MathDB
Problems
Contests
National and Regional Contests
Russia Contests
All-Russian Olympiad
1966 All Russian Mathematical Olympiad
1966 All Russian Mathematical Olympiad
Part of
All-Russian Olympiad
Subcontests
(12)
077
1
Hide problems
ASU 077 All Russian MO 1966 s=\pm a1\pm a2\pm ...\pm a_n
Given the numbers
a
1
,
a
2
,
.
.
.
,
a
n
a_1, a_2, ... , a_n
a
1
,
a
2
,
...
,
a
n
such that
0
≤
a
1
≤
a
2
≤
2
a
1
,
a
2
≤
a
3
≤
2
a
2
,
.
.
.
,
a
n
−
1
≤
a
n
≤
2
a
n
−
1
0\le a_1\le a_2\le 2a_1 , a_2\le a_3\le 2a_2 , ... , a_{n-1}\le a_n\le 2a_{n-1}
0
≤
a
1
≤
a
2
≤
2
a
1
,
a
2
≤
a
3
≤
2
a
2
,
...
,
a
n
−
1
≤
a
n
≤
2
a
n
−
1
Prove that in the sum
s
=
±
a
1
±
a
2
±
.
.
.
±
a
n
s=\pm a1\pm a2\pm ...\pm a_n
s
=
±
a
1
±
a
2
±
...
±
a
n
You can choose appropriate signs to make
0
≤
s
≤
a
1
0\le s\le a_1
0
≤
s
≤
a
1
.
083
1
Hide problems
ASU 083 All Russian MO 1966 game strategy, sings between 1,2,...,20
20
20
20
numbers are written on the board
1
,
2
,
.
.
.
,
20
1, 2, ... ,20
1
,
2
,
...
,
20
. Two players are putting signs before the numbers in turn (
+
+
+
or
−
-
−
). The first wants to obtain the minimal possible absolute value of the sum. What is the maximal value of the absolute value of the sum that can be achieved by the second player?
082
1
Hide problems
ASU 082 All Russian MO 1966 min speed of plane, angle changed
The distance from
A
A
A
to
B
B
B
is
d
d
d
kilometres. A plane flying with the constant speed in the constant direction along and over the line
(
A
B
)
(AB)
(
A
B
)
is being watched from those points. Observers have reported that the angle to the plane from the point
A
A
A
has changed by
α
\alpha
α
degrees and from
B
B
B
— by
β
\beta
β
degrees within one second. What can be the minimal speed of the plane?
081
1
Hide problems
ASU 081 All Russian MO 1966 100 points on plane, family circles
Given
100
100
100
points on the plane. Prove that you can cover them with a family of circles with the sum of their diameters less than
100
100
100
and the distance between any two of the circles more than one.
080
1
Hide problems
ASU 080 All Russian MO 1966 tetrahedrons with given smallest height
Given a triangle
A
B
C
ABC
A
BC
. Consider all the tetrahedrons
P
A
B
C
PABC
P
A
BC
with
P
H
PH
P
H
-- the smallest of all tetrahedron's heights. Describe the set of all possible points
H
H
H
.
079
1
Hide problems
ASU 079 All Russian MO 1966 three crossroads in a city
For three arbitrary crossroads
A
,
B
,
C
A,B,C
A
,
B
,
C
in a certain city there exist a way from
A
A
A
to
B
B
B
not coming through
C
C
C
. Prove that for every couple of the crossroads there exist at least two non-intersecting ways connecting them. (there are at least two crossroads in the city)
078
1
Hide problems
ASU 078 All Russian MO 1966 circle of radius S/P inside a convex polygon
Prove that you can always pose a circle of radius
S
/
P
S/P
S
/
P
inside a convex polygon with the perimeter
P
P
P
and area
S
S
S
.
076
1
Hide problems
ASU 076 All Russian MO 1966 shortest path on cross-lined paper
A rectangle
A
B
C
D
ABCD
A
BC
D
is drawn on the cross-lined paper with its sides laying on the lines, and
∣
A
D
∣
|AD|
∣
A
D
∣
is
k
k
k
times more than
∣
A
B
∣
|AB|
∣
A
B
∣
(
k
k
k
is an integer). All the shortest paths from
A
A
A
to
C
C
C
coming along the lines are considered. Prove that the number of those with the first link on
[
A
D
]
[AD]
[
A
D
]
is
k
k
k
times more then of those with the first link on
[
A
B
]
[AB]
[
A
B
]
.
075
1
Hide problems
ASU 075 All Russian MO 1966 arranging soldiers by height
a) Pupils of the
8
8
8
-th form are standing in a row. There is the pupil of the
7
7
7
-th form in before each, and he is smaller (in height) than the elder. Prove that if you will sort the pupils in each of rows with respect to their height, every 8-former will still be taller than the
7
7
7
-former standing before him. b) An infantry detachment soldiers stand in the rectangle, being arranged in columns with respect to their height. Prove that if you rearrange them with respect to their height in every separate row, they will still be staying with respect to their height in columns.
074
1
Hide problems
ASU 074 All Russian MO 1966 (x^2+y) and (y^2+x) perfect squares
Can both
(
x
2
+
y
)
(x^2+y)
(
x
2
+
y
)
and
(
y
2
+
x
)
(y^2+x)
(
y
2
+
x
)
be exact squares for natural
x
x
x
and
y
y
y
?
073
1
Hide problems
ASU 073 All Russian MO 1966 |PA|+|PD|>|PB|+|PC|
a) Points
B
B
B
and
C
C
C
are inside the segment
[
A
D
]
[AD]
[
A
D
]
.
∣
A
B
∣
=
∣
C
D
∣
|AB|=|CD|
∣
A
B
∣
=
∣
C
D
∣
. Prove that for all of the points P on the plane holds inequality
∣
P
A
∣
+
∣
P
D
∣
>
∣
P
B
∣
+
∣
P
C
∣
|PA|+|PD|>|PB|+|PC|
∣
P
A
∣
+
∣
P
D
∣
>
∣
PB
∣
+
∣
PC
∣
b) Given four points
A
,
B
,
C
,
D
A,B,C,D
A
,
B
,
C
,
D
on the plane. For all of the points
P
P
P
on the plane holds inequality
∣
P
A
∣
+
∣
P
D
∣
>
∣
P
B
∣
+
∣
P
C
∣
.
|PA|+|PD| > |PB|+|PC|.
∣
P
A
∣
+
∣
P
D
∣
>
∣
PB
∣
+
∣
PC
∣.
Prove that points
B
B
B
and C are inside the segment
[
A
D
]
[AD]
[
A
D
]
and
∣
A
B
∣
=
∣
C
D
∣
|AB|=|CD|
∣
A
B
∣
=
∣
C
D
∣
.
072
1
Hide problems
ASU 072 All Russian MO 1966 one astronomer on every planet
There is exactly one astronomer on every planet of a certain system. He watches the closest planet. The number of the planets is odd and all of the distances are different. Prove that there one planet being not watched.