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Problems
Contests
National and Regional Contests
Russia Contests
All-Russian Olympiad
2007 All-Russian Olympiad
2007 All-Russian Olympiad
Part of
All-Russian Olympiad
Subcontests
(8)
8
2
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decimal fractions of 1/k!
Dima has written number
1
/
80
!
,
1
/
81
!
,
…
,
1
/
99
!
1/80!,\,1/81!,\,\dots,1/99!
1/80
!
,
1/81
!
,
…
,
1/99
!
on
20
20
20
infinite pieces of papers as decimal fractions (the following is written on the last piece: \frac {1}{99!} \equal{} 0{,}{00\dots 00}10715\dots, 155 0-s before 1). Sasha wants to cut a fragment of
N
N
N
consecutive digits from one of pieces without the comma. For which maximal
N
N
N
he may do it so that Dima may not guess, from which piece Sasha has cut his fragment? A. Golovanov
each cyle contains edges of both colors
Given an undirected graph with
N
N
N
vertices. For any set of
k
k
k
vertices, where
1
≤
k
≤
N
1\le k\le N
1
≤
k
≤
N
, there are at most
2
k
−
2
2k-2
2
k
−
2
edges, which join vertices of this set. Prove that the edges may be coloured in two colours so that each cycle contains edges of both colours. (Graph may contain multiple edges). I. Bogdanov, G. Chelnokov
6
3
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an altitude of an acute triangle
Let
A
B
C
ABC
A
BC
be an acute triangle. The points
M
M
M
and
N
N
N
are midpoints of
A
B
AB
A
B
and
B
C
BC
BC
respectively, and
B
H
BH
B
H
is an altitude of
A
B
C
ABC
A
BC
. The circumcircles of
A
H
N
AHN
A
H
N
and
C
H
M
CHM
C
H
M
meet in
P
P
P
where
P
≠
H
P\ne H
P
=
H
. Prove that
P
H
PH
P
H
passes through the midpoint of
M
N
MN
MN
. V. Filimonov
two circles and their common tangents
Two circles
ω
1
\omega_{1}
ω
1
and
ω
2
\omega_{2}
ω
2
intersect in points
A
A
A
and
B
B
B
. Let
P
Q
PQ
PQ
and
R
S
RS
RS
be segments of common tangents to these circles (points
P
P
P
and
R
R
R
lie on
ω
1
\omega_{1}
ω
1
, points
Q
Q
Q
and
S
S
S
lie on
ω
2
\omega_{2}
ω
2
). It appears that
R
B
∥
P
Q
RB\parallel PQ
RB
∥
PQ
. Ray
R
B
RB
RB
intersects
ω
2
\omega_{2}
ω
2
in a point
W
≠
B
W\ne B
W
=
B
. Find
R
B
/
B
W
RB/BW
RB
/
B
W
. S. Berlov
polynomial has exactly $n$ integral roots?
Do there exist non-zero reals
a
a
a
,
b
b
b
,
c
c
c
such that, for any
n
>
3
n>3
n
>
3
, there exists a polynomial
P
n
(
x
)
=
x
n
+
⋯
+
a
x
2
+
b
x
+
c
P_{n}(x) = x^{n}+\dots+a x^{2}+bx+c
P
n
(
x
)
=
x
n
+
⋯
+
a
x
2
+
b
x
+
c
, which has exactly
n
n
n
(not necessary distinct) integral roots? N. Agakhanov, I. Bogdanov
7
4
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5
3
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Maykop and Belorechensk
The distance between Maykop and Belorechensk is
24
24
24
km. Two of three friends need to reach Belorechensk from Maykop and another friend wants to reach Maykop from Belorechensk. They have only one bike, which is initially in Maykop. Each guy may go on foot (with velocity at most
6
6
6
kmph) or on a bike (with velocity at most
18
18
18
kmph). It is forbidden to leave a bike on a road. Prove that all of them may achieve their goals after
2
2
2
hours
40
40
40
minutes. (Only one guy may seat on the bike simultaneously).Folclore
two numbers in each vertice of a convex $100$-gon
Two numbers are written on each vertex of a convex
100
100
100
-gon. Prove that it is possible to remove a number from each vertex so that the remaining numbers on any two adjacent vertices are different. F. Petrov
a set of $n>2$ planar vectors
Given a set of
n
>
2
n>2
n
>
2
planar vectors. A vector from this set is called long, if its length is not less than the length of the sum of other vectors in this set. Prove that if each vector is long, then the sum of all vectors equals to zero. N. Agakhanov
4
4
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3
2
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rhombus
Given a rhombus
A
B
C
D
ABCD
A
BC
D
. A point
M
M
M
is chosen on its side
B
C
BC
BC
. The lines, which pass through
M
M
M
and are perpendicular to
B
D
BD
B
D
and
A
C
AC
A
C
, meet line
A
D
AD
A
D
in points
P
P
P
and
Q
Q
Q
respectively. Suppose that the lines
P
B
,
Q
C
,
A
M
PB,QC,AM
PB
,
QC
,
A
M
have a common point. Find all possible values of a ratio
B
M
M
C
\frac{BM}{MC}
MC
BM
. S. Berlov, F. Petrov, A. Akopyan
draw diagonals in a regular (2n+1)-gon
Two players by turns draw diagonals in a regular
(
2
n
+
1
)
(2n+1)
(
2
n
+
1
)
-gon (
n
>
1
n>1
n
>
1
). It is forbidden to draw a diagonal, which was already drawn, or intersects an odd number of already drawn diagonals. The player, who has no legal move, loses. Who has a winning strategy? K. Sukhov
2
4
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1
4
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