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Contests
National and Regional Contests
Russia Contests
239 Open Math Olympiad
2016 239 Open Mathematical Olympiad
2016 239 Open Mathematical Olympiad
Part of
239 Open Math Olympiad
Subcontests
(8)
8
2
Hide problems
table $(2k + 1) \times (2k + 1)$ is filled with real numbers not exceeding 1 in
Given a natural number
k
>
1
k>1
k
>
1
. Find the smallest number
α
\alpha
α
satisfying the following condition. Suppose that the table
(
2
k
+
1
)
×
(
2
k
+
1
)
(2k + 1) \times (2k + 1)
(
2
k
+
1
)
×
(
2
k
+
1
)
is filled with real numbers not exceeding
1
1
1
in absolute value, and the sums of the numbers in all lines are equal to zero. Then you can rearrange the numbers so that each number remains in its row and all the sums over the columns will be at most
α
\alpha
α
.
There are n triangles inscribed in a circle and all 3n of their vertices are dif
There are
n
n
n
triangles inscribed in a circle and all
3
n
3n
3
n
of their vertices are different. Prove that it is possible to put a boy in one of the vertices in each triangle, and a girl in the other, so that boys and girls alternate on a circle.
7
2
Hide problems
$f(xy+x+y)=(f(x)-f(y))f(y-x-1)$
Find all functions
f
:
R
+
→
R
+
f:\mathbb{R^+}\to\mathbb{R^+}
f
:
R
+
→
R
+
satisfying
f
(
x
y
+
x
+
y
)
=
(
f
(
x
)
−
f
(
y
)
)
f
(
y
−
x
−
1
)
f(xy+x+y)=(f(x)-f(y))f(y-x-1)
f
(
x
y
+
x
+
y
)
=
(
f
(
x
)
−
f
(
y
))
f
(
y
−
x
−
1
)
for all
x
>
0
,
y
>
x
+
1
x>0, y>x+1
x
>
0
,
y
>
x
+
1
.
six pair-wise coprime numbers
A set is called
s
i
x
s
q
u
a
r
e
six\ square
s
i
x
s
q
u
a
re
if it has six pair-wise coprime numbers and for any partition of it into two set with three elements, the sum of the numbers in one of them is perfect square. Prove that there exist infinitely many
s
i
x
s
q
u
a
r
e
six\ square
s
i
x
s
q
u
a
re
.
6
2
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A finite family of finite sets $F$
A finite family of finite sets
F
F
F
is given, satisfying two conditions: (i) if
A
,
B
∈
F
A, B \in F
A
,
B
∈
F
, then
A
∪
B
∈
F
A \cup B \in F
A
∪
B
∈
F
; (ii) if
A
∈
F
A \in F
A
∈
F
, then the number of elements
∣
A
∣
| A |
∣
A
∣
is not a multiple of
3
3
3
. Prove that you can specify at most two elements so that every set of the family
F
F
F
contains at least one of them.
blue path with 100 edges
A graph is called
7
−
c
h
i
p
7-chip
7
−
c
hi
p
if it obtained by removing at most three edges that have no vertex in common from a complete graph with seven vertices. Consider a complete graph
G
G
G
with
v
v
v
vertices which each edge of its is colored blue or red. Prove that there is either a blue path with
100
100
100
edges or a red
7
−
c
h
i
p
7-chip
7
−
c
hi
p
.
5
2
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hexagon $ AC_1BA_1CB_1 $
Through point
P
P
P
inside triangle
A
B
C
ABC
A
BC
, straight lines were drawn, parallel to the sides, until they intersect with the sides. In the three resulting parallelograms, diagonals that do not contain point
P
P
P
, are drawn. Points
A
1
A_1
A
1
,
B
1
B_1
B
1
and
C
1
C_1
C
1
are the intersection points of the lines containing these diagonals such that
A
1
A_1
A
1
and
A
A
A
are in different sides of line
B
C
BC
BC
and
B
1
B_1
B
1
and
C
1
C_1
C
1
are similar. Prove that if hexagon
A
C
1
B
A
1
C
B
1
AC_1BA_1CB_1
A
C
1
B
A
1
C
B
1
is inscribed and convex, then point
P
P
P
is the orthocenter of triangle
A
1
B
1
C
1
A_1B_1C_1
A
1
B
1
C
1
.
$AP + 2PB = CP$.
