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Contests
National and Regional Contests
Russia Contests
All-Russian Olympiad
1981 All Soviet Union Mathematical Olympiad
1981 All Soviet Union Mathematical Olympiad
Part of
All-Russian Olympiad
Subcontests
(23)
326
1
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ASU 326 All Soviet Union MO 1981 locus in a right triangle prism
The segments
[
A
D
]
,
[
B
E
]
[AD], [BE]
[
A
D
]
,
[
BE
]
and
[
C
F
]
[CF]
[
CF
]
are the side edges of the right triangle prism. (the equilateral triangle is a base) Find all the points in its base
A
B
C
ABC
A
BC
, situated on the equal distances from the
(
A
E
)
,
(
B
F
)
(AE), (BF)
(
A
E
)
,
(
BF
)
and
(
C
D
)
(CD)
(
C
D
)
lines.
325
1
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ASU 325 All Soviet Union MO 1981 min of 4 +x^2y^4 + x^4y^2 - 3x^2y^2
a) Find the minimal value of the polynomial
P
(
x
,
y
)
=
4
+
x
2
y
4
+
x
4
y
2
−
3
x
2
y
2
P(x,y) = 4 + x^2y^4 + x^4y^2 - 3x^2y^2
P
(
x
,
y
)
=
4
+
x
2
y
4
+
x
4
y
2
−
3
x
2
y
2
b) Prove that it cannot be represented as a sum of the squares of some polynomials of
x
,
y
x,y
x
,
y
.
324
1
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ASU 324 All Soviet Union MO 1981 6 points inside 3x4 rectangle
Six points are marked inside the
3
×
4
3\times 4
3
×
4
rectangle. Prove that there is a pair of marked points with the distance between them not greater than
5
\sqrt5
5
.
323
1
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ASU 323 All Soviet Union MO 1981 piles with cards of no 100 to 999
The natural numbers from
100
100
100
to
999
999
999
are written on separate cards. They are gathered in one pile with their numbers down in arbitrary order. Let us open them in sequence and divide into
10
10
10
piles according to the least significant digit. The first pile will contain cards with
0
0
0
at the end, ... , the tenth -- with
9
9
9
. Then we shall gather
10
10
10
piles in one pile, the first -- down, then the second, ... and the tenth -- up. Let us repeat the procedure twice more, but the next time we shall divide cards according to the second digit, and the last time -- to the most significant one. What will be the order of the cards in the obtained pile?
322
1
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ASU 322 All Soviet Union 1981 greater gcd of each n,(n+1),...,(n+20) than 30030
Find
n
n
n
such that each of the numbers
n
,
(
n
+
1
)
,
.
.
.
,
(
n
+
20
)
n,(n+1),...,(n+20)
n
,
(
n
+
1
)
,
...
,
(
n
+
20
)
has the common divider greater than one with the number
30030
=
2
⋅
3
⋅
5
⋅
7
⋅
11
⋅
13
30030 = 2\cdot 3\cdot 5\cdot 7\cdot 11\cdot 13
30030
=
2
⋅
3
⋅
5
⋅
7
⋅
11
⋅
13
.
321
1
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ASU 321 All Soviet Union MO 1981 numbers in vertices of a cube
A number is written in the each vertex of a cube. It is allowed to add one to two numbers written in the ends of one edge. Is it possible to obtain the cube with all equal numbers if the numbers were initially as on the pictures:
320
1
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ASU 320 All Soviet Union MO 1981 d < 4p, copy of convex polygon
A pupil has tried to make a copy of a convex polygon, drawn inside the unit circle. He draw one side, from its end -- another, and so on. Having finished, he has noticed that the first and the last vertices do not coincide, but are situated
d
d
d
units of length far from each other. The pupil draw angles precisely, but made relative error less than
p
p
p
in the lengths of sides. Prove that
d
<
4
p
d < 4p
d
<
4
p
.
319
1
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ASU 319 All Soviet Union MO 1981 0<=x,y and x^3+y^3=x-y => x^2+y^2<1.
