
16
16
moves? (A. Tolpygo, Kiev)
(300) 1
1
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TOT 300 1991 Autumn O J1 AB= BC wanted, 2 circles
The centre of circle
1
1
1
lies on circle
2
2
2
.
A
A
A
and
B
B
B
are the intersection points of the circles. The tangent line to circle
2
2
2
at point
B
B
B
intersects circle
1
1
1
at point
C
C
C
. Prove that
A
B
=
B
C
AB = BC
A
B
=
BC
.(V. Prasovov, Moscow)
(299) 6
1
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TOT 299 1991 Spring A S6 15 day tournament for 32 boxers
There are
32
32
32
boxers in a tournament. Each boxer can fight no more often than once per day. It is known that the boxers are of different strength, and the stronger man always wins. Prove that a
15
15
15
day tournament can be organised so as to determine their classification (put them in the order of strength). The schedule of fights for each day is fixed on the evening before and cannot be changed during the day. (A. Andjans, Riga)
(298) 5
1
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TOT 298 1991 Spring A S5 system of roads fro 16 cities
There are
16
16
16
cities in a certain kingdom. The king wants to have a system of roads constructed so that one can go along those roads from any city to any other one without going through more than one intermediate city and so that no more than
5
5
5
roads go out of any city. (a) Prove that this is possible. (b) Prove that if we replace the number
5
5
5
by the number
4
4
4
in the statement of the problem the king’s desire will become unrealizable.(D. Fomin, Leningrad)
(296) 3
1
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TOT 296 1991 Spring A S3 sum x_i=0, sum x_i^2=1
The numbers
x
1
,
x
2
,
x
3
,
.
.
.
,
x
n
x_1,x_2,x_3, ..., x_n
x
1
,
x
2
,
x
3
,
...
,
x
n
satisfy the two conditions
∑
i
=
1
n
x
i
=
0
,
∑
i
=
1
n
x
i
2
=
1
\sum^n_{i=1}x_i=0 \,\, , \,\,\,\,\sum^n_{i=1}x_i^2=1
i
=
1
∑
n
x
i
=
0
,
i
=
1
∑
n
x
i
2
=
1
Prove that there are two numbers among them whose product is no greater than
−
1
/
n
- 1/n
−
1/
n
. (Stolov, Kharkov)
(295) 2
1
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TOT 295 1991 Spring A S2 fixed point for lines
The chord
M
N
MN
MN
on the circle is fixed. For every diameter
A
B
AB
A
B
of the circle consider the intersection point
C
C
C
of the lines
A
M
AM
A
M
and
B
N
BN
BN
and construct the line
ℓ
\ell
ℓ
passing through
C
C
C
perpendicularly to
A
B
AB
A
B
. Prove that all the lines
ℓ
\ell
ℓ
pass through a fixed point.(E. Kulanin, Moscow)
(294) 4
1
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TOT 294 1991 Spring O S4 five wooden cubes in space
(a) Is it possible to place five wooden cubes in space so that each of them has a part of its face touching each of the others?(b) Answer the same question, but with
6
6
6
cubes.
(293) 3
1
Hide problems
TOT 293 1991 Spring O S3 a, b -> a+b+ab, last no from 100 remaining
100
100
100
numbers
1
1
1
,
1
/
2
1/2
1/2
,
1
/
3
1/3
1/3
,
.
.
.
...
...
,
1
/
100
1/100
1/100
are written on the blackboard. One may delete two arbitrary numbers
a
a
a
and
b
b
b
among them and replace them by the number
a
+
b
+
a
b
a + b + ab
a
+
b
+
ab
. After
99
99
99
such operations only one number is left. What is this final number?(D. Fomin, Leningrad)
(292) 2
1
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TOT 292 1991 Spring O S2 triangle construction
Two points
K
K
K
and
L
L
L
are given on a circle. Construct a triangle
A
B
C
ABC
A
BC
so that its vertex
C
C
C
and the intersection points of its medians
A
K
AK
A
K
and
B
L
BL
B
L
both lie on the circle,
K
K
K
and
L
L
L
being the midpoints of its sides
B
C
BC
BC
and
A
C
AC
A
C
.
(291) 1
1
Hide problems
TOT 291 1991 Spring O S1 sum x^{2^n} = sum y^{2^n}
Find all natural numbers
n
n
n
, and all integers
x
,
y
x,y
x
,
y
(
x
≠
y
x\ne y
x
=
y
) for which the following equation is satisfied:
x
+
x
2
+
x
4
+
.
.
.
+
x
2
n
=
y
+
y
2
+
y
4
+
.
