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Contests
National and Regional Contests
Russia Contests
All-Russian Olympiad
1962 All Russian Mathematical Olympiad
1962 All Russian Mathematical Olympiad
Part of
All-Russian Olympiad
Subcontests
(14)
026
1
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ASU 026 All Russian MO 1962 10.5 table
Given positive numbers
a
1
,
a
2
,
.
.
.
,
a
m
,
b
1
,
b
2
,
.
.
.
,
b
n
a_1, a_2, ..., a_m, b_1, b_2, ..., b_n
a
1
,
a
2
,
...
,
a
m
,
b
1
,
b
2
,
...
,
b
n
. Is known that
a
1
+
a
2
+
.
.
.
+
a
m
=
b
1
+
b
2
+
.
.
.
+
b
n
.
a_1+a_2+...+a_m=b_1+b_2+...+b_n.
a
1
+
a
2
+
...
+
a
m
=
b
1
+
b
2
+
...
+
b
n
.
Prove that you can fill an empty table with
m
m
m
rows and
n
n
n
columns with no more than
(
m
+
n
−
1
)
(m+n-1)
(
m
+
n
−
1
)
positive number in such a way, that for all
i
,
j
i,j
i
,
j
the sum of the numbers in the
i
i
i
-th row will equal to
a
i
a_i
a
i
, and the sum of the numbers in the
j
j
j
-th column -- to
b
j
b_j
b
j
.
025
1
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ASU 025 All Russian MO 1962 10.4 a_{k-1}-2a_k+a_{k+1} >=0
Given
a
0
,
a
1
,
.
.
.
,
a
n
a_0, a_1, ... , a_n
a
0
,
a
1
,
...
,
a
n
. It is known that
a
0
=
a
n
=
0
,
a
k
−
1
−
2
a
k
+
a
k
+
1
≥
0
a_0=a_n=0, a_{k-1}-2a_k+a_{k+1}\ge 0
a
0
=
a
n
=
0
,
a
k
−
1
−
2
a
k
+
a
k
+
1
≥
0
for all
k
=
1
,
2
,
.
.
.
,
k
−
1
k = 1, 2, ... , k-1
k
=
1
,
2
,
...
,
k
−
1
.Prove that all the numbers are nonnegative.
024
1
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ASU 024 All Russian MO 1962 10.3 (x-y)^5+(y-z)^5+(z-x)^5
Given
x
,
y
,
z
x,y,z
x
,
y
,
z
, three different integers. Prove that
(
x
−
y
)
5
+
(
y
−
z
)
5
+
(
z
−
x
)
5
(x-y)^5+(y-z)^5+(z-x)^5
(
x
−
y
)
5
+
(
y
−
z
)
5
+
(
z
−
x
)
5
is divisible by
5
(
x
−
y
)
(
y
−
z
)
(
z
−
x
)
5(x-y)(y-z)(z-x)
5
(
x
−
y
)
(
y
−
z
)
(
z
−
x
)
023
1
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ASU 023 All Russian MO 1962 10.2 max triangle area
What maximal area can have a triangle if its sides
a
,
b
,
c
a,b,c
a
,
b
,
c
satisfy inequality
0
≤
a
≤
1
≤
b
≤
2
≤
c
≤
3
0\le a\le 1\le b\le 2\le c\le 3
0
≤
a
≤
1
≤
b
≤
2
≤
c
≤
3
?
022
1
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ASU 022 All Russian MO 1962 10.1 perpendicularity in isosceles
The
M
M
M
point is the midpoint of the base
[
A
C
]
[AC]
[
A
C
]
of an isosceles triangle
A
B
C
ABC
A
BC
.
[
M
H
]
[MH]
[
M
H
]
is orthogonal to
[
B
C
]
[BC]
[
BC
]
side. Point
P
P
P
is the midpoint of the segment
[
M
H
]
[MH]
[
M
H
]
. Prove that
[
A
H
]
[AH]
[
A
H
]
is orthogonal to
[
B
P
]
[BP]
[
BP
]
.
021
1
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ASU 021 All Russian MO 1962 9.4 sum of digits
Given
1962
1962
1962
-digit number. It is divisible by
9
9
9
. Let
x
x
x
be the sum of its digits. Let the sum of the digits of
x
x
x
be
y
y
y
. Let the sum of the digits of
y
y
y
be
z
z
z
. Find
z
z
z
.
020
1
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ASU 020 All Russian MO 1962 9.3 min, max inside a regular pentagon
Given regular pentagon
A
B
C
D
E
ABCDE
A
BC
D
E
.
M
M
M
is an arbitrary point inside
A
B
C
D
E
ABCDE
A
BC
D
E
or on its side. Let the distances
∣
M
A
∣
,
∣
M
B
∣
,
.
.
.
,
∣
M
E
∣
|MA|, |MB|, ... , |ME|
∣
M
A
∣
,
∣
MB
∣
,
...
