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Problems
Contests
National and Regional Contests
Kazakhstan Contests
Kazakhstan National Olympiad
2014 Kazakhstan National Olympiad
2014 Kazakhstan National Olympiad
Part of
Kazakhstan National Olympiad
Subcontests
(3)
1
2
Hide problems
the greatest number of perfect squares
a
1
,
a
2
,
.
.
.
,
a
2014
a_1,a_2,...,a_{2014}
a
1
,
a
2
,
...
,
a
2014
is a permutation of
1
,
2
,
3
,
.
.
.
,
2014
1,2,3,...,2014
1
,
2
,
3
,
...
,
2014
. What is the greatest number of perfect squares can have a set
a
1
2
+
a
2
,
a
2
2
+
a
3
,
a
3
2
+
a
4
,
.
.
.
,
a
2013
2
+
a
2014
,
a
2014
2
+
a
1
?
{ a_1^2+a_2,a_2^2+a_3,a_3^2+a_4,...,a_{2013}^2+a_{2014},a_{2014}^2+a_1 }?
a
1
2
+
a
2
,
a
2
2
+
a
3
,
a
3
2
+
a
4
,
...
,
a
2013
2
+
a
2014
,
a
2014
2
+
a
1
?
Kazakhstan National Olympiad 2014 P4 D2 11 grade
Given a scalene triangle
A
B
C
ABC
A
BC
. Incircle of
△
A
B
C
\triangle{ABC{}}
△
A
BC
touches the sides
A
B
AB
A
B
and
B
C
BC
BC
at points
C
1
C_1
C
1
and
A
1
A_1
A
1
respectively, and excircle of
△
A
B
C
\triangle{ABC}
△
A
BC
(on side
A
C
AC
A
C
) touches
A
B
AB
A
B
and
B
C
BC
BC
at points
C
2
C_2
C
2
and
A
2
A_2
A
2
respectively.
B
N
BN
BN
is bisector of
∠
A
B
C
\angle{ABC}
∠
A
BC
(
N
N
N
lies on
B
C
BC
BC
). Lines
A
1
C
1
A_1C_1
A
1
C
1
and
A
2
C
2
A_2C_2
A
2
C
2
intersects the line
A
C
AC
A
C
at points
K
1
K_1
K
1
and
K
2
K_2
K
2
respectively. Let circumcircles of
△
B
K
1
N
\triangle{BK_1N}
△
B
K
1
N
and
△
B
K
2
N
\triangle{BK_2N}
△
B
K
2
N
intersect circumcircle of a
△
A
B
C
\triangle{ABC}
△
A
BC
at points
P
1
P_1
P
1
and
P
2
P_2
P
2
respectively. Prove that
A
P
1
AP_1
A
P
1
=
C
P
2
CP_2
C
P
2
2
2
Hide problems
Relatively prime for all n
Do there exist positive integers
a
a
a
and
b
b
b
such that
a
n
+
n
b
a^n+n^b
a
n
+
n
b
and
b
n
+
n
a
b^n+n^a
b
n
+
n
a
are relatively prime for all natural
n
n
n
?
Kazakhstan National Olympiad 2014 P5 D2 11 grade
Q
\mathbb{Q}
Q
is set of all rational numbers. Find all functions
f
:
Q
×
Q
→
Q
f:\mathbb{Q}\times\mathbb{Q}\rightarrow\mathbb{Q}
f
:
Q
×
Q
→
Q
such that for all
x
x
x
,
y
y
y
,
z
z
z
∈
Q
\in\mathbb{Q}
∈
Q
satisfy
f
(
x
,
y
)
+
f
(
y
,
z
)
+
f
(
z
,
x
)
=
f
(
0
,
x
+
y
+
z
)
f(x,y)+f(y,z)+f(z,x)=f(0,x+y+z)
f
(
x
,
y
)
+
f
(
y
,
z
)
+
f
(
z
,
x
)
=
f
(
0
,
x
+
y
+
z
)
3
2
Hide problems
Parallel lines..
The triangle
A
B
C
ABC
A
BC
is inscribed in a circle
w
1
w_1
w
1
. Inscribed in a triangle circle touchs the sides
B
C
BC
BC
in a point
N
N
N
.
w
2
w_2
w
2
— the circle inscribed in a segment
B
A
C
BAC
B
A
C
circle of
w
1
w_1
w
1
, and passing through a point
N
N
N
. Let points
O
O
O
and
J
J
J
— the centers of circles
w
2
w_2
w
2
and an extra inscribed circle (touching side
B
C
BC
BC
) respectively. Prove, that lines
A
O
AO
A
O
and
J
N
JN
J
N
are parallel.
Kazakhstan National Olympiad 2014 P3 D2 10 grade
Prove that, for all
n
∈
N
n\in\mathbb{N}
n
∈
N
, on
[
n
−
4
n
,
n
+
4
n
]
[n-4\sqrt{n}, n+4\sqrt{n}]
[
n
−
4
n
,
n
+
4
n
]
exists natural number
k
=
x
3
+
y
3
k=x^3+y^3
k
=
x
3
+
y
3
where
x
x
x
,
y
y
y
are nonnegative integers.