Triangle
A
B
C
ABC
A
BC
in which
A
B
<
B
C
AB <BC
A
B
<
BC
, is inscribed in a circle
ω
\omega
ω
and circumscribed about a circle
γ
\gamma
γ
with center
I
I
I
. The line
ℓ
\ell
ℓ
parallel to
A
C
AC
A
C
, touches the circle
γ
\gamma
γ
and intersects the arcs
B
A
C
BAC
B
A
C
and
B
C
A
BCA
BC
A
at points
P
P
P
and
Q
Q
Q
, respectively. It is known that
P
Q
=
2
B
I
PQ = 2BI
PQ
=
2
B
I
. Prove that
A
P
+
2
P
B
=
C
P
AP + 2PB = CP
A
P
+
2
PB
=
CP
.
4
2
Hide problems
$$p_1 = 1, q_1 = 1, p_{n + 1} = 2q_n^2-p_n^2$$
The sequences of natural numbers
p
n
p_n
p
n
and
q
n
q_n
q
n
are given such that
p
1
=
1
,
q
1
=
1
,
p
n
+
1
=
2
q
n
2
−
p
n
2
,
q
n
+
1
=
2
q
n
2
+
p
n
2
p_1 = 1,\ q_1 = 1,\ p_{n + 1} = 2q_n^2-p_n^2,\ q_{n + 1} = 2q_n^2+p_n^2
p
1
=
1
,
q
1
=
1
,
p
n
+
1
=
2
q
n
2
−
p
n
2
,
q
n
+
1
=
2
q
n
2
+
p
n
2
Prove that
p
n
p_n
p
n
and
q
m
q_m
q
m
are coprime for any m and n.
$a+b+c+\frac{3}{ab+bc+ca}\geq4$
Positive real numbers
a
,
b
,
c
a,b,c
a
,
b
,
c
are given such that
a
b
c
=
1
abc=1
ab
c
=
1
. Prove that
a
+
b
+
c
+
3
a
b
+
b
c
+
c
a
≥
4.
a+b+c+\frac{3}{ab+bc+ca}\geq4.
a
+
b
+
c
+
ab
+
b
c
+
c
a
3
≥
4.
3
2
Hide problems
$$2(a+b+c)+\frac{9}{(ab+bc+ca)^2}\geq7$$.
Positive real numbers
a
a
a
,
b
b
b
,
c
c
c
are given such that
a
b
c
=
1
abc=1
ab
c
=
1
.Prove that
2
(
a
+
b
+
c
)
+
9
(
a
b
+
b
c
+
c
a
)
2
≥
7.
2(a+b+c)+\frac{9}{(ab+bc+ca)^2}\geq7.
2
(
a
+
b
+
c
)
+
(
ab
+
b
c
+
c
a
)
2
9
≥
7.
A regular hexagon with a side of $50$
A regular hexagon with a side of
50
50
50
was divided to equilateral triangles with unit side, parallel to the sides of the hexagon. It is allowed to delete any three nodes of the resulting lattice defining a segment of length
2
2
2
. As a result of several such operations, exactly one node remains. How many ways is this possible?
2
2
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angle $ARC$ is obtuse.
In triangle
A
B
C
ABC
A
BC
, the incircle touches sides
A
B
AB
A
B
and
B
C
BC
BC
at points
P
P
P
and
Q
Q
Q
, respectively. Median of triangle
A
B
C
ABC
A
BC
from vertex
B
B
B
meets segment
P
Q
P Q
PQ
at point
R
R
R
. Prove that angle
A
R
C
ARC
A
RC
is obtuse.
triangles $BTP$ and $DTQ$ are similar
In a convex quadrilateral
A
B
C
D
ABCD
A
BC
D
rays
A
B
AB
A
B
and
D
C
DC
D
C
intersect at point
P
P
P
, and rays
B
C
BC
BC
and
A
D
AD
A
D
at point
Q
Q
Q
. There is a point
T
T
T
on the diagonal
A
C
AC
A
C
such that the triangles
B
T
P
BTP
BTP
and
D
T
Q
DTQ
D
TQ
are similar, in that order. Prove that
B
D
∥
P
Q
BD \Vert PQ
B
D
∥
PQ
.
1
1
Hide problems
The sum of some divisor of $k$ and some divisor of $k - 1$ is equal to $a$,
A natural number
k
>
1
k>1
k
>
1
is given. The sum of some divisor of
k
k
k
and some divisor of
k
−
1
k - 1
k
−
1
is equal to
a
a
a
,where
a
>
k
+
1
a>k + 1
a
>
k
+
1
. Prove that at least one of the numbers
a
−
1
a - 1
a
−
1
or
a
+
1
a + 1
a
+
1
composite.