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Problems
Contests
National and Regional Contests
Russia Contests
239 Open Math Olympiad
2004 239 Open Mathematical Olympiad
2004 239 Open Mathematical Olympiad
Part of
239 Open Math Olympiad
Subcontests
(8)
6
2
Hide problems
Subset in The Form of Union of Segments
Do there exist a set
A
⊂
[
0
,
1
]
A\subset [0,1]
A
⊂
[
0
,
1
]
such that
(
a
)
(a)
(
a
)
A
A
A
is a finite union of segments of total length
1
2
\frac{1}{2}
2
1
,
(
b
)
(b)
(
b
)
The symmetric difference of
A
A
A
and
B
:
=
A
/
2
∪
(
A
/
2
+
1
/
2
)
B:=A/2\cup(A/2+1/2)
B
:=
A
/2
∪
(
A
/2
+
1/2
)
is a union of segments of the total length less than
1
10000
\frac{1}{10000}
10000
1
?
Divisibility of LCM
Given distinct positive integers
a
1
,
a
2
,
…
,
a
n
a_1,\,a_2,\,\dots,a_n
a
1
,
a
2
,
…
,
a
n
. Let
b
i
=
(
a
i
−
a
1
)
(
a
i
−
a
2
)
…
(
a
i
−
a
i
−
1
)
(
a
i
−
a
i
+
1
)
…
(
a
i
−
a
n
)
b_i = (a_i - a_1) (a_i-a_2) \dots (a_i-a_{i-1}) (a_i-a_{i+1})\dots(a_i-a_n)
b
i
=
(
a
i
−
a
1
)
(
a
i
−
a
2
)
…
(
a
i
−
a
i
−
1
)
(
a
i
−
a
i
+
1
)
…
(
a
i
−
a
n
)
. Prove that the least common multiple
[
b
1
,
b
2
,
…
,
b
n
]
[b_1,b_2,\dots,b_n]
[
b
1
,
b
2
,
…
,
b
n
]
is divisible by
(
n
−
1
)
!
.
(n-1)!.
(
n
−
1
)!
.
8
1
Hide problems
upgrade of problem 7 for grade 8-9, 239MO 2004
Given a triangle
A
B
C
ABC
A
BC
. A point
X
X
X
is chosen on a side
A
C
AC
A
C
. Some circle passes through
X
X
X
, touches the side
A
C
AC
A
C
and intersects the circumcircle of triangle
A
B
C
ABC
A
BC
in points
M
M
M
and
N
N
N
such that the segment
M
N
MN
MN
bisects
B
X
BX
BX
and intersects sides
A
B
AB
A
B
and
B
C
BC
BC
in points
P
P
P
and
Q
Q
Q
. Prove that the circumcircle of triangle
P
B
Q
PBQ
PBQ
passes through a fixed point different from
B
B
B
. proposed by Sergej Berlov
5
1
Hide problems
prove that points A_3, B_3 and C_3 lie on a line
The incircle of triangle
A
B
C
ABC
A
BC
touches its sides
A
B
,
B
C
,
C
A
AB, BC, CA
A
B
,
BC
,
C
A
in points
C
1
,
A
1
,
B
1
C_1, A_1, B_1
C
1
,
A
1
,
B
1
respectively. The point
B
2
B_2
B
2
is symmetric to
B
1
B_1
B
1
with respect to line
A
1
C
1
A_1C_1
A
1
C
1
, lines
B
B
2
BB_2
B
B
2
and
A
C
AC
A
C
meet in point
B
3
B_3
B
3
. points
A
3
A_3
A
3
and
C
3
C_3
C
3
may be defined analogously. Prove that points
A
3
,
B
3
A_3, B_3
A
3
,
B
3
and
C
3
C_3
C
3
lie on a line, which passes through the circumcentre of a triangle
A
B
C
ABC
A
BC
. proposed by L. Emelyanov
7
2
Hide problems
200n diagonals are drawn in a convex n-gon
200
n
200n
200
n
diagonals are drawn in a convex
n
n
n
-gon. Prove that one of them intersects at least 10000 others. proposed by D. Karpov, S. Berlov
PBQ passes through the circumcentre of triangle ABC
