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Contests
National and Regional Contests
Russia Contests
All-Russian Olympiad
1961 All Russian Mathematical Olympiad
1961 All Russian Mathematical Olympiad
Part of
All-Russian Olympiad
Subcontests
(12)
001
1
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ASU 001 All Russian MO 1961 8.1 curve intersecting 16 segments
Given a figure, containing
16
16
16
segments. You should prove that there is no curve, that intersect each segment exactly once. The curve may be not closed, may intersect itself, but it is not allowed to touch the segments or to pass through the vertices. [asy] draw((0,0)--(6,0)--(6,3)--(0,3)--(0,0)); draw((0,3/2)--(6,3/2)); draw((2,0)--(2,3/2)); draw((4,0)--(4,3/2)); draw((3,3/2)--(3,3)); [/asy]
005
1
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ASU 005 All Russian MO 1961 8.5a 10.5b combinatorics
a) Given a quartet of positive numbers
(
a
,
b
,
c
,
d
)
(a,b,c,d)
(
a
,
b
,
c
,
d
)
. It is transformed to the new one according to the rule:
a
′
=
a
b
,
b
′
=
b
c
,
c
′
=
c
d
,
d
′
=
d
a
a'=ab, b' =bc, c'=cd,d'=da
a
′
=
ab
,
b
′
=
b
c
,
c
′
=
c
d
,
d
′
=
d
a
. The second one is transformed to the third according to the same rule and so on. Prove that if at least one initial number does not equal 1, than You can never obtain the initial set. b) Given a set of
2
k
2^k
2
k
(
k
k
k
-th power of two) numbers, equal either to
1
1
1
or to
−
1
-1
−
1
. It is transformed as that was in the a) problem (each one is multiplied by the next, and the last -- by the first. Prove that You will always finally obtain the set of positive units.
012
1
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ASU 012 All Russian MO 1961 10.2 squares in a rectangle
Given
120
120
120
unit squares arbitrarily situated in the
20
×
25
20\times 25
20
×
25
rectangle. Prove that you can place a circle with the unit diameter without intersecting any of the squares.
011
1
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ASU 011 All Russian MO1961 10.1 sequences
Prove that for three arbitrary infinite sequences, of natural numbers
a
1
,
a
2
,
.
.
.
,
a
n
,
.
.
.
a_1,a_2,...,a_n,...
a
1
,
a
2
,
...
,
a
n
,
...
,
b
1
,
b
2
,
.
.
.
,
b
n
,
.
.
.
b_1,b_2,...,b_n,...
b
1
,
b
2
,
...
,
b
n
,
...
,
c
1
,
c
2
,
.
.
.
,
c
n
,
.
.
.
c_1,c_2,...,c_n,...
c
1
,
c
2
,
...
,
c
n
,
...
there exist numbers
p
p
p
and
q
q
q
such, that
a
p
≥
a
q
a_p \ge a_q
a
p
≥
a
q
,
b
p
≥
b
q
b_p \ge b_q
b
p
≥
b
q
and
c
p
≥
c
q
c_p \ge c_q
c
p
≥
c
q
.
010
1
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ASU 010 All Russian MO 1961 9.5 game strategy
Nicholas and Peter are dividing
(
2
n
+
1
)
(2n+1)
(
2
n
+
1
)
nuts. Each wants to get more. Three ways for that were suggested. (Each consist of three stages.) First two stages are common. 1 stage: Peter divides nuts onto
2
2
2
heaps, each contain not less than
2
2
2
nuts. 2 stage: Nicholas divides both heaps onto
2
2
2
heaps, each contain not less than
1
1
1
nut. 3 stage:1 way: Nicholas takes the biggest and the least heaps. 2 way: Nicholas takes two middle size heaps. 3 way: Nicholas takes either the biggest and the least heaps or two middle size heaps, but gives one nut to the Peter for the right of choice.Find the most and the least profitable method for the Nicholas.
009
1
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ASU 009 All Russian MO 1961 9.4 number theory
Given
a
,
b
,
p
a, b, p
a
,
b
,
p
arbitrary integers. Prove that there always exist relatively prime (i.e. that have no common divisor)
k
k
k
and
l
l
l
, that
(
a
k
+
b
l
)
(ak + bl)
(
ak
+
b
l
)
is divisible by
p
p
p
.
008
1
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ASU 008 All Russian MO 1961 9.3 combinatorial geometry
Given
n
n
n
points, some of them connected by non-intersecting segments. You can reach every point from every one, moving along the segments, and there is no couple, connected by two different ways. Prove that the total number of the segments is
(
n
−
1
)
(n-1)
(
n
−
1
)
.
007
1
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ASU 007 All Russian MO 1961 9.2 & 10.3 numbers in a table
Given some
m
×
n
m\times n
m
×
n
table, and some numbers in its fields. You are allowed to change the sign in one row or one column simultaneously. Prove that you can obtain a table, with nonnegative sums over each row and over each column.
006
1
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ASU 006 All Russian MO 1961 9.1a 10.4b equilateral
a) Points
A
A
A
and
B
B
B
move uniformly and with equal angle speed along the circumferences with
O
a
O_a
O
a
and
O
b
O_b
O
b
centres (both clockwise). Prove that a vertex
C
C
C
of the equilateral triangle
A
B
C
ABC
A
BC
also moves along a certain circumference uniformly.b) The distance from the point
P
P
P
to the vertices of the equilateral triangle
A
B
C
ABC
A
BC
equal
∣
A
P
∣
=
2
,
∣
B
P
∣
=
3
|AP|=2, |BP|=3
∣
A
P
∣
=
2
,
∣
BP
∣
=
3
. Find the maximal value of
C
P
CP
CP
.
004
1
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ASU 004 All Soviet Union MO 1961 8.4 stars in a table
Given a table
4
×
4
4\times 4
4
×
4
. a) Find, how
7
7
7
stars can be put in its fields in such a way, that erasing of two arbitrary lines and two columns will always leave at list one of the stars. b) Prove that if there are less than
7
7
7
stars, You can always find two columns and two rows, such, that if you erase them, no star will remain in the table.
003
1
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ASU 003 All Russian MO 1961 8.3 number theory
Prove that among
39
39
39
sequential natural numbers there always is a number with the sum of its digits divisible by
11
11
11
.
002
1
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ASU 002 All Russian MO 1961 8.2 geometry
Given a rectangle
A
1
A
2
A
3
A
4
A_1A_2A_3A_4
A
1
A
2
A
3
A
4
. Four circles with
A
i
A_i
A
i
as their centres have their radiuses
r
1
,
r
2
,
r
3
,
r
4
r_1, r_2, r_3, r_4
r
1
,
r
2
,
r
3
,
r
4
; and
r
1
+
r
3
=
r
2
+
r
4
<
d
r_1+r_3=r_2+r_4<d
r
1
+
r
3
=
r
2
+
r
4
<
d
, where d is a diagonal of the rectangle. Two pairs of the outer common tangents to {the first and the third} and {the second and the fourth} circumferences make a quadrangle. Prove that you can inscribe a circle into that quadrangle.