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Problems
Contests
National and Regional Contests
Kazakhstan Contests
Kazakhstan National Olympiad
2017 Kazakhstan National Olympiad
2017 Kazakhstan National Olympiad
Part of
Kazakhstan National Olympiad
Subcontests
(6)
5
1
Hide problems
Kazakhstan National Olympiad 2017, Final Round, 11-Grade, P5
Consider all possible sets of natural numbers
(
x
1
,
x
2
,
.
.
.
,
x
100
)
(x_1, x_2, ..., x_{100})
(
x
1
,
x
2
,
...
,
x
100
)
such that
1
≤
x
i
≤
2017
1\leq x_i \leq 2017
1
≤
x
i
≤
2017
for every
i
=
1
,
2
,
.
.
.
,
100
i = 1,2, ..., 100
i
=
1
,
2
,
...
,
100
. We say that the set
(
y
1
,
y
2
,
.
.
.
,
y
100
)
(y_1, y_2, ..., y_{100})
(
y
1
,
y
2
,
...
,
y
100
)
is greater than the set
(
z
1
,
z
2
,
.
.
.
,
z
100
)
(z_1, z_2, ..., z_{100})
(
z
1
,
z
2
,
...
,
z
100
)
if
y
i
>
z
i
y_i> z_i
y
i
>
z
i
for every
i
=
1
,
2
,
.
.
.
,
100
i = 1,2, ..., 100
i
=
1
,
2
,
...
,
100
. What is the largest number of sets that can be written on the board, so that any set is not more than the other set?
3
1
Hide problems
Kazakhstan National Olympiad 2017, Final Round, 11-Grade, P3
{
a
n
}
\{a_n\}
{
a
n
}
is an infinite, strictly increasing sequence of positive integers and
a
a
n
≤
a
n
+
a
n
+
3
a_{a_n}\leq a_n+a_{n+3}
a
a
n
≤
a
n
+
a
n
+
3
for all
n
≥
1
n\geq 1
n
≥
1
. Prove that, there are infinitely many triples
(
k
,
l
,
m
)
(k,l,m)
(
k
,
l
,
m
)
of positive integers such that
k
<
l
<
m
k<l<m
k
<
l
<
m
and
a
k
+
a
m
=
2
a
l
a_k+a_m=2a_l
a
k
+
a
m
=
2
a
l
1
1
Hide problems
Kazakhstan National Olympiad 2017, Final Round, 11-Grade, P1
The non-isosceles triangle
A
B
C
ABC
A
BC
is inscribed in the circle
ω
\omega
ω
. The tangent line to this circle at the point
C
C
C
intersects the line
A
B
AB
A
B
at the point
D
D
D
. Let the bisector of the angle
C
D
B
CDB
C
D
B
intersect the segments
A
C
AC
A
C
and
B
C
BC
BC
at the points
K
K
K
and
L
L
L
, respectively. The point
M
M
M
is on the side
A
B
AB
A
B
such that
A
K
B
L
=
A
M
B
M
\frac{AK}{BL} = \frac{AM}{BM}
B
L
A
K
=
BM
A
M
. Let the perpendiculars from the point
M
M
M
to the straight lines
K
L
KL
K
L
and
D
C
DC
D
C
intersect the lines
A
C
AC
A
C
and
D
C
DC
D
C
at the points
P
P
P
and
Q
Q
Q
respectively. Prove that
2
∠
C
Q
P
=
∠
A
C
B
2\angle CQP=\angle ACB
2∠
CQP
=
∠
A
CB
4
1
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Kazakhstan National Olympiad 2017, Final Round, 10-Grade, P4
The acute triangle
A
B
C
ABC
A
BC
(
A
C
>
B
C
)
(AC> BC)
(
A
C
>
BC
)
is inscribed in a circle with the center at the point
O
O
O
, and
C
D
CD
C
D
is the diameter of this circle. The point
K
K
K
is on the continuation of the ray
D
A
DA
D
A
beyond the point
A
A
A
. And the point
L
L
L
is on the segment
B
D
BD
B
D
(
D
L
>
L
B
)
(DL> LB)
(
D
L
>
L
B
)
so that
∠
O
K
D
=
∠
B
A
C
\angle OKD = \angle BAC
∠
O
KD
=
∠
B
A
C
,
∠
O
L
D
=
∠
A
B
C
\angle OLD = \angle ABC
∠
O
L
D
=
∠
A
BC
. Prove that the line
K
L
KL
K
L
passes through the midpoint of the segment
A
B
AB
A
B
.
6
1
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Number Theory, Problem 6
Show that there exist infinitely many composite positive integers
n
n
n
such that
n
n
n
divides
2
n
−
1
2
+
1
2^{\frac{n-1}{2}}+1
2
2
n
−
1
+
1
2
1
Hide problems
Inequality, Problem 2
For positive reals
x
,
y
,
z
≥
1
2
x,y,z\ge \frac{1}{2}
x
,
y
,
z
≥
2
1
with
x
2
+
y
2
+
z
2
=
1
x^2+y^2+z^2=1
x
2
+
y
2
+
z
2
=
1
, prove this inequality holds
(
1
x
+
1
y
−
1
z
)
(
1
x
−
1
y
+
1
z
)
≥
2
(\frac{1}{x}+\frac{1}{y}-\frac{1}{z})(\frac{1}{x}-\frac{1}{y}+\frac{1}{z})\ge 2
(
x
1
+
y
1
−
z
1
)
(
x
1
−
y
1
+
z
1
)
≥
2