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Contests
National and Regional Contests
Russia Contests
All-Russian Olympiad
1986 All Soviet Union Mathematical Olympiad
1986 All Soviet Union Mathematical Olympiad
Part of
All-Russian Olympiad
Subcontests
(23)
434
1
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ASU 434 All Soviet Union MO 1986 vector sum zero in n-gon A_1A_2...A_n
Given a regular
n
n
n
-gon
A
1
A
2
.
.
.
A
n
A_1A_2...A_n
A
1
A
2
...
A
n
. Prove that if a)
n
n
n
is even number, than for the arbitrary point
M
M
M
in the plane, it is possible to choose signs in an expression
±
M
A
1
→
±
M
A
2
→
±
.
.
.
±
M
A
n
→
\pm \overrightarrow{MA_1} \pm \overrightarrow{MA_2} \pm ... \pm \overrightarrow{MA_n}
±
M
A
1
±
M
A
2
±
...
±
M
A
n
to make it equal to the zero vector . b)
n
n
n
is odd, than the abovementioned expression equals to the zero vector for the finite set of
M
M
M
points only.
440
1
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ASU 440 All Soviet Union MO 1986 sum of angles of tetrahedron fixed
Consider all the tetrahedrons
A
X
B
Y
AXBY
A
XB
Y
, circumscribed around the sphere. Let
A
A
A
and
B
B
B
points be fixed. Prove that the sum of angles in the non-plane quadrangle
A
X
B
Y
AXBY
A
XB
Y
doesn't depend on points
X
X
X
and
Y
Y
Y
.
439
1
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ASU 439 All Soviet Union MO 1986 P(2) = n, coefficients are 0, 1, 2 ,3.
Let us call a polynomial admissible if all it's coefficients are
0
,
1
,
2
0, 1, 2
0
,
1
,
2
or
3
3
3
. For given
n
n
n
find the number of all the admissible polynomials
P
P
P
such, that
P
(
2
)
=
n
P(2) = n
P
(
2
)
=
n
.
438
1
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ASU 438 All Soviet Union MO 1986 triangle, square around unit circle, areas
A triangle and a square are circumscribed around the unit circle. Prove that the intersection area is more than
3.4
3.4
3.4
. Is it possible to assert that it is more than
3.5
3.5
3.5
?
437
1
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ASU 437 All Soviet Union MO 1986 sum of all 1/mn not an integer
Prove that the sum of all numbers representable as
1
m
n
\frac{1}{mn}
mn
1
, where
m
,
n
m,n
m
,
n
-- natural numbers,
1
≤
m
<
n
≤
1986
1 \le m < n \le1986
1
≤
m
<
n
≤
1986
, is not an integer.
436
1
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ASU 436 All Soviet Union MO 1986 |sin1|+|sin2|+|sin(3n-1)|+|sin(3n)|>8n/5.
Prove that for every natural
n
n
n
the following inequality is valid
∣
sin
1
∣
+
∣
sin
2
∣
+
∣
sin
(
3
n
−
1
)
∣
+
∣
sin
3
n
∣
>
8
n
5
|\sin 1| + |\sin 2| + |\sin (3n-1)| + |\sin 3n| > \frac{8n}{5}
∣
sin
1∣
+
∣
sin
2∣
+
∣
sin
(
3
n
−
1
)
∣
+
∣
sin
3
n
∣
>
5
8
n
435
1
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ASU 435 All Soviet Union MO 1986 nxn table with +1 and -1, man and max wanted
All the fields of a square
n
×
n
n\times n
n
×
n
(n>2) table are filled with
+
1
+1
+
1
or
−
1
-1
−
1
according to the rules: At the beginning
−
1
-1
−
1
are put in all the boundary fields. The number put in the field in turn (the field is chosen arbitrarily) equals to the product of the closest, from the different sides, numbers in its row or in its column. a) What is the minimal b) What is the maximal possible number of
+
1
+1
+
1
in the obtained table?
433
1
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ASU 433 All Soviet Union MO 1986 100x99,101x99 chessboard diagonals
Find the relation of the black part length and the white part length for the main diagonal of the a)
100
×
99
100\times 99
100
×
99
chess-board; b)
101
×
99
101\times 99
101
×
99
chess-board.
432
1
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ASU 432 All Soviet Union MO 1986 elf with 30 equal cups with milk
Given
30
30
30
equal cups with milk. An elf tries to make the amount of milk equal in all the cups. He takes a pair of cups and aligns the milk level in two cups. Can there be such an initial distribution of milk in the cups, that the elf will not be able to achieve his goal in a finite number of operations?
431
1
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ASU 431 All Soviet Union MO 1986 distances in a dodecagon
Given two points inside a convex dodecagon (twelve sides) situated
10
10
10
cm far from each other. Prove that the difference between the sum of distances, from the point to all the vertices, is less than
1
1
1
m for those points.
430
1
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ASU 430 All Soviet Union MO 1986 a^2 + b = c, find three no with equal digits
The decimal notation of three natural numbers consists of equal digits:
n
n
n
digits
x
x
x
for
a
a
a
,
n
n
n
digits
y
y
y
for
b
b
b
and
2
n
2n
2
n
digits
z
z
z
for
c
c
c
. For every
n
>
1
n > 1
n
>
1
find all the possible triples of digits
x
,
y
,
z
x,y,z
x
,
y
,
z
such, that
a
2
+
b
=
c
a^2 + b = c
a
2
+
b
=
c
429
1
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ASU 429 All Soviet Union MO 1986 n^3 diff. integers on n^3 unit cubes
A cube with edge of length
n
n
n
(
n
≥
3
n\ge 3
n
≥
3
) consists of
n
3
n^3
n
3
unit cubes. Prove that it is possible to write different
n
3
n^3
n
3
integers on all the unit cubes to provide the zero sum of all integers in the every row parallel to some edge.
