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Problems
Contests
International Contests
Tournament Of Towns
1989 Tournament Of Towns
1989 Tournament Of Towns
Part of
Tournament Of Towns
Subcontests
(40)
(242) 6
1
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TOT 242 1989 Autumn A S6 at least 1 star in each column in m x n array
A rectangular array has
m
m
m
rows and
n
n
n
columns, where
m
<
n
m < n
m
<
n
. Some cells of the array contain stars, in such a way that there is at least one star in each column. Prove that there is at least one such star such that the row containing it has more stars than the column containing it.(A. Razborov, Moscow)
(241) 5
1
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TOT 241 1989 Autumn A S5 100 points, N convex N-gon, 100-N interior
We are given
100
100
100
points.
N
N
N
of these are vertices of a convex
N
N
N
-gon and the other
100
−
N
100 - N
100
−
N
of these are inside this
N
N
N
-gon. The labels of these points make it impossible to tell whether or not they are vertices of the
N
N
N
-gon. It is known that no three points are collinear and that no
4
4
4
points belong to two parallel lines. It has been decided to ask questions of the following type: What is the area of the triangle
X
Y
Z
XYZ
X
Y
Z
, where
X
,
Y
X, Y
X
,
Y
and
Z
Z
Z
are labels representing three of the
100
100
100
given points? Prove that
300
300
300
such questions are sufficient in order to clarify which points are vertices and to determine the area of the
N
N
N
-gon. (D. Fomin, Leningrad)
(240) 4
1
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TOT 240 1989 Autumn A S4 sum 1/d_i <0,9, infinite arithm, progressions
The set of natural numbers is represented as a union of pairwise disjoint subsets, whose elements form infinite arithmetic progressions with positive differences
d
1
,
d
2
,
d
3
,
.
.
.
d_1,d_2,d_3,...
d
1
,
d
2
,
d
3
,
...
. Is it possible that the sum
1
d
1
+
1
d
1
+
1
d
3
+
.
.
.
\frac{1}{d_1}+\frac{1}{d_1}+\frac{1}{d_3}+...
d
1
1
+
d
1
1
+
d
3
1
+
...
does not exceed
0.9
0.9
0.9
? Consider the cases where (a) the total number of progressions is finite, and (b) the number of progressions is infinite. (In this case the condition that
1
d
1
+
1
d
1
+
1
d
3
+
.
.
.
\frac{1}{d_1}+\frac{1}{d_1}+\frac{1}{d_3}+...
d
1
1
+
d
1
1
+
d
3
1
+
...
does not exceed
0.9
0.9
0.9
should be taken to mean that the sum of any finite number of terms does not exceed 0.9.)(A. Tolpugo, Kiev)
(239) 3
1
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TOT 239 1989 Autumn A S3 area of cross wanted, rotate _|_lines, circle
Choose a point
A
A
A
inside a circle of radius
R
R
R
. Construct a pair of perpendicular lines through
A
A
A
. Then rotate these lines through the same angle
V
V
V
about
A
A
A
. The figure formed inside the circle, as the lines move from their initial to their final position, is in the form of a cross with its centre at
A
A
A
. Find the area of this cross.(Problem from Latvia)
(238) 2
1
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TOT 238 1989 Autumn A S2 sum of squares of products of subsets 1,2,..,N
Consider all the possible subsets of the set
{
1
,
2
,
.
.
.
,
N
}
\{1,2,..., N\}
{
1
,
2
,
...
,
N
}
which do not contain any consecutive numbers. Prove that the sum of the squares of the products of the numbers in these subsets is
(
N
+
1
)
!
−
1
(N + 1)! - 1
(
N
+
1
)!
−
1
. (Based on idea of R.P. Stanley)
(237) 1
1
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TOT 237 1989 Autumn A S1 sphere, triangular pyramid, plane
Is it possible to choose a sphere, a triangular pyramid and a plane so that every plane, parallel to the chosen one, intersects the sphere and the pyramid in sections of equal area?(Problem from Latvia)
(236) 4
1
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TOT 236 1989 Autumn O S4 digits of 2^{1989} ,5^{1989} in a row
The numbers
2
1989
2^{1989}
2
1989
and
5
1989
5^{1989}
5
1989
are written out one after the other (in decimal notation). How many digits are written altogether?(G. Galperin)
(235) 3
1
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TOT 235 1989 Autumn O S3 sum of any 1000 000 never perfect square
Do there exist
1000000
1000 000
1000000
distinct positive integers such that the sum of any collection of these numbers is never an exact square?