.
.
+
y
2
n
.
x + x^2 + x^4 + ...+ x^{2^n} = y + y^2 + y^4 + ... + y^{2^n} .
x
+
x
2
+
x
4
+
...
+
x
2
n
=
y
+
y
2
+
y
4
+
...
+
y
2
n
.
(285) 1
1
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TOT 285 1991 Spring A J1 product of 99 fractions >2/3
Prove that the product of the
99
99
99
fractions
k
3
−
1
k
3
+
1
,
k
=
2
,
3
,
.
.
.
,
100
\frac{k^3-1}{k^3+1} \,\, , \,\,\,\,\,\, k=2,3,...,100
k
3
+
1
k
3
−
1
,
k
=
2
,
3
,
...
,
100
is greater than
2
/
3
2/3
2/3
. (D. Fomin, Leningrad)
(290) 6
1
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TOT 290 1991 Spring A J6 10 day tournament for 16 boxers
There are 16 boxers in a tournament. Each boxer can fight no more often than once per day. It is known that the boxers are of different strength, and the stronger man always wins. Prove that a 1
0
0
0
day tournament can be organised so as to determine their classification (put them in the order of strength). The schedule of fights for each day is fixed on the evening before and cannot be changed during the day. (A. Andjans, Riga)
(289) 5
1
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TOT 289 1991 Spring A J5 system of road for 8 cities
There are
8
8
8
cities in a certain kingdom. The king wants to have a system of roads constructed so that one can go along those roads from any city to any other one without going through more than one intermediate city and so that no more than
k
k
k
roads go out of any city. For what values of
k
k
k
is this possible?(D. Fomin, Leningrad)
(288) 4
1
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TOT 288 1991 Spring A J4 perpendicular, rotate an arc
A circle is divided by the chord
A
B
AB
A
B
into two segments and one of them is rotated about the point
A
A
A
by a certain angle, the point
B
B
B
being taken to
B
′
B'
B
′
. Prove that the line segments joining the midpoints of the two arcs (i.e. the arc
A
B
AB
A
B
which had not been rotated and the rotated arc
A
B
′
AB'
A
B
′
) with the midpoint of
B
B
′
BB'
B
B
′
are perpendicular.(F. Nazyrov, 11th form student, Obninsk)
(287) 3
1
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TOT 287 1991 Spring A J3 numbers ending with digit 5
We are looking for numbers ending with the digit
5
5
5
such that in their decimal expansion each digit beginning with the second digit is no less than the previous one. Moreover the squares of these numbers must also possess the same property. (a) Find four such numbers. (b) Prove that there are infinitely many.(A. Andjans, Riga)
(286) 2
1
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TOT 286 1991 Spring A J2 peprendicular in tangential pentagon
The pentagon
A
B
C
D
E
ABCDE
A
BC
D
E
has an inscribed circle and the diagonals
A
D
AD
A
D
and
C
E
CE
CE
intersect in its centre
O
O
O
. Prove that the segment
B
O
BO
BO
and the side
D
E
DE
D
E
are perpendicular.(Folklore)
(284) 4
1
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TOT 284 1991 Spring O J4 123+102k
The number
123
123
123
is shown on the screen of a computer. Each minute the computer adds
102
102
102
to the number on the screen. The computer expert Misha may change the order of digits in the number on the screen whenever he wishes. Can he ensure that no four-digit number ever appears on the screen?(F.L. Nazarov, Leningrad)
(283) 3
1
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TOT 283 1991 Spring O J3 30 boots n a row
We are given
30
30
30
boots standing in a row,
15
15
15
of which are for right feet and
15
15
15
for the left. Prove that there are ten successive boots somewhere in this row with
5
5
5
right and
5
5
5
left boots among them.(D. Fomin, Leningrad)
(282) 2
1
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TOT 282 1991 Spring O J2 apollonius problem CCC, circle ext.tangent to 3 circles
Each of three given circles with radii
1
1
1
,
r
r
r
and
r
r
r
touches the others from the outside. For what values of
r
r
r
does there exist a triangle “circumscribed” to these circles? (This means the circles lie inside the triangle, each circle touching two sides of the triangle and each side of the triangle touching two circles.)(N.B. Vasiliev, Moscow)
(281) 1
1
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TOT 281 1991 Spring O J1 sum of their squares is divisible by N
N
N
N
integers are given. Prove that the sum of their squares is divisible by
N
N
N
if it is known that the difference between the product of any
N
−
1
N - 1
N
−
1
of them and the last one is divisible by
N
N
N
.(D. Fomin, Leningrad)