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National and Regional Contests
Russia Contests
All-Russian Olympiad
1970 All Soviet Union Mathematical Olympiad
1970 All Soviet Union Mathematical Olympiad
Part of
All-Russian Olympiad
Subcontests
(15)
143
1
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ASU 143 All Soviet Union MO 1970 coloring vertices
The vertices of the regular
n
n
n
-gon are marked with some colours (each vertex -- with one colour) in such a way, that the vertices of one colour represent the right polygon. Prove that there are two equal ones among the smaller polygons.
142
1
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ASU 142 All Soviet Union MO 1970 even sum of digits, odd sum of digits
All natural numbers containing not more than
n
n
n
digits are divided onto two groups. The first contains the numbers with the even sum of the digits, the second -- with the odd sum. Prove that if
0
<
k
<
n
0<k<n
0
<
k
<
n
than the sum of the
k
k
k
-th powers of the numbers in the first group equals to the sum of the
k
k
k
-th powers of the numbers in the second group.
141
1
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ASU 141 All Soviet Union MO 1970 from 11111 to 99999 in random order
All the
5
5
5
-digit numbers from
11111
11111
11111
to
99999
99999
99999
are written on the cards. Those cards lies in a line in an arbitrary order. Prove that the resulting
444445
444445
444445
-digit number is not a power of two.
140
1
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ASU 140 All Soviet Union MO 1970 equal rectangles 8 point intersection
Two equal rectangles are intersecting in
8
8
8
points. Prove that the common part area is greater than the half of the rectangle's area.
139
1
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ASU 139 All Soviet Union MO 1970 equal sums of digits
Prove that for every natural number
k
k
k
there exists an infinite set of such natural numbers
t
t
t
, that the decimal notation of
t
t
t
does not contain zeroes and the sums of the digits of the numbers
t
t
t
and
k
t
kt
k
t
are equal.
138
1
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ASU 138 All Soviet Union MO 1970 distance equals inradius
Given triangle
A
B
C
ABC
A
BC
, midpoint
M
M
M
of the side
[
B
C
]
[BC]
[
BC
]
, the centre
O
O
O
of the inscribed circle. The line
(
M
O
)
(MO)
(
MO
)
crosses the height
A
H
AH
A
H
in the point
E
E
E
. Prove that the distance
∣
A
E
∣
|AE|
∣
A
E
∣
equals the inscribed circle radius.
137
1
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ASU 137 All Soviet Union MO 1970 100 from 200 integers with sum mult. 100
Prove that from every set of
200
200
200
integers you can choose a subset of
100
100
100
with the total sum divisible by
100
100
100
.
136
1
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ASU 136 All Soviet Union MO 1970 m same digits in 5 n-digit numbers
Given five
n
n
n
-digit binary numbers. For each two numbers their digits coincide exactly on
m
m
m
places. There is no place with the common digit for all the five numbers. Prove that
2
/
5
≤
m
/
n
≤
3
/
5
2/5 \le m/n \le 3/5
2/5
≤
m
/
n
≤
3/5
135
1
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ASU 135 All Soviet Union MO 1970 median, bisector, height concurrent
The angle bisector
[
A
D
]
[AD]
[
A
D
]
, the median
[
B
M
]
[BM]
[
BM
]
and the height
[
C
H
]
[CH]
[
C
H
]
of the acute-angled triangle
A
B
C
ABC
A
BC
intersect in one point. Prove that the
∠
B
A
C
>
4
5
o
\angle BAC> 45^o
∠
B
A
C
>
4
5
o
.
134
1
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ASU 134 All Soviet Union MO 1970 every 3 of five triangles construct a triangle
Given five segments. It is possible to construct a triangle of every subset of three of them. Prove that at least one of those triangles is acute-angled.
133
1
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ASU 133 All Soviet Union MO 1970 91 castle rooms and triangle split
a) A castle is equilateral triangle with the side of
100
100
100
metres. It is divided onto
100
100
100
triangle rooms. Each wall between the rooms is
10
10
10
metres long and contain one door. You are inside and are allowed to pass through every door not more than once. Prove that you can visit not more than
91
91
91
room (not exiting the castle). b) Every side of the triangle is divided onto
k
k
k
parts by the lines parallel to the sides. And the triangle is divided onto
k
2
k^2
k
2
small triangles. Let us call the "chain" such a sequence of triangles, that every triangle in it is included only once, and the consecutive triangles have the common side. What is the greatest possible number of the triangles in the chain?
132
1
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ASU 132 All Soviet Union MO 1970 rearranging 17-digit number
The digits of the
17
17
17
-digit number are rearranged in the reverse order. Prove that at list one digit of the sum of the new and the initial number is even.
131
1
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ASU 131 All Soviet Union MO 1970 side of convex polygon = longest diagonal
How many sides of the convex polygon can equal its longest diagonal?
130
1
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ASU 130 All Soviet Union MO 1970 abc=1, a+b+c > (1/a+1/b+1/c) >0
The product of three positive numbers equals to one, their sum is strictly greater than the sum of the inverse numbers. Prove that one and only one of them is greater than one.
129
1
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ASU 129 All Soviet Union MO 1970 two points construction
Given a circle, its diameter
[
A
B
]
[AB]
[
A
B
]
and a point
C
C
C
on it. Construct (with the help of compasses and ruler) two points
X
X
X
and
Y
Y
Y
, that are symmetric with respect to
(
A
B
)
(AB)
(
A
B
)
line, such that
(
Y
C
)
(YC)
(
Y
C
)
is orthogonal to
(
X
A
)
(XA)
(
X
A
)
.