(234) 2
1
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TOT 234 1989 Autumn O S2 quadrilateral construction, given 3 midpoints
Three points
K
,
L
K, L
K
,
L
and
M
M
M
are given in the plane. It is known that they are the midpoints of three successive sides of an erased quadrilateral and that these three sides have the same length. Reconstruct the quadrilateral.
(233) 1
1
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TOT 233 1989 Autumn O S1 10friends send cards to each other
Ten friends send greeting cards to each other, each sending
5
5
5
cards. Prove that at least two of them sent cards to each other.(Folklore)
(232) 6
1
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TOT 232 1989 Autumn A J6 regular hexagon is cut into N # of equal area
A regular hexagon is cut up into
N
N
N
parallelograms of equal area. Prove that
N
N
N
is divisible by three.(V. Prasolov, I. Sharygin, Moscow)
(231) 5
1
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TOT 231 1989 Autumn A J5 1x2 in MxN board
A rectangular
M
×
N
M \times N
M
×
N
board is divided into
1
×
1 \times
1
×
cells. There are also many domino pieces of size
1
×
2
1 \times 2
1
×
2
. These pieces are placed on a board so that each piece occupies two cells. The board is not entirely covered, but it is impossible to move the domino pieces (the board has a frame, so that the pieces cannot stick out of it). Prove that the number of uncovered cells is (a) less than
1
4
M
N
\frac14 MN
4
1
MN
,(b) less than
1
5
M
N
\frac15 MN
5
1
MN
.
(230) 4
1
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TOT 230 1989 Autumn A J4 no of triples (a, b, c) , a + b + c = N
Given the natural number N, consider triples of different positive integers
(
a
,
b
,
c
)
(a, b, c)
(
a
,
b
,
c
)
such that
a
+
b
+
c
=
N
a + b + c = N
a
+
b
+
c
=
N
. Take the largest possible system of these triples such that no two triples of the system have any common elements. Denote the number of triples of this system by
K
(
N
)
K(N)
K
(
N
)
. Prove that: (a)
K
(
N
)
>
N
6
−
1
K(N) >\frac{N}{6}-1
K
(
N
)
>
6
N
−
1
(b)
K
(
N
)
<
2
N
9
K(N) <\frac{2N}{9}
K
(
N
)
<
9
2
N
(L.D. Kurliandchik, Leningrad)
(229) 3
1
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TOT 229 1989 Autumn A J3 equilateral triangles by 3families of // lines
The plane is cut up into equilateral triangles by three families of parallel lines. Is it possible to find
4
4
4
vertices of these triangles which form a square?
(228) 2
1
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TOT 228 1989 Autumn A J2 cyclic hexagon, AB=BC,CD=DE, EF=FA
The hexagon
A
B
C
D
E
F
ABCDEF
A
BC
D
EF
is inscribed in a circle,
A
B
=
B
C
=
a
,
C
D
=
D
E
=
b
AB = BC = a, CD = DE = b
A
B
=
BC
=
a
,
C
D
=
D
E
=
b
, and
E
F
=
F
A
=
c
EF = FA = c
EF
=
F
A
=
c
. Prove that the area of triangle
B
D
F
BDF
B
D
F
equals half the area of the hexagon.(I.P. Nagel, Yevpatoria).
(227) 1
1
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TOT 227 1989 Autumn A J1 [ x / 2 ] = [ x / 11 ] +1
Find the number of solutions in positive integers of the equation
⌊
x
2
⌋
=
⌊
x
11
⌋
+
1
\lfloor \frac{x}{2} \rfloor = \lfloor \frac{x}{11} \rfloor +1
⌊
2
x
⌋
=
⌊
11
x
⌋
+
1
where
⌊
A
⌋
\lfloor A\rfloor
⌊
A
⌋
denotes the integer part of the number
A
A
A
, e.g.