Given an isosceles triangle
A
B
C
ABC
A
BC
(with
A
B
=
B
C
AB=BC
A
B
=
BC
). A point
X
X
X
is chosen on a side
A
C
AC
A
C
. Some circle passes through
X
X
X
, touches the side
A
C
AC
A
C
and intersects the circumcircle of triangle
A
B
C
ABC
A
BC
in points
M
M
M
and
N
N
N
such that the segment
M
N
MN
MN
bisects
B
X
BX
BX
and intersects sides
A
B
AB
A
B
and
B
C
BC
BC
in points
P
P
P
and
Q
Q
Q
. Prove that the circumcircle of triangle
P
B
Q
PBQ
PBQ
passes through the circumcentre of triangle
A
B
C
ABC
A
BC
. proposed by Sergej Berlov
4
1
Hide problems
ab / (c+ab) symmetric
Let the sum of positive reals
a
,
b
,
c
a,b,c
a
,
b
,
c
be equal to 1. Prove an inequality
a
b
c
+
a
b
+
b
c
a
+
b
c
+
a
c
b
+
a
c
≤
3
/
2
\sqrt{{ab}\over {c+ab}}+\sqrt{{bc}\over {a+bc}}+\sqrt{{ac}\over {b+ac}}\le 3/2
c
+
ab
ab
+
a
+
b
c
b
c
+
b
+
a
c
a
c
≤
3/2
. proposed by Fedor Petrov
3
1
Hide problems
a^{2^n}+2^n is not prime
Prove that for any integer
a
a
a
there exist infinitely many positive integers
n
n
n
such that
a
2
n
+
2
n
a^{2^n}+2^n
a
2
n
+
2
n
is not a prime. proposed by S. Berlov
2
2
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colour plane and find triangle
Do there exist such a triangle
T
T
T
, that for any coloring of a plane in two colors one may found a triangle
T
′
T'
T
′
, equal to
T
T
T
, such that all vertices of
T
′
T'
T
′
have the same color. proposed by S. Berlov
The incircle of a triangle ABC has centre I
The incircle of a triangle
A
B
C
ABC
A
BC
has centre
I
I
I
and touches sides
A
B
,
B
C
,
C
A
AB, BC, CA
A
B
,
BC
,
C
A
in points
C
1
,
A
1
,
B
1
C_1, A_1, B_1
C
1
,
A
1
,
B
1
respectively. Denote by
L
L
L
the foot of a bissector of angle
B
B
B
, and by
K
K
K
the point of intersecting of lines
B
1
I
B_1I
B
1
I
and
A
1
C
1
A_1C_1
A
1
C
1
. Prove that
K
L
∥
B
B
1
KL\parallel BB_1
K
L
∥
B
B
1
. proposed by L. Emelyanov, S. Berlov
1
2
Hide problems
prove that at least one of three trinomials has a real root
Given non-constant linear functions
p
(
x
)
,
q
(
x
)
,
r
(
x
)
p(x), q(x), r(x)
p
(
x
)
,
q
(
x
)
,
r
(
x
)
. Prove that at least one of three trinomials
p
q
+
r
,
p
r
+
q
,
q
r
+
p
pq+r, pr+q, qr+p
pq
+
r
,
p
r
+
q
,
q
r
+
p
has a real root. proposed by S. Berlov
given non-constant linear functions
Given non-constant linear functions
p
1
(
x
)
,
p
2
(
x
)
,
…
p
n
(
x
)
p_1(x), p_2(x), \dots p_n(x)
p
1
(
x
)
,
p
2
(
x
)
,
…
p
n
(
x
)
. Prove that at least
n
−
2
n-2
n
−
2
of polynomials
p
1
p
2
…
p
n
−
1
+
p
n
,
p
1
p
2
…
p
n
−
2
p
n
+
p
n
−
1
,
…
p
2
p
3
…
p
n
+
p
1
p_1p_2\dots p_{n-1}+p_n, p_1p_2\dots p_{n-2} p_n + p_{n-1},\dots p_2p_3\dots p_n+p_1
p
1
p
2
…
p
n
−
1
+
p
n
,
p
1
p
2
…
p
n
−
2
p
n
+
p
n
−
1
,
…
p
2
p
3
…
p
n
+
p
1
have a real root.