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International Contests
International Zhautykov Olympiad
2011 International Zhautykov Olympiad
2011 International Zhautykov Olympiad
Part of
International Zhautykov Olympiad
Subcontests
(3)
3
2
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International Zhautykov Olympiad 2011 - Problem 3
Let
N
\mathbb{N}
N
denote the set of all positive integers. An ordered pair
(
a
;
b
)
(a;b)
(
a
;
b
)
of numbers
a
,
b
∈
N
a,b\in\mathbb{N}
a
,
b
∈
N
is called interesting, if for any
n
∈
N
n\in\mathbb{N}
n
∈
N
there exists
k
∈
N
k\in\mathbb{N}
k
∈
N
such that the number
a
k
+
b
a^k+b
a
k
+
b
is divisible by
2
n
2^n
2
n
. Find all interesting ordered pairs of numbers.
International Zhautykov Olympiad 2011 - Problem 6
Diagonals of a cyclic quadrilateral
A
B
C
D
ABCD
A
BC
D
intersect at point
K
.
K.
K
.
The midpoints of diagonals
A
C
AC
A
C
and
B
D
BD
B
D
are
M
M
M
and
N
,
N,
N
,
respectively. The circumscribed circles
A
D
M
ADM
A
D
M
and
B
C
M
BCM
BCM
intersect at points
M
M
M
and
L
.
L.
L
.
Prove that the points
K
,
L
,
M
,
K ,L ,M,
K
,
L
,
M
,
and
N
N
N
lie on a circle. (all points are supposed to be different.)
2
2
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International Zhautykov Olympiad 2011 - Problem 2
Find all functions
f
:
R
→
R
f:\mathbb{R}\rightarrow\mathbb{R}
f
:
R
→
R
which satisfy the equality,
f
(
x
+
f
(
y
)
)
=
f
(
x
−
f
(
y
)
)
+
4
x
f
(
y
)
f(x+f(y))=f(x-f(y))+4xf(y)
f
(
x
+
f
(
y
))
=
f
(
x
−
f
(
y
))
+
4
x
f
(
y
)
for any
x
,
y
∈
R
x,y\in\mathbb{R}
x
,
y
∈
R
.
International Zhautykov Olympiad 2011 - Problem 5
Let
n
n
n
be integer,
n
>
1.
n>1.
n
>
1.
An element of the set
M
=
{
1
,
2
,
3
,
…
,
n
2
−
1
}
M=\{ 1,2,3,\ldots,n^2-1\}
M
=
{
1
,
2
,
3
,
…
,
n
2
−
1
}
is called good if there exists some element
b
b
b
of
M
M
M
such that
a
b
−
b
ab-b
ab
−
b
is divisible by
n
2
.
n^2.
n
2
.
Furthermore, an element
a
a
a
is called very good if
a
2
−
a
a^2-a
a
2
−
a
is divisible by
n
2
.
n^2.
n
2
.
Let
g
g
g
denote the number of good elements in
M
M
M
and
v
v
v
denote the number of very good elements in
M
.
M.
M
.
Prove that
v
2
+
v
≤
g
≤
n
2
−
n
.
v^2+v \leq g \leq n^2-n.
v
2
+
v
≤
g
≤
n
2
−
n
.
1
2
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International Zhautykov Olympiad 2011 - Problem 1
Given is trapezoid
A
B
C
D
ABCD
A
BC
D
,
M
M
M
and
N
N
N
being the midpoints of the bases of
A
D
AD
A
D
and
B
C
BC
BC
, respectively. a) Prove that the trapezoid is isosceles if it is known that the intersection point of perpendicular bisectors of the lateral sides belongs to the segment
M
N
MN
MN
. b) Does the statement of point a) remain true if it is only known that the intersection point of perpendicular bisectors of the lateral sides belongs to the line
M
N
MN
MN
?
International Zhautykov Olympiad 2011- Problem 4
Find the maximum number of sets which simultaneously satisfy the following conditions:i) any of the sets consists of
4
4
4
elements,ii) any two different sets have exactly
2
2
2
common elements,iii) no two elements are common to all the sets.