Positive numbers
x
,
y
x,y
x
,
y
satisfy equality
x
3
+
y
3
=
x
−
y
x^3+y^3=x-y
x
3
+
y
3
=
x
−
y
Prove that
x
2
+
y
2
<
1
x^2+y^2<1
x
2
+
y
2
<
1
318
1
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ASU 318 All Soviet Union MO 1981 P/2 < p < 3P/4 perimeter inequality
The points
C
1
,
A
1
,
B
1
C_1, A_1, B_1
C
1
,
A
1
,
B
1
belong to
[
A
B
]
,
[
B
C
]
,
[
C
A
]
[AB], [BC], [CA]
[
A
B
]
,
[
BC
]
,
[
C
A
]
sides, respectively, of the triangle
A
B
C
ABC
A
BC
.
∣
A
C
1
∣
∣
C
1
B
∣
=
∣
B
A
1
∣
∣
A
1
C
∣
=
∣
C
B
1
∣
∣
B
1
A
∣
=
1
3
\frac{|AC_1|}{|C_1B| }=\frac{ |BA_1|}{|A_1C| }= \frac{|CB_1|}{|B_1A| }= \frac{1}{3}
∣
C
1
B
∣
∣
A
C
1
∣
=
∣
A
1
C
∣
∣
B
A
1
∣
=
∣
B
1
A
∣
∣
C
B
1
∣
=
3
1
Prove that the perimeter
P
P
P
of the triangle
A
B
C
ABC
A
BC
and the perimeter
p
p
p
of the triangle
A
1
B
1
C
1
A_1B_1C_1
A
1
B
1
C
1
, satisfy inequality
P
2
<
p
<
3
P
4
\frac{P}{2} < p < \frac{3P}{4}
2
P
<
p
<
4
3
P
317
1
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ASU 317 All Soviet Union MO 1981 18 soccer teams, 8 tours of 1-round
Eighteen soccer teams have played
8
8
8
tours of a one-round tournament. Prove that there is a triple of teams, having not met each other yet.
316
1
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ASU 316 All Soviet Union MO 1981 solve x^3 - y^3 = xy + 61 in N
Find the natural solutions of the equation
x
3
−
y
3
=
x
y
+
61
x^3 - y^3 = xy + 61
x
3
−
y
3
=
x
y
+
61
.
315
1
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ASU 315 All Soviet Union MO 1981 4 parallelograms => 5th one
The quadrangles
A
M
B
E
,
A
H
B
T
,
B
K
X
M
AMBE, AHBT, BKXM
A
MBE
,
A
H
BT
,
B
K
XM
, and
C
K
X
P
CKXP
C
K
XP
are parallelograms. Prove that the quadrangle
A
B
T
E
ABTE
A
BTE
is also parallelogram. (the vertices are mentioned counterclockwise)
314
1
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ASU 314 All Soviet Union MO 1981 rectangular table bw squares, 75% related
Is it possible to fill a rectangular table with black and white squares (only) so, that the number of black squares will equal to the number of white squares, and each row and each column will have more than
75
%
75\%
75%
squares of the same colour?
313
1
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ASU 313 All Soviet Union MO 1981 k_n<=n \sqrt n, (m-n) | (k_n - k_m)
Find all the sequences of natural
k
n
k_n
k
n
with two properties:a)
k
n
≤
n
n
k_n \le n \sqrt {n}
k
n
≤
n
n
for all
n
n
n
b)
(
k
n
−
k
m
)
(k_n - k_m)
(
k
n
−
k
m
)
is divisible by
(
m
−
n
)
(m-n)
(
m
−
n
)
for all
m
>
n
m>n
m
>
n
312
1
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ASU 312 All Soviet Union MO 1981 (KLMN) =1/2 (ABCD), areas
The points
K
K
K
and
M
M
M
are the centres of the
A
B
AB
A
B
and
C
D
CD
C
D
sides of the convex quadrangle
A
B
C
D
ABCD
A
BC
D
. The points
L
L
L
and
M
M
M
belong to two other sides and
K
L
M
N
KLMN
K
L
MN
is a rectangle. Prove that
K
L
M
N
KLMN
K
L
MN
area is a half of
A
B
C
D
ABCD
A
BC
D
area.
