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Contests
National and Regional Contests
Russia Contests
All-Russian Olympiad
1979 All Soviet Union Mathematical Olympiad
1979 All Soviet Union Mathematical Olympiad
Part of
All-Russian Olympiad
Subcontests
(15)
283
1
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ASU 283 All Soviet Union MO 1979 n points on a line
Given
n
n
n
points (in sequence)
A
1
,
A
2
,
.
.
.
,
A
n
A_1, A_2, ... , A_n
A
1
,
A
2
,
...
,
A
n
on a line. All the segments
A
1
A
2
A_1A_2
A
1
A
2
,
A
2
A
3
A_2A_3
A
2
A
3
,
.
.
.
...
...
,
A
n
−
1
A
n
A_{n-1}A_n
A
n
−
1
A
n
are shorter than
1
1
1
. We need to mark
(
k
−
1
)
(k-1)
(
k
−
1
)
points so that the difference of every two segments, with the ends in the marked points, is shorter than
1
1
1
. Prove that it is possible a) for
k
=
3
k=3
k
=
3
,b) for every
k
k
k
less than
(
n
−
1
)
(n-1)
(
n
−
1
)
.
282
1
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ASU 282 All Soviet Union MO 1979 4 equal incircles , criterion for diamond
The convex quadrangle is divided by its diagonals onto four triangles. The circles inscribed in those triangles are equal. Prove that the given quadrangle is a diamond.
281
1
Hide problems
ASU 281 All Soviet Union MO 1979 odd sum a_1a_{k+1}+ ... + a_{n-k}a_n
The finite sequence
a
1
,
a
2
,
.
.
.
,
a
n
a_1, a_2, ... , a_n
a
1
,
a
2
,
...
,
a
n
of ones and zeroes should satisfy a condition:for every
k
k
k
from
0
0
0
to
(
n
−
1
)
(n-1)
(
n
−
1
)
the sum a_1a_{k+1} + a_2a_{k+2} + ... + a_{n-k}a_n should be odd. a) Construct such a sequence for
n
=
25
n=25
n
=
25
. b) Prove that there exists such a sequence for some
n
>
1000
n > 1000
n
>
1000
.
280
1
Hide problems
ASU 280 All Soviet Union MO 1979 1979 lines in space
Given the point
O
O
O
in the space and
1979
1979
1979
straight lines
l
1
,
l
2
,
.
.
.
,
l
1979
l_1, l_2, ... , l_{1979}
l
1
,
l
2
,
...
,
l
1979
containing it. Not a pair of lines is orthogonal. Given a point
A
1
A_1
A
1
on
l
1
l_1
l
1
that doesn't coincide with
O
O
O
. Prove that it is possible to choose the points
A
i
A_i
A
i
on
l
i
l_i
l
i
(
i
=
2
,
3
,
.
.
.
,
1979
i = 2, 3, ... , 1979
i
=
2
,
3
,
...
,
1979
) in so that
1979
1979
1979
pairs will be orthogonal:
A
1
A
3
A_1A_3
A
1
A
3
and
l
2
l_2
l
2
,
A
2
A
4
A_2A_4
A
2
A
4
and
l
3
l_3
l
3
,
.
.
.
...
...
,
A
i
−
1
A
i
+
1
A_{i-1}A_{i+1}
A
i
−
1
A
i
+
1
and
l
i
l_i
l
i
,
.
.
.
...
...
,
A
1977
A
1979
A_{1977}A_{1979}
A
1977
A
1979
and
l
1978
l_{1978}
l
1978
,
A
1978
A
1
A_{1978}A_1
A
1978
A
1
and
l
1979
l_{1979}
l
1979
,
A
1979
A
2
A_{1979}A_2
A
1979
A
2
and
l
1
l_1
l
1
279
1
Hide problems
ASU 279 All Soviet Union MO 1979 divide [0,1] onto (p+q) equal segments
Natural
p
p
p
and
q
q
q
are relatively prime. The
[
0
,
1
]
[0,1]
[
0
,
1
]
is divided onto
(
p
+
q
)
(p+q)
(
p
+
q
)
equal segments. Prove that every segment except two marginal contain exactly one from the
(
p
+
q
−
2
)
(p+q-2)
(
p
+
q
−
2
)
numbers
{
1
/
p
,
2
/
p
,
.
.
.
,
(
p
−
1
)
/
p
,
1
/
q
,
2
/
q
,
.
.
.
,
(
q
−
1
)
/
q
}
\{1/p, 2/p, ... , (p-1)/p, 1/q, 2/q, ... , (q-1)/q\}
{
1/
p
,
2/
p
,
...
,
(
p
−
1
)
/
p
,
1/
q
,
2/
q
,
...
,
(
q
−
1
)
/
q
}
278
1
Hide problems
ASU 278 All Soviet Union MO 1979 (x_1 +...+ x_n + 1)^2 \ge 4(x_1^2 +...+ x_n^2)
Prove that for the arbitrary numbers
x
1
,
x
2
,
.
.
.
,
x
n
x_1, x_2, ... , x_n
x
1
,
x
2
,
...
,
x
n
from the
[
0
,
1
]
[0,1]
[
0
,
1
]
segment
(
x
1
+
x
2
+
.
.
.