,
∣
ME
∣
be renumerated and denoted with
r
1
≤
r
2
≤
r
3
≤
r
4
≤
r
5
.
r_1\le r_2\le r_3\le r_4\le r_5.
r
1
≤
r
2
≤
r
3
≤
r
4
≤
r
5
.
Find all the positions of the
M
M
M
, giving
r
3
r_3
r
3
the minimal possible value. Find all the positions of the
M
M
M
, giving
r
3
r_3
r
3
the maximal possible value.
019
1
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ASU 019 All Russian MO 1962 9.2 4-variable inequality
Given a quartet of positive numbers
a
,
b
,
c
,
d
a,b,c,d
a
,
b
,
c
,
d
, and is known, that
a
b
c
d
=
1
abcd=1
ab
c
d
=
1
. Prove that
a
2
+
b
2
+
c
2
+
d
2
+
a
b
+
a
c
+
a
d
+
b
c
+
b
d
+
d
c
≥
10
a^2+b^2+c^2+d^2+ab+ac+ad+bc+bd+dc \ge 10
a
2
+
b
2
+
c
2
+
d
2
+
ab
+
a
c
+
a
d
+
b
c
+
b
d
+
d
c
≥
10
018
1
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ASU 018 All Russian MO 1962 9.1 triangle construction
Given two sides of the triangle. Construct that triangle, if medians to those sides are orthogonal.
017
1
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ASU 017 All Russian MO 1962 8.5 & 9.5 numbers in a table
Given a
n
×
n
n\times n
n
×
n
table, where
n
n
n
is odd. There is either
1
1
1
or
−
1
-1
−
1
in its every field. A product of the numbers in the column is written under every column. A product of the numbers in the row is written to the right of every row. Prove that the sum of
2
n
2n
2
n
products doesn't equal to
0
0
0
.
016
1
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ASU 016 All Russian MO 1962 8.4 polynomial integer
Prove that there are no integers
a
,
b
,
c
,
d
a,b,c,d
a
,
b
,
c
,
d
such that the polynomial
a
x
3
+
b
x
2
+
c
x
+
d
ax^3+bx^2+cx+d
a
x
3
+
b
x
2
+
c
x
+
d
equals
1
1
1
at
x
=
19
x=19
x
=
19
, and equals
2
2
2
at
x
=
62
x=62
x
=
62
.
015
1
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ASU 015 All Russian MO d 1962 8.3 a_{100}=3a_{99}-2a_{98}
Given positive numbers
a
1
,
a
2
,
.
.
.
,
a
99
,
a
100
a_1,a_2,...,a_{99},a_{100}
a
1
,
a
2
,
...
,
a
99
,
a
100
. It is known, that
a
1
>
a
0
,
a
2
=
3
a
1
−
2
a
0
,
a
3
=
3
a
2
−
2
a
1
,
.
.
.
,
a
100
=
3
a
99
−
2
a
98
a_1>a_0, a_2=3a_1-2a_0, a_3=3a_2-2a_1, ..., a_{100}=3a_{99}-2a_{98}
a
1
>
a
0
,
a
2
=
3
a
1
−
2
a
0
,
a
3
=
3
a
2
−
2
a
1
,
...
,
a
100
=
3
a
99
−
2
a
98
Prove that
a
100
>
2
99
.
a_{100}>2^{99}.
a
100
>
2
99
.
014
1
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ASU 014 All Russian MO 1962 8.2 locus
Given the circumference
s
s
s
and the straight line
l
l
l
, passing through the centre
O
O
O
of
s
s
s
. Another circumference
s
′
s'
s
′
passes through the point
O
O
O
and has its centre on the
l
l
l
. Describe the set of the points
M
M
M
, where the common tangent of
s
s
s
and
s
′
s'
s
′
touches
s
′
s'
s
′
.
013
1
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ASU 013 All Russian MO 1962 8.1 areas
Given points
A
′
,
B
′
,
C
′
,
D
′
,
A' ,B' ,C' ,D',
A
′
,
B
′
,
C
′
,
D
′
,
on the extension of the
[
A
B
]
,
[
B
C
]
,
[
C
D
]
,
[
D
A
]
[AB], [BC], [CD], [DA]
[
A
B
]
,
[
BC
]
,
[
C
D
]
,
[
D
A
]
sides of the convex quadrangle
A
B
C
D
ABCD
A
BC
D
, such, that the following pairs of vectors are equal:
[
B
B
′
]
=
[
A
B
]
,
[
C
C
′
]
=
[
B
C
]
,
[
D
D
′
]
=
[
C
D
]
,
[
A
A
′
]
=
[
D
A
]
.
[BB']=[AB], [CC']=[BC], [DD']=[CD], [AA']=[DA].
[
B
B
′
]
=
[
A
B
]
,
[
C
C
′
]
=
[
BC
]
,
[
D
D
′
]
=
[
C
D
]
,
[
A
A
′
]
=
[
D
A
]
.
Prove that the quadrangle
A
′
B
′
C
′
D
′
A'B'C'D'
A
′
B
′
C
′
D
′
area is five times more than the quadrangle
A
B
C
D
ABCD
A
BC
D
area.