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National and Regional Contests
Russia Contests
All-Russian Olympiad
1967 All Soviet Union Mathematical Olympiad
1967 All Soviet Union Mathematical Olympiad
Part of
All-Russian Olympiad
Subcontests
(10)
093
1
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ASU 093 All Soviet Union MO 1967 k divisor of 99
Given natural number
k
k
k
with a property "if
n
n
n
is divisible by
k
k
k
, than the number, obtained from
n
n
n
by reversing the order of its digits is also divisible by
k
k
k
". Prove that the
k
k
k
is a divisor of
99
99
99
.
092
1
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ASU 092 All Soviet Union MO 1967 rhombus
Three vertices
K
L
M
KLM
K
L
M
of the rhombus (diamond)
K
L
M
N
KLMN
K
L
MN
lays on the sides
[
A
B
]
,
[
B
C
]
[AB], [BC]
[
A
B
]
,
[
BC
]
and
[
C
D
]
[CD]
[
C
D
]
of the given unit square. Find the area of the set of all the possible vertices
N
N
N
.
091
1
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ASU 091 All Soviet Union MO 1967 499 white castles and 1 black king
"KING-THE SUICIDER" Given a chess-board
1000
×
1000
1000\times 1000
1000
×
1000
,
499
499
499
white castles and a black king. Prove that it does not matter neither the initial situation nor the way white plays, but the king can always enter under the check in a finite number of moves.
090
1
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ASU 090 All Soviet Union MO 1967 max length of integer sequence
In the sequence of the natural (i.e. positive integers) numbers every member from the third equals the absolute value of the difference of the two previous. What is the maximal possible length of such a sequence, if every member is less or equal to
1967
1967
1967
?
089
1
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ASU 089 All Soviet Union MO 1967 diophantine x^2+x=y^4+y^3+y^2+y
Find all the integers
x
,
y
x,y
x
,
y
satisfying equation
x
2
+
x
=
y
4
+
y
3
+
y
2
+
y
x^2+x=y^4+y^3+y^2+y
x
2
+
x
=
y
4
+
y
3
+
y
2
+
y
.
088
1
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ASU 088 All Soviet Union MO 1967 no divisible by 5^{1000}
Prove that there exists a number divisible by
5
1000
5^{1000}
5
1000
not containing a single zero in its decimal notation.
087
1
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ASU 087 All Soviet Union MO 1967 0,1,...,9 around a circle
a) Can you pose the numbers
0
,
1
,
.
.
.
,
9
0,1,...,9
0
,
1
,
...
,
9
on the circumference in such a way, that the difference between every two neighbours would be either
3
3
3
or
4
4
4
or
5
5
5
? b) The same question, but about the numbers
0
,
1
,
.
.
.
,
13
0,1,...,13
0
,
1
,
...
,
13
.
086
1
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ASU 086 All Soviet Union MO 1967 8 lamps in 8 points of space
a) A lamp of a lighthouse enlights an angle of
90
90
90
degrees. Prove that you can turn the lamps of four arbitrary posed lighthouses so, that all the plane will be enlightened. b) There are eight lamps in eight points of the space. Each can enlighten an octant (three-faced space polygon with three mutually orthogonal edges). Prove that you can turn them so, that all the space will be enlightened.
085
1
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ASU 085 All Soviet Union MO 1967 rearranging digits of a natural
a) The digits of a natural number were rearranged. Prove that the sum of given and obtained numbers can't equal
999...9
999...9
999...9
(
1967
1967
1967
of nines). b) The digits of a natural number were rearranged. Prove that if the sum of the given and obtained numbers equals
1010
1010
1010
, than the given number was divisible by
10
10
10
.
084
1
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ASU 084 All Soviet Union MO 1967 height AH=BM median
a) The maximal height
∣
A
H
∣
|AH|
∣
A
H
∣
of the acute-angled triangle
A
B
C
ABC
A
BC
equals the median
∣
B
M
∣
|BM|
∣
BM
∣
. Prove that the angle
A
B
C
ABC
A
BC
isn't greater than
60
60
60
degrees. b) The height
∣
A
H
∣
|AH|
∣
A
H
∣
of the acute-angled triangle
A
B
C
ABC
A
BC
equals the median
∣
B
M
∣
|BM|
∣
BM
∣
and bisectrix
∣
C
D
∣
|CD|
∣
C
D
∣
. Prove that the angle
A
B
C
ABC
A
BC
is equilateral.