⌊
2.031
⌋
=
2
\lfloor 2.031\rfloor = 2
⌊
2.031
⌋
=
2
,
⌊
2
⌋
=
2
\lfloor 2\rfloor = 2
⌊
2
⌋
=
2
, etc.
(226) 4
1
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TOT 226 1989 Autumn O J4 x+ 1 / (y+ 1/z) = 10 / 7
Find the positive integer solutions of the equation
x
+
1
y
+
1
z
=
10
7
x+ \frac{1}{y+ \frac{1}{z}}= \frac{10}{7}
x
+
y
+
z
1
1
=
7
10
(G. Galperin)
(225) 3
1
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TOT 225 1989 Autumn O J3 sum of any 10 of 1989 is positive
A set of
1989
1989
1989
numbers is given. It is known that the sum of any
10
10
10
of them is positive. Prove that the sum of all these numbers is positive. (Folklore)
(224) 2
1
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TOT 224 1989 Autumn O J2 successive integers as sidelengths
The lengths of the sides of an acute angled triangle are successive integers. Prove that the altitude to the second longest side divides this side into two segments whose difference in length equals
4
4
4
.
(223) 1
1
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TOT 223 1989 Autumn O J1 three runeers in a race
Three runners,
X
,
Y
X, Y
X
,
Y
and
Z
Z
Z
, participated in a race.
Z
Z
Z
got held up at the start and began running last, while
Y
Y
Y
was second from the start. During the race
Z
Z
Z
exchanged positions with other contestants
6
6
6
times, while
X
X
X
did that
5
5
5
times. It is known that
Y
Y
Y
finished ahead of
X
X
X
. In what order did they finish?
(219) 3
1
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TOT 219 1989 Spring S3 composition of 1000 linear functions
Given
1000
1000
1000
linear functions
f
k
(
x
)
=
p
k
x
+
q
k
f_k(x)=p_k x + q_k
f
k
(
x
)
=
p
k
x
+
q
k
where
k
=
1
,
2
,
.
.
.
,
1000
k = 1 , 2 ,... , 1000
k
=
1
,
2
,
...
,
1000
, it is necessary to evaluate their composite
f
(
x
)
=
f
1
(
f
2
(
f
3
.
.
.
f
1000
(
x
)
.
.
.
)
)
f(x) =f_1 (f_2(f_3 ... f_{1000}(x)...))
f
(
x
)
=
f
1
(
f
2
(
f
3
...
f
1000
(
x
)
...
))
at the point
x
0
x_0
x
0
. Prove that this can be done in no more than
30
30
30
steps, where at each step one may execute simultaneously any number of arithmetic operations on pairs of numbers obtained from the previous step (at the first step one may use the numbers
p
1
,
p
2
,
.
.
.
,
p
1000
,
q
l
,
q
2
,
.
.
.
,
q
1000
,
x
o
p_1 , p_2 ,... ,p_{1000}, q_l , q_2 ,... ,q_{1000} , x_o
p
1
,
p
2
,
...
,
p
1000
,
q
l
,
q
2
,
...