428
1
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ASU 428 All Soviet Union MO 1986 equal angles impossible
A line is drawn through the
A
A
A
vertex of triangle
A
B
C
ABC
A
BC
with
∣
A
B
∣
≠
∣
A
C
∣
|AB|\ne|AC|
∣
A
B
∣
=
∣
A
C
∣
. Prove that the line can not contain more than one point
M
M
M
such, that
M
M
M
is not a triangle vertex, and
∠
A
B
M
=
∠
A
C
M
\angle ABM = \angle ACM
∠
A
BM
=
∠
A
CM
. What lines do not contain such a point
M
M
M
at all?
427
1
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ASU 427 All Soviet Union 1986 1/a1+2/(a1+a2)+...+n/(a1+...+an)<4(1/a1+...+1/an)
Prove that the following inequality holds for all positive
{
a
i
}
\{a_i\}
{
a
i
}
:
1
a
1
+
2
a
1
+
a
2
+
.
.
.
+
n
a
1
+
.
.
.
+
a
n
<
4
(
1
a
1
+
.
.
.
+
1
a
n
)
\frac{1}{a_1} + \frac{2}{a_1+a_2} + ... +\frac{ n}{a_1+...+a_n} < 4\left(\frac{1}{a_1} + ... + \frac{1}{a_n}\right)
a
1
1
+
a
1
+
a
2
2
+
...
+
a
1
+
...
+
a
n
n
<
4
(
a
1
1
+
...
+
a
n
1
)
425
1
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ASU 425 All Soviet Union MO 1986 1000 equal segment each side of hexagon, color
Given right hexagon. Each side is divided onto
1000
1000
1000
equal segments. All the points of division are connected with the segments, parallel to sides. Let us paint in turn the triples of unpainted nodes of obtained net, if they are vertices of the unilateral triangle, doesn't matter of what size an orientation. Suppose, we have managed to paint all the vertices except one. Prove that the unpainted node is not a hexagon vertex.
426
1
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ASU 426 All Soviet Union MO 1986 natural numbers = square of its divisors
Find all the natural numbers equal to the square of its divisors number.
424
1
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ASU 424 All Soviet Union MO 1986 locus starti ng with intersecting circles
Two circumferences, with the distance
d
d
d
between centres, intersect in points
P
P
P
and
Q
Q
Q
. Two lines are drawn through the point
A
A
A
on the first circumference (
Q
≠
A
≠
P
Q\ne A\ne P
Q
=
A
=
P
) and points
P
P
P
and
Q
Q
Q
. They intersect the second circumference in the points
B
B
B
and
C
C
C
. a) Prove that the radius of the circle, circumscribed around the triangle
A
B
C
ABC
A
BC
, equals
d
d
d
. b) Describe the set of the new circle's centres, if thepoint
A
A
A
moves along all the first circumference.
423
1
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ASU 423 All Soviet Union MO 1986 filling mxn tables with perfect squares
Prove that the rectangle
m
×
n
m\times n
m
×
n
table can be filled with exact squares so, that the sums in the rows and the sums in the columns will be exact squares also.
422
1
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ASU 422 All Soviet Union MO 1986 integer coordinates impossible
Prove that it is impossible to draw a convex quadrangle, with one diagonal equal to doubled another, the angle between them
45
45
45
degrees, on the coordinate plane, so, that all the vertices' coordinates would be integers.
421
1
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ASU 421 All Soviet Union MO 1986 n cities, n-1 roads, shortest distances
Certain king of a certain state wants to build
n
n
n
cities and
n
−
1
n-1
n
−
1
roads, connecting them to provide a possibility to move from every city to every city. (Each road connects two cities, the roads do not intersect, and don't come through another city.) He wants also, to make the shortests distances between the cities, along the roads, to be
1
,
2
,
3
,
.
.
.
,
n
(
n
−
1
)
/
2
1,2,3,...,n(n-1)/2
1
,
2
,
3
,
...
,
n
(
n
−
1
)
/2
kilometres. Is it possible for a)
n
=
6
n=6
n
=
6
b)
n
=
1986
n=1986
n
=
1986
?
420
1
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ASU 420 All Soviet Union MO 1986 min area of circle's intersection
The point
M
M
M
belongs to the side
[
A
C
]
[AC]
[
A
C
]
of the acute-angle triangle
A
B
C
ABC
A
BC
. Two circles are circumscribed around triangles
A
B
M
ABM
A
BM
and
B
C
M
BCM
BCM
. What
M
M
M
position corresponds to the minimal area of those circles intersection?
419
1
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ASU 419 All Soviet Union MO 1986 intersecting equal squares
Two equal squares, one with red sides, another with blue ones, give an octagon in intersection. Prove that the sum of red octagon sides lengths is equal to the sum of blue octagon sides lengths.
418
1
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ASU 418 All Soviet Union MO 1986 x^2+ax+b+1 natural roots a^2+b^2 composite
The square polynomial
x
2
+
a
x
+
b
+
1
x^2+ax+b+1
x
2
+
a
x
+
b
+
1
has natural roots. Prove that
(
a
2
+
b
2
)
(a^2+b^2)
(
a
2
+
b
2
)
is a composite number.