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Problems
Contests
National and Regional Contests
Russia Contests
239 Open Math Olympiad
2010 239 Open Mathematical Olympiad
2010 239 Open Mathematical Olympiad
Part of
239 Open Math Olympiad
Subcontests
(8)
8
2
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Coloring the vertices of graph
Consider the graph
G
G
G
with
100
100
100
vertices, and the minimum odd cycle goes through
13
13
13
vertices. Prove that the vertices of the graph can be colored in
6
6
6
colors in a way that no two adjacent vertices have the same color.
x+y^2+z^3=1 inequality
For positive numbers
x
x
x
,
y
y
y
, and
z
z
z
, we know that
x
+
y
2
+
z
3
=
1
x + y^2 + z^3 = 1
x
+
y
2
+
z
3
=
1
. Prove that
x
2
+
x
y
+
y
2
+
y
z
+
z
2
+
z
x
>
1
2
.
\frac{x}{2 + xy} + \frac{y}{2 + yz} + \frac{z}{2 + zx} > \frac{1}{2} .
2
+
x
y
x
+
2
+
yz
y
+
2
+
z
x
z
>
2
1
.
7
2
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Dividing the perimeter with points
You are given a convex polygon with perimeter
24
3
+
4
π
24\sqrt{3} + 4\pi
24
3
+
4
π
. If there exists a pair of points dividing the perimeter in half such that the distance between them is equal to
24
24
24
, Prove that there exists a pair of points dividing the perimeter in half such that the distance between them does not exceed
12
12
12
.
Concurrent lines in convex quadrilateral
In a convex quadrilateral
A
B
C
D
ABCD
A
BC
D
, We have
∠
B
=
∠
D
=
12
0
∘
\angle{B} = \angle{D} = 120^{\circ}
∠
B
=
∠
D
=
12
0
∘
. Points
A
′
A'
A
′
,
B
′
B'
B
′
and
C
′
C'
C
′
are symmetric to
D
D
D
relative to
B
C
BC
BC
,
C
A
CA
C
A
and
A
B
AB
A
B
, respectively. Prove that lines
A
A
′
AA'
A
A
′
,
B
B
′
BB'
B
B
′
and
C
C
′
CC'
C
C
′
are concurrent.
6
2
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Six positive numbers with product of 1
We have six positive numbers
a
1
,
a
2
,
…
,
a
6
a_1, a_2, \ldots , a_6
a
1
,
a
2
,
…
,
a
6
such that
a
1
a
2
…
a
6
=
1
a_1a_2\ldots a_6 =1
a
1
a
2
…
a
6
=
1
. Prove that:
1
a
1
(
a
2
+
1
)
+
1
a
2
(
a
3
+
1
)
+
…
+
1
a
6
(
a
1
+
1
)
≥
3.
\frac{1}{a_1(a_2 + 1)} + \frac{1}{a_2(a_3 + 1)} + \ldots + \frac{1}{a_6(a_1 + 1)} \geq 3.
a
1
(
a
2
+
1
)
1
+
a
2
(
a
3
+
1
)
1
+
…
+
a
6
(
a
1
+
1
)
1
≥
3.
Similar numbers
We call natural numbers
n
n
n
and
k
k
k
are similar if they are multiples of square of a number greater than
1
1
1
. Let
f
(
n
)
f(n)
f
(
n
)
denote the number of numbers from
1
1
1
to
n
n
n
similar to
n
n
n
(for example,
f
(
16
)
=
4
f(16)=4
f
(
16
)
=
4
, since the number
16
16
16
is similar to
4
4
4
,
8
8
8
,
12
12
12
and
16
16
16
). What integer values can the quotient
n
f
(
n
)
\frac{n}{f(n)}
f
(
n
)
n
take?
5
2
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Large coprime numbers are perfect squares
Given three natural numbers greater than
100
100
100
, that are pairwise coprime and such that the square of the difference of any two of them is divisible by the third and any of them is less than the product of the other two. Prove that these numbers are squares of some natural numbers.
Souvenirs from Chernobyl
Among
33
33
33
balls, there are
2
2
2
radioactive ones. You can put several balls in the detector and it will show if the both radioactive balls are among the balls. What is the smallest number that we have to use the detector so that one can certainly find at least one of the radioactive balls?
4
1
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Five circles and three points
Consider three pairwise intersecting circles
ω
1
\omega_1
ω
1
,
ω
2
\omega_2
ω
2
and
ω
3
\omega_3
ω
3
. Let their three common chords intersect at point
R
R
R
. We denote by
O
1
O_1
O
1
the center of the circumcircle of a triangle formed by some triple common points of
ω
1
\omega_1
ω
1
&
ω
2
\omega_2
ω
2
,
ω
2
\omega_2
ω
2
&
ω
3
\omega_3
ω
3
and
ω
3
\omega_3
ω
3
&
ω
1
\omega_1
ω
1
. and we denote by
O
2
O_2
O
2
the center of the circumcircle of the triangle formed by the second intersection points of the same pairs of circles. Prove that points
R
R
R
,
O
1
O_1
O
1
and
O
2
O_2
O
2
are collinear.
3
1
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Sabotage of natural numbers
Grisha wrote
n
n
n
different natural numbers, the sum of which does not exceed
S
S
S
. The saboteur added to each of them a number from the half-interval
[
0
,
1
)
[0, 1)
[
0
,
1
)
. The sabotage is successful if there exists two subsets, the sums of the numbers in which differ by no more than
1
1
1
. At what minimum
S
S
S
can Grisha ensure that the sabotage will definitely not be succeeded?
2
2
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Incircle and B-Excircle
The incircle of the triangle
A
B
C
ABC
A
BC
touches the sides
A
C
AC
A
C
and
B
C
BC
BC
at points
K
K
K
and
L
L
L
, respectively. the
B
B
B
-excircle touches the side
A
C
AC
A
C
of this triangle at point
P
P
P
. The segment
A
L
AL
A
L
intersects the inscribed circle for the second time at point
S
S
S
. Line
K
L
KL
K
L
intersects the circumscribed circle of triangle
A
S
K
ASK
A
S
K
for the second at point
M
M
M
. Prove that
P
L
=
P
M
PL = PM
P
L
=
PM
.
Incircle and B-Excircle
The incircle of the triangle
A
B
C
ABC
A
BC
touches the sides
A
C
AC
A
C
and
B
C
BC
BC
at points
K
K
K
and
L
L
L
, respectively. the
B
B
B
-excircle touches the side
A
C
AC
A
C
of this triangle at point
P
P
P
. Line
K
L
KL
K
L
intersects with the line passing through
A
A
A
and parallel to
B
C
BC
BC
at point
M
M
M
. Prove that
P
L
=
P
M
PL = PM
P
L
=
PM
.
1
1
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Chess on multi-colored board
Each square of the chessboard was painted in one of eight colors in such a way that the number of squares colored by all the colors are equal. Is it always possible to put
8
8
8
rooks not threatening each other on multi-colored cells?