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National and Regional Contests
Russia Contests
All-Russian Olympiad
1984 All Soviet Union Mathematical Olympiad
1984 All Soviet Union Mathematical Olympiad
Part of
All-Russian Olympiad
Subcontests
(24)
393
1
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ASU 393 All Soviet Union MO 1984 tangential quadrilateral wanted, 3 circles
Given three circles
c
1
,
c
2
,
c
3
c_1,c_2,c_3
c
1
,
c
2
,
c
3
with
r
1
,
r
2
,
r
3
r_1,r_2,r_3
r
1
,
r
2
,
r
3
radiuses,
r
1
>
r
2
,
r
1
>
r
3
r_1 > r_2, r_1 > r_3
r
1
>
r
2
,
r
1
>
r
3
. Each lies outside of two others. The A point -- an intersection of the outer common tangents to
c
1
c_1
c
1
and
c
2
c_2
c
2
-- is outside
c
3
c_3
c
3
. The
B
B
B
point -- an intersection of the outer common tangents to
c
1
c_1
c
1
and
c
3
c_3
c
3
-- is outside
c
2
c_2
c
2
. Two pairs of tangents -- from
A
A
A
to
c
3
c_3
c
3
and from
B
B
B
to
c
2
c_2
c
2
-- are drawn. Prove that the quadrangle, they make, is circumscribed around some circle and find its radius.
391
1
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ASU 391 All Soviet Union MO 1984 +1,-1 on a 3x3 chessboard
The white fields of
3
x
3
3x3
3
x
3
chess-board are filled with either
+
1
+1
+
1
or
−
1
-1
−
1
. For every field, let us calculate the product of neighbouring numbers. Then let us change all the numbers by the respective products. Prove that we shall obtain only
+
1
+1
+
1
's, having repeated this operation finite number of times.
394
1
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ASU 394 All Soviet Union MO 1984 area of every cube's cross-section
Prove that every cube's cross-section, containing its centre, has the area not less then its face's area.
392
1
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ASU 392 All Soviet Union MO 1984 compare 2/201 with ln(101/100)
What is more
2
201
\frac{2}{201}
201
2
or
ln
101
100
\ln\frac{101}{100}
ln
100
101
? (No differential calculus allowed).
390
1
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ASU 390 All Soviet Union MO 1984 +1,-1 in 1983 x 1984 chessboard
The white fields of
1983
×
1984
1983\times 1984
1983
×
1984
1983x1984 are filled with either
+
1
+1
+
1
or
−
1
-1
−
1
. For every black field, the product of neighbouring numbers is
+
1
+1
+
1
. Prove that all the numbers are
+
1
+1
+
1
.
389
1
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ASU 389 All Soviet Union MO 1984 x_1=x_2=1, x_{n+2}=x^2_{n+1}-x_n/2
Given a sequence
{
x
n
}
\{x_n\}
{
x
n
}
,
x
1
=
x
2
=
1
,
x
n
+
2
=
x
n
+
1
2
−
x
n
2
x_1 = x_2 = 1, x_{n+2} = x^2_{n+1} - \frac{x_n}{2}
x
1
=
x
2
=
1
,
x
n
+
2
=
x
n
+
1
2
−
2
x
n
Prove that the sequence has limit and find it.
388
1
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ASU 388 All Soviet Union MO 1984 |AE| + |ED| + ||AB|-|CD|| > |BE| + |CE|
The
A
,
B
,
C
A,B,C
A
,
B
,
C
and
D
D
D
points (from left to right) belong to the straight line. Prove that every point
E
E
E
, that doesn't belong to the line satisfy:
∣
A
E
∣
+
∣
E
D
∣
+
∣
∣
A
B
∣
−
∣
C
D
∣
∣
>
∣
B
E
∣
+
∣
C
E
∣
|AE| + |ED| + | |AB| - |CD| | > |BE| + |CE|
∣
A
E
∣
+
∣
E
D
∣
+
∣∣
A
B
∣
−
∣
C
D
∣∣
>
∣
BE
∣
+
∣
CE
∣
387
1
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ASU 387 All Soviet Union MO 1984 xx...x6yy...y4 perfect square
The
x
x
x
and
y
y
y
figures satisfy a condition: for every
n
≥
1
n\ge1
n
≥
1
the number
x
x
.
.
.
x
6
y
y
.
.
