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National and Regional Contests
Russia Contests
All-Russian Olympiad
1964 All Russian Mathematical Olympiad
1964 All Russian Mathematical Olympiad
Part of
All-Russian Olympiad
Subcontests
(15)
055
1
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ASU 055 All Russian MO 1964 11.2 tangential trapezoid
Let
A
B
C
D
ABCD
A
BC
D
be an tangential trapezoid,
E
E
E
is a point of its diagonals intersection,
r
1
,
r
2
,
r
3
,
r
4
r_1,r_2,r_3,r_4
r
1
,
r
2
,
r
3
,
r
4
-- the radiuses of the circles inscribed in the triangles
A
B
E
ABE
A
BE
,
B
C
E
BCE
BCE
,
C
D
E
CDE
C
D
E
,
D
A
E
DAE
D
A
E
respectively. Prove that
1
/
(
r
1
)
+
1
/
(
r
3
)
=
1
/
(
r
2
)
+
1
/
(
r
4
)
.
1/(r_1)+1/(r_3) = 1/(r_2)+1/(r_4).
1/
(
r
1
)
+
1/
(
r
3
)
=
1/
(
r
2
)
+
1/
(
r
4
)
.
054
1
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ASU 054 All Russian MO 1964 11.1 perfect square
Find the smallest exact square with last digit not
0
0
0
, such that after deleting its last two digits we shall obtain another exact square.
053
1
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ASU 053 All Russian MO 1964 10.5 11.4 divide cube in tetrahedron
We have to divide a cube onto
k
k
k
non-overlapping tetrahedrons. For what smallest
k
k
k
is it possible?
052
1
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ASU 052 All Russian MO 1964 10.4 11.3 divisions x_1:x_2:...:x_n
Given an expression
x
1
:
x
2
:
.
.
.
:
x
n
x_1 : x_2 : ... : x_n
x
1
:
x
2
:
...
:
x
n
(
:
:
:
means division). We can put the braces as we want. How many expressions can we obtain?
045
1
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ASU 045 All Russian MO 1964 8.5a 10.3ab hexagon
a) Given a convex hexagon
A
B
C
D
E
F
ABCDEF
A
BC
D
EF
with all the equal angles. Prove that
∣
A
B
∣
−
∣
D
E
∣
=
∣
E
F
∣
−
∣
B
C
∣
=
∣
C
D
∣
−
∣
F
A
∣
|AB|-|DE| = |EF|-|BC| = |CD|-|FA|
∣
A
B
∣
−
∣
D
E
∣
=
∣
EF
∣
−
∣
BC
∣
=
∣
C
D
∣
−
∣
F
A
∣
b) The opposite problem: Prove that it is possible to construct a convex hexagon with equal angles of six segments
a
1
,
a
2
,
.
.
.
,
a
6
a_1,a_2,...,a_6
a
1
,
a
2
,
...
,
a
6
, whose lengths satisfy the condition
a
1
−
a
4
=
a
5
−
a
2
=
a
3
−
a
6
a_1-a_4 = a_5-a_2 = a_3-a_6
a
1
−
a
4
=
a
5
−
a
2
=
a
3
−
a
6
051
1
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ASU 051 All Russian MO 1964 10.2 a-k^n is divisible by b-k
Given natural
a
,
b
,
n
a,b,n
a
,
b
,
n
. It is known, that for every natural
k
k
k
(
k
≠
b
k\ne b
k
=
b
) the number
a
−
k
n
a-k^n
a
−
k
n
is divisible by
b
−
k
b-k
b
−
k
. Prove that
a
=
b
n
a=b^n
a
=
b
n
050
1
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ASU 050 All Russian MO 1964 10.1 tangential
The quadrangle
A
B
C
D
ABCD
A
BC
D
is circumscribed around the circle with the centre
O
O
O
. Prove that
∠
A
O
B
+
∠
C
O
D
=
18
0
o
.
\angle AOB+ \angle COD=180^o.
∠
A
OB
+
∠
CO
D
=
18
0
o
.
049
1
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ASU 049 All Russian MO 1964 9.5 shortest path in hexagon
A honeybug crawls along the honeycombs with the unite length of their hexagons. He has moved from the node
A
A
A
to the node
B
B
B
along the shortest possible trajectory. Prove that the half of his way he moved in one direction.
048
1
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ASU 048 All Russian MO 1964 9.4 n! not divisible by n^2
Find all the natural
n
n
n
such that
n
!
n!
n
!
is not divisible by
n
2
n^2
n
2
.
047
1
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ASU 047 All Russian MO 1964 9.3 similar quadrilateral
Four perpendiculars are drawn from the vertices of a convex quadrangle to its diagonals. Prove that their bases make a quadrangle similar to the given one.
046
1
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ASU 046 All Russian MO 1964 9.2 nested radicals in ZxZ
Find integer solutions
(
x
,
y
)
(x,y)
(
x
,
y
)
of the equation (
1964
1964
1964
times "
\sqrt{}
"):
x
+
x
+
.
.
.
.
x
+
x
=
y
\sqrt{x+\sqrt{x+\sqrt{....\sqrt{x+\sqrt{x}}}}}=y
x
+
x
+
....
x
+
x
=
y
044
1
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ASU 044 All Russian MO 1964 8.4 sets of halfs of integers
Given an arbitrary set of
2
k
+
1
2k+1
2
k
+
1
integers
{
a
1
,
a
2
,
.
.
.
,
a
2
k
+
1
}
\{a_1,a_2,...,a_{2k+1}\}
{
a
1
,
a
2
,
...
,
a
2
k
+
1
}
. We make a new set
{
(
a
1
+
a
2
)
/
2
,
(
a
2
+
a
3
)
/
2
,
(
a
2
k
+
a
2
k
+
1
)
/
2
,
(
a
2
k
+
1
+
a
1
)
/
2
}
\{(a_1+a_2)/2, (a_2+a_3)/2, (a_{2k}+a_{2k+1})/2, (a_{2k+1}+a_1)/2\}
{(
a
1
+
a
2
)
/2
,
(
a
2
+
a
3
)
/2
,
(
a
2
k
+
a
2
k
+
1
)
/2
,
(
a
2
k
+
1
+
a
1
)
/2
}
and a new one, according to the same rule, and so on... Prove that if we obtain integers only, the initial set consisted of equal integers only.
043
1
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ASU 043 All Russian MO 1964 8.3 sum of digits
Given
1000000000
1000000000
1000000000
first natural numbers. We change each number with the sum of its digits and repeat this procedure until there will remain
1000000000
1000000000
1000000000
one digit numbers. Is there more "
1
1
1
"-s or "
2
2
2
"-s?
042
1
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ASU 042 All Russian MO 1964 8.2 m(m+1) not a perfect power
Prove that for no natural
m
m
m
a number
m
(
m
+
1
)
m(m+1)
m
(
m
+
1
)
is a power of an integer.
041
1
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ASU 041 All Russian MO 1964 8.1& 9.1 height not less than sides
The two heights in the triangle are not less than the respective sides. Find the angles.