+
x
n
+
1
)
2
≥
4
(
x
1
2
+
x
2
2
+
.
.
.
+
x
n
2
)
(x_1 + x_2 + ...+ x_n + 1)^2 \ge 4(x_1^2 + x_2^2 + ... + x_n^2)
(
x
1
+
x
2
+
...
+
x
n
+
1
)
2
≥
4
(
x
1
2
+
x
2
2
+
...
+
x
n
2
)
277
1
Hide problems
ASU 277 All Soviet Union MO 1979 square carpets with the total area 4
Given some square carpets with the total area
4
4
4
. Prove that they can fully cover the unit square.
276
1
Hide problems
ASU 276 All Soviet Union MO 1979 system (x-y\sqrt{x^2-y^2})/(\sqrt{1-x^2+y^2})=b
Find
x
x
x
and
y
y
y
(
a
a
a
and
b
b
b
parameters):
{
x
−
y
x
2
−
y
2
1
−
x
2
+
y
2
=
a
y
−
x
x
2
−
y
2
1
−
x
2
+
y
2
=
b
\begin{cases} \dfrac{x-y\sqrt{x^2-y^2}}{\sqrt{1-x^2+y^2}} = a\\ \\ \dfrac{y-x\sqrt{x^2-y^2}}{\sqrt{1-x^2+y^2}} = b\end{cases}
⎩
⎨
⎧
1
−
x
2
+
y
2
x
−
y
x
2
−
y
2
=
a
1
−
x
2
+
y
2
y
−
x
x
2
−
y
2
=
b
275
1
Hide problems
ASU 275 All Soviet Union MO 1979 checkers in chessboard
What is the least possible number of the checkers being required a) for the
8
×
8
8\times 8
8
×
8
chess-board,b) for the
n
×
n
n\times n
n
×
n
chess-board,to provide the property: Every line (of the chess-board fields) parallel to the side or diagonal is occupied by at least one checker ?
274
1
Hide problems
ASU 274 All Soviet Union MO 1979 vectors in plane
Given some points in the plane. For some pairs
A
,
B
A,B
A
,
B
the vector
A
B
AB
A
B
is chosen. For every point the number of the chosen vectors starting in that point equal to the number of the chosen vectors ending in that point. Prove that the sum of the chosen vectors equals to zero vector.
273
1
Hide problems
ASU 273 All Soviet Union MO 1979 x_1+x_4/2+x_9/3+...+x_n^2/n \le 1
For every
n
n
n
, the decreasing sequence
{
x
k
}
\{x_k\}
{
x
k
}
satisfies a condition
x
1
+
x
4
/
2
+
x
9
/
3
+
.
.
.
+
x
n
2
/
n
≤
1
x_1+x_4/2+x_9/3+...+x_n^2/n \le 1
x
1
+
x
4
/2
+
x
9
/3
+
...
+
x
n
2
/
n
≤
1
Prove that for every
n
n
n
, it also satisfies
x
1
+
x
2
/
2
+
x
3
/
3
+
.
.
.
+
x
n
/
n
≤
3
x_1+x_2/2+x_3/3+...+x_n/n\le 3
x
1
+
x
2
/2
+
x
3
/3
+
...
+
x
n
/
n
≤
3
272
1
Hide problems
ASU 272 All Soviet Union MO 1979 list of random numbers, arithmetic mean
Some numbers are written in the notebook. We can add to that list the arithmetic mean of some of them, if it doesn't equal to the number, already having been included in it. Let us start with two numbers,
0
0
0
and
1
1
1
. Prove that it is possible to obtain :a)
1
/
5
1/5
1/5
,b) an arbitrary rational number between
0
0
0
and
1
1
1
.
271
1
Hide problems
ASU 271 All Soviet Union MO 1979 enemies of members of parliament
Every member of a certain parliament has not more than
3
3
3
enemies. Prove that it is possible to divide it onto two subparliaments so, that everyone will have not more than one enemy in his subparliament. (
A
A
A
is the enemy of
B
B
B
if and only if
B
B
B
is the enemy of
A
A
A
.)
270
1
Hide problems
ASU 270 All Soviet Union MO 1979 grasshopper hopping on cartesian
A grasshopper is hopping in the angle
x
≥
0
,
y
≥
0
x\ge 0, y\ge 0
x
≥
0
,
y
≥
0
of the coordinate plane (that means that it cannot land in the point with negative coordinate). If it is in the point
(
x
,
y
)
(x,y)
(
x
,
y
)
, it can either jump to the point
(
x
+
1
,
y
−
1
)
(x+1,y-1)
(
x
+
1
,
y
−
1
)
, or to the point
(
x
−
5
,
y
+
7
)
(x-5,y+7)
(
x
−
5
,
y
+
7
)
. Draw a set of such an initial points
(
x
,
y
)
(x,y)
(
x
,
y
)
, that having started from there, a grasshopper cannot reach any point farther than
1000
1000
1000
from the point
(
0
,
0
)
(0,0)
(
0
,
0
)
. Find its area.
269
1
Hide problems
ASU 269 All Soviet Union MO 1979 isosceles triangle inscribed in isosceles
What is the least possible ratio of two isosceles triangles areas, if three vertices of the first one belong to three different sides of the second one?