,
q
1000
,
x
o
).{S. Fomin, Leningrad)
(222) 6
1
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TOT 222 1989 Spring S6 101 rectangles with integer sides <100
We are given
101
101
101
rectangles with sides of integer lengths not exceeding
100
100
100
. Prove that among these
101
101
101
rectangles there are
3
3
3
rectangles, say
A
,
B
A , B
A
,
B
and
C
C
C
such that
A
A
A
will fit inside
B
B
B
and
B
B
B
inside
C
C
C
. ( N . Sedrakyan, Yerevan)
(221) 5
1
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TOT 221 1989 Spring S5 integers on regions of planes by N lines
We are given
N
N
N
lines (
N
>
1
N > 1
N
>
1
) in a plane, no two of which are parallel and no three of which have a point in common. Prove that it is possible to assign, to each region of the plane determined by these lines, a non-zero integer of absolute value not exceeding
N
N
N
, such that the sum of the integers o n either side of any of the given lines is equal to
0
0
0
. (S . Fomin, Leningrad)
(220) 4
1
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TOT 220 1989 Spring S4 a club of 11 people has a committee
A club of
11
11
11
people has a committee. At every meeting of the committee a new committee is formed which differs by
1
1
1
person from its predecessor (either one new member is included or one member is removed). The committee must always have at least three members and , according to the club rules, the committee membership at any stage must differ from its membership at every previous stage. Is it possible that after some time all possible compositions of the committee will have already occurred?(S. Fomin , Leningrad)
(218) 2
1
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TOT 218 1989 Spring S2 incenter wanted, <BMC = 90^o + 1/2 < BAC
The point
M
M
M
, inside
△
A
B
C
\vartriangle ABC
△
A
BC
, satisfies the conditions that
∠
B
M
C
=
9
0
o
+
1
2
∠
B
A
C
\angle BMC = 90^o +\frac12 \angle BAC
∠
BMC
=
9
0
o
+
2
1
∠
B
A
C
and that the line
A
M
AM
A
M
contains the centre of the circumscribed circle of
△
B
M
C
\vartriangle BMC
△
BMC
. Prove that
M
M
M
is the centre of the inscribed circle of
△
A
B
C
\vartriangle ABC
△
A
BC
.
(217) 1
1
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TOT 217 1989 Spring S1 pair of 2 six-digit numbers wanted
Find a pair of
2
2
2
six-digit numbers such that, if they are written down side by side to form a twelve-digit number , this number is divisible by the product of the two original numbers. Find all such pairs of six-digit numbers.( M . N . Gusarov, Leningrad)
(216) 4
1
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TOT 216 1989 Spring Train S4 non-intersecting path on Rubik 's cube
Is it possible to mark a diagonal on each little square on the surface of a Rubik 's cube so that one obtains a non-intersecting path? (S . Fomin, Leningrad)
(215) 3
1
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TOT 215 1989 Spring Train S3 product of any 2 of 6 is divisible by their sum
Find six distinct positive integers such that the product of any two of them is divisible by their sum.(D. Fomin, Leningrad)
(214) 2
1
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TOT 214 1989 Spring Train S2 tangent incircles in tangential trapezoid
It is known that a circle can be inscribed in a trapezium
A
B
C
D
ABCD
A
BC
D
. Prove that the two circles, constructed on its oblique sides as diameters, touch each other.(D. Fomin, Leningrad)
(213) 1
1
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TOT 213 1989 Spring Train S1 a^2 + 3b^2 + 5c^2 + 7 d^2 >= 1
The positive numbers
a
,
b
,
c
a, b, c
a
,
b
,
c
and
d
d
d
satisfy
a
≤
b
≤
c
≤
d
a\le b\le c\le d
a
≤
b
≤
c
≤
d
and
a
+
b
+
c
+
d
≤
1
a + b + c + d \le 1
a
+
b
+
c
+
d
≤
1
. Prove that
a
2
+
3
b
2
+
5
c
2
+
7
d
2
≥
1
a^2 + 3b^2 + 5c^2 + 7 d^2 \ge 1
a
2
+
3
b
2
+
5
c
2
+
7
d
2
≥
1
.
(212) 6
1
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TOT 212 1989 Spring J6 3n + 1 stars in the cells of a 2n x 2n array
(a) Prove that if 3n stars are placed in
3
n
3n
3
n
cells of a
2
n
×
2
n
2n \times 2n
2
n
×
2
n
array, then it is possible to remove
n
n
n
rows and
n
n
n
columns in such away that all stars will be removed . (b) Prove that it is possible to place
3
n
+
1
3n + 1
3
n
+
1
stars in the cells of a
2
n
×
2
n
2n \times 2n
2
n
×
2
n
array in such a way that after removing any
n
n
n
rows and
n
n
n
columns at least one star remains. (K . P. Kohas, Leningrad)
(211) 5
1
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TOT 211 1989 Spring J5 red and blue equal arcs
The centre of a circle is the origin of
N
N
N
vectors whose ends divide the circle in
N
N
N
equal arcs . Some of the vectors are blue and some are red . We calculate the sum of the angles formed between each pair consisting of a red vector and a blue vector (the angle being measured anticlockwise from red to blue) and divide this sum by the total number of such angles . Prove that the "mean angle" thus obtained is
18
0
o
180^o
18
0
o
. (V. P roizvolov)
(210) 4
1
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TOT 210 1989 Spring J4 sum of no set of successive numbers divisible by k
Prove that if
k
k
k
is an even positive integer then it is possible to write the integers from
1
1
1
to
k
−
1
k-1
k
−
1
in such an order that the sum of no set of successive numbers is divisible by
k
k
k
.