.
y
4
xx...x6yy...y4
xx
...
x
6
yy
...
y
4
(
n
n
n
times
x
x
x
and
n
n
n
times
y
y
y
) is a perfect square. Find all possible
x
x
x
and
y
y
y
.
386
1
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ASU 386 All Soviet Union MO 1984 absolutely prime numbers
Let us call "absolutely prime" the prime number, if having transposed its digits in an arbitrary order, we obtain prime number again. Prove that its notation cannot contain more than three different digits.
385
1
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ASU 385 All Soviet Union MO 1984 n+1 integer weights with sum 2n
There are scales and
(
n
+
1
)
(n+1)
(
n
+
1
)
weights with the total weight
2
n
2n
2
n
. Each weight is an integer. We put all the weights in turn on the lighter side of the scales, starting from the heaviest one, and if the scales is in equilibrium -- on the left side. Prove that when all the weights will be put on the scales, they will be in equilibrium.
384
1
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ASU 384 All Soviet Union MO 1984 coin moves around a polygon, areas
The centre of the coin with radius
r
r
r
is moved along some polygon with the perimeter
P
P
P
, that is circumscribed around the circle with radius
R
R
R
(
R
>
r
R>r
R
>
r
). Find the coin trace area (a sort of polygon ring).
383
1
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ASU 383 All Soviet Union MO 1984 x^2 + 10x + 20 coefficient game
The teacher wrote on a blackboard:
x
2
+
10
x
+
20
x^2 + 10x + 20
x
2
+
10
x
+
20
Then all the pupils in the class came up in turn and either decreased or increased by
1
1
1
either the free coefficient or the coefficient at
x
x
x
, but not both. Finally they have obtained:
x
2
+
20
x
+
10
x^2 + 20x + 10
x
2
+
20
x
+
10
Is it true that some time during the process there was written the square polynomial with the integer roots?
382
1
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ASU 382 All Soviet Union MO 1984 3x3 non linear system
Positive
x
,
y
,
z
x,y,z
x
,
y
,
z
satisfy a system:
{
x
2
+
x
y
+
y
2
/
3
=
25
y
2
/
3
+
z
2
=
9
z
2
+
z
x
+
x
2
=
16
\begin{cases} x^2 + xy + y^2/3= 25 \\ y^2/ 3 + z^2 = 9 \\ z^2 + zx + x^2 = 16 \end{cases}
⎩
⎨
⎧
x
2
+
x
y
+
y
2
/3
=
25
y
2
/3
+
z
2
=
9
z
2
+
z
x
+
x
2
=
16
Find the value of expression
x
y
+
2
y
z
+
3
z
x
xy + 2yz + 3zx
x
y
+
2
yz
+
3
z
x
.
381
1
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ASU 381 All Soviet Union MO 1984 cevians, circumcircle, congruent triangles
Given triangle
A
B
C
ABC
A
BC
. From the
P
P
P
point three lines
(
P
A
)
,
(
P
B
)
,
(
P
C
)
(PA),(PB),(PC)
(
P
A
)
,
(
PB
)
,
(
PC
)
are drawn. They cross the circumscribed circle at
A
1
,
B
1
,
C
1
A_1, B_1,C_1
A
1
,
B
1
,
C
1
points respectively. It comes out that the
A
1
B
1
C
1
A_1B_1C_1
A
1
B
1
C
1
triangle equals to the initial one. Prove that there are not more than eight such a points
P
P
P
in a plane.
380
1
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ASU 380 All Soviet Union MO 1984 n reals n increasing order in a line, lines
n
n
n
real numbers are written in increasing order in a line. The same numbers are written in the second line below in unknown order. The third line contains the sums of the pairs of numbers above from two previous lines. It comes out, that the third line is arranged in increasing order. Prove that the second line coincides with the first one.
379
1
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ASU 379 All Soviet Union MO 1984 diophant (5+3\sqrt2)^m=(3+5\sqrt2)^n
Find integers
m
m
m
and
n
n
n
such that
(
5
+
3
2
)
m
=
(
3
+
5
2
)
n
(5 + 3 \sqrt2)^m = (3 + 5 \sqrt2)^n
(
5
+
3
2
)
m
=
(
3
+
5
2
)
n
.