310
1
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ASU 310 All Soviet Union MO 1981 90 inhabitants out of 1000, news in 10 days
There are
1000
1000
1000
inhabitants in a settlement. Every evening every inhabitant tells all his friends all the news he had heard the previous day. Every news becomes finally known to every inhabitant. Prove that it is possible to choose
90
90
90
of inhabitants so, that if you tell them a news simultaneously, it will be known to everybody in
10
10
10
days.
311
1
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ASU 311 All Soviet Union MO 1981 acos(x)+b cos(3x)<=1 =>|b|\le 1
It is known about real
a
a
a
and
b
b
b
that the inequality
a
cos
x
+
b
cos
(
3
x
)
>
1
a \cos x + b \cos (3x) > 1
a
cos
x
+
b
cos
(
3
x
)
>
1
has no real solutions. Prove that
∣
b
∣
≤
1
|b|\le 1
∣
b
∣
≤
1
.
309
1
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ASU 309 All Soviet Union MO 1981 3 equilateral with 2 equal vectors
Three equilateral triangles
A
B
C
,
C
D
E
,
E
H
K
ABC, CDE, EHK
A
BC
,
C
D
E
,
E
HK
(the vertices are mentioned counterclockwise) are lying in the plane so, that the vectors
A
D
→
\overrightarrow{AD}
A
D
and
D
K
→
\overrightarrow{DK}
DK
are equal. Prove that the triangle
B
H
D
BHD
B
HD
is also equilateral
308
1
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ASU 308 All Soviet Union MO min area of rectangle inside y\le-x^2,y\ge x^2-2x+a
Given real
a
a
a
. Find the least possible area of the rectangle with the sides parallel to the coordinate axes and containing the figure determined by the system of inequalities
y
≤
−
x
2
a
n
d
y
≥
x
2
−
2
x
+
a
y \le -x^2 \,\,\, and \,\,\, y \ge x^2 - 2x + a
y
≤
−
x
2
an
d
y
≥
x
2
−
2
x
+
a
307
1
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ASU 307 All Soviet Union MO 1981 4 rows in a rectangular table
The rectangular table has four rows. The first one contains arbitrary natural numbers (some of them may be equal). The consecutive lines are filled according to the rule: we look through the previous row from left to the certain number
n
n
n
and write the number
k
k
k
if
n
n
n
was met
k
k
k
times. Prove that the second row coincides with the fourth one.
306
1
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ASU 306 All Soviet Union MO 1981 product of consecutive naturals
Let us say, that a natural number has the property
P
(
k
)
P(k)
P
(
k
)
if it can be represented as a product of
k
k
k
succeeding natural numbers greater than
1
1
1
. a) Find k such that there exists n which has properties
P
(
k
)
P(k)
P
(
k
)
and
P
(
k
+
2
)
P(k+2)
P
(
k
+
2
)
simultaneously. b) Prove that there is no number having properties
P
(
2
)
P(2)
P
(
2
)
and
P
(
4
)
P(4)
P
(
4
)
simultaneously
305
1
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ASU 305 All Soviet Union MO 1981 perpendicular chords inside a cyclic
Given points
A
,
B
,
M
,
N
A,B,M,N
A
,
B
,
M
,
N
on the circumference. Two chords
[
M
A
1
]
[MA_1]
[
M
A
1
]
and
[
M
A
2
]
[MA_2]
[
M
A
2
]
are orthogonal to lines
(
N
A
)
(NA)
(
N
A
)
and
(
N
B
)
(NB)
(
NB
)
respectively. Prove that
(
A
A
1
)
(AA_1)
(
A
A
1
)
and
(
B
B
1
)
(BB_1)
(
B
B
1
)
lines are parallel.
304
1
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ASU 304 All Soviet Union MO 1981 rotating 45^o 1 of 2 same chessboards, area
Two equal chess-boards (
8
×
8
8\times 8
8
×
8
) have the same centre, but one is rotated by
45
45
45
degrees with respect to another. Find the total area of black fields intersection, if the fields have unit length sides.