(206) 4
1
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TOT 206 1989 Spring Train J4 closed path on Rubik's cube
Can one draw , on the surface of a Rubik's cube , a closed path which crosses each little square exactly once and does not pass through any vertex of a square? (S . Fomin, Leningrad)
(205) 3
1
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TOT 205 1989 Spring Train J3 888...88?999...99 divisible by 7
What digit must be put in place of the "
?
?
?
" in the number
888...88
?
999...99
888...88?999...99
888...88
?
999...99
(where the
8
8
8
and
9
9
9
are each written
50
50
50
times) in order that the resulting number is divisible by
7
7
7
? (M . I. Gusarov)
(204) 2
1
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TOT 204 1989 Spring Train J2 double inradii, triangles by median
In the triangle
A
B
C
ABC
A
BC
the median
A
M
AM
A
M
is drawn. Is it possible that the radius of the circle inscribed in
△
A
B
M
\vartriangle ABM
△
A
BM
could be twice as large as the radius of the circle inscribed in
△
A
C
M
\vartriangle ACM
△
A
CM
?( D . Fomin , Leningrad)
(203) 1
1
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TOT 203 1989 Spring Train J1 a^2 + 3b^2 + 5c^2 <= 1
The positive numbers
a
,
b
a, b
a
,
b
and
c
c
c
satisfy
a
≥
b
≥
c
a \ge b \ge c
a
≥
b
≥
c
and
a
+
b
+
c
≤
1
a + b + c \le 1
a
+
b
+
c
≤
1
. Prove that
a
2
+
3
b
2
+
5
c
2
≤
1
a^2 + 3b^2 + 5c^2 \le 1
a
2
+
3
b
2
+
5
c
2
≤
1
. (F . L . Nazarov)
(209) 3
1
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TOT 209 1989 Spring J3 convex ABCD , PQRS from paper and cardboard
The convex quadrilaterals
A
B
C
D
ABCD
A
BC
D
and
P
Q
R
S
PQRS
PQRS
are made respectively from paper and cardboard. We say that they suit each other if the following two conditions are met : ( 1 ) It is possible to put the cardboard quadrilateral on the paper one so that the vertices of the first lie on the sides of the second, one vertex per side, and (2) If, after this, we can fold the four non-covered triangles of the paper quadrilateral on to the cardboard one, covering it exactly. ( a) Prove that if the quadrilaterals suit each other, then the paper one has either a pair of opposite sides parallel or (a pair of) perpendicular diagonals. (b) Prove that if
A
B
C
D
ABCD
A
BC
D
is a parallelogram, then one can always make a cardboard quadrilateral to suit it.(N. Vasiliev)
(208) 2
1
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TOT 208 1989 Spring J2 game strategy with a pawn on chessboard
On a square of a chessboard there is a pawn . Two players take turns to move it to another square, subject to the rule that , at each move the distance moved is strictly greater than that of the previous move. A player loses when unable to make a move on his turn. Who wins if the players always choose the best strategy? (The pawn is always placed in the centre of its square. )( F . L . Nazarov)
(207) 1
1
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TOT 207 1989 Spring J1 staircase with 100 steps
A staircase has
100
100
100
steps. Kolya wishes to descend the staircase by alternately jumping down some steps and then up some. The possible jumps he can do are through
6
6
6
(i.e. over
5
5
5
and landing on the
6
6
6
th) ,
7
7
7
or
8
8
8
steps . He also does not wish to land twice on the same step . Can he descend the staircase in this way?( S . Fomin, Leningrad)