378
1
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ASU 378 All Soviet Union MO 1984 concurrecncy, incircle related
The circle with the centre
O
O
O
is inscribed in the triangle
A
B
C
ABC
A
BC
. The circumference touches its sides
[
B
C
]
,
[
C
A
]
,
[
A
B
]
[BC], [CA], [AB]
[
BC
]
,
[
C
A
]
,
[
A
B
]
in
A
1
,
B
1
,
C
1
A_1, B_1, C_1
A
1
,
B
1
,
C
1
points respectively. The
[
A
O
]
,
[
B
O
]
,
[
C
O
]
[AO], [BO], [CO]
[
A
O
]
,
[
BO
]
,
[
CO
]
segments cross the circumference in
A
2
,
B
2
,
C
2
A_2, B_2, C_2
A
2
,
B
2
,
C
2
points respectively. Prove that lines
(
A
1
A
2
)
,
(
B
1
B
2
)
(A_1A_2),(B_1B_2)
(
A
1
A
2
)
,
(
B
1
B
2
)
and
(
C
1
C
2
)
(C_1C_2)
(
C
1
C
2
)
intersect in one point.
377
1
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ASU 377 All Soviet Union MO 1984 narural no ariund a circle, sum of 2
n
n
n
natural numbers (
n
>
3
n>3
n
>
3
) are written on the circumference. The relation of the two neighbours sum to the number itself is a whole number. Prove that the sum of those relations is a) not less than
2
n
2n
2
n
b) less than
3
n
3n
3
n
376
1
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ASU 376 All Soviet Union MO 1984 a cube and two colours, game
Given a cube and two colours. Two players paint in turn a triple of arbitrary unpainted edges with his colour. (Everyone makes two moves.) The first wins if he has painted all the edges of some face with his colour. Can he always win?
375
1
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ASU 375 All Soviet Union MO 1984 x^((sin a)^2) * y^((cos a)^2) < x + y
Prove that every positive
x
,
y
x,y
x
,
y
and real
a
a
a
satisfy inequality
x
sin
2
a
y
cos
2
a
<
x
+
y
x^{\sin ^2a} y^{\cos^2a} < x + y
x
s
i
n
2
a
y
c
o
s
2
a
<
x
+
y
.
374
1
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ASU 374 All Soviet Union MO 1984 4 colours and enough square plates 1x1
Given four colours and enough square plates
1
×
1
1\times 1
1
×
1
. We have to paint four edges of every plate with four different colours and combine plates, putting them with the edges of the same colour together. Describe all the pairs
m
,
n
m,n
m
,
n
, such that we can combine those plates in a
n
×
m
n\times m
n
×
m
rectangle, that has every edge of one colour, and its four edges have different colours.
373
1
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ASU 373 All Soviet Union MO 1984 2 equilaterals and equal vectors
Given two equilateral triangles
A
1
B
1
C
1
A_1B_1C_1
A
1
B
1
C
1
and
A
2
B
2
C
2
A_2B_2C_2
A
2
B
2
C
2
in the plane. (The vertices are mentioned counterclockwise.) We draw vectors
O
A
→
,
O
B
→
,
O
C
→
\overrightarrow{OA}, \overrightarrow{OB}, \overrightarrow{OC}
O
A
,
OB
,
OC
, from the arbitrary point
O
O
O
, equal to
A
1
A
2
→
,
B
1
B
2
→
,
C
1
C
2
→
\overrightarrow{A_1A_2}, \overrightarrow{B_1B_2}, \overrightarrow{C_1C_2}
A
1
A
2
,
B
1
B
2
,
C
1
C
2
respectively. Prove that the triangle
A
B
C
ABC
A
BC
is equilateral.
372
1
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ASU 372 All Soviet Union MO 1984 (a+b)^2/2+(a+b)/4>= a\sqrt(b)+b\sqrt(a)
Prove that every positive
a
a
a
and
b
b
b
satisfy inequality
(
a
+
b
)
2
2
+
a
+
b
4
≥
a
b
+
b
a
\frac{(a+b)^2}{2} + \frac{a+b}{4} \ge a\sqrt b + b\sqrt a
2
(
a
+
b
)
2
+
4
a
+
b
≥
a
b
+
b
a
371
1
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ASU 371 All Soviet Union MO 1984 n integers with product n and sum 0
a) The product of
n
n
n
integers equals
n
n
n
, and their sum is zero. Prove that
n
n
n
is divisible by
4
4
4
. b) Let
n
n
n
is divisible by
4
4
4
. Prove that there exist
n
n
n
integers such, that their product equals
n
n
n
, and their sum is zero.