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Contests
National and Regional Contests
Kazakhstan Contests
Kazakhstan National Olympiad
2018 Kazakhstan National Olympiad
2018 Kazakhstan National Olympiad
Part of
Kazakhstan National Olympiad
Subcontests
(6)
6
1
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Nairi Sedrakyans new geometric ineqality
Inside of convex quadrilateral
A
B
C
D
ABCD
A
BC
D
found a point
M
M
M
such that
∠
A
M
B
=
∠
A
D
M
+
∠
B
C
M
\angle AMB=\angle ADM+\angle BCM
∠
A
MB
=
∠
A
D
M
+
∠
BCM
and
∠
A
M
D
=
∠
A
B
M
+
∠
D
C
M
\angle AMD=\angle ABM+\angle DCM
∠
A
M
D
=
∠
A
BM
+
∠
D
CM
.Prove that
A
M
⋅
C
M
+
B
M
⋅
D
M
≥
A
B
⋅
B
C
⋅
C
D
⋅
D
A
.
AM\cdot CM+BM\cdot DM\ge \sqrt{AB\cdot BC\cdot CD\cdot DA}.
A
M
⋅
CM
+
BM
⋅
D
M
≥
A
B
⋅
BC
⋅
C
D
⋅
D
A
.
5
1
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Kazakhstan MO 2018 realy nice
Given set
S
=
{
x
y
(
x
+
y
)
∣
x
,
y
∈
N
}
S = \{ xy\left( {x + y} \right)\; |\; x,y \in \mathbb{N}\}
S
=
{
x
y
(
x
+
y
)
∣
x
,
y
∈
N
}
.Let
a
a
a
and
n
n
n
natural numbers such that
a
+
2
k
∈
S
a+2^k\in S
a
+
2
k
∈
S
for all
k
=
1
,
2
,
3
,
.
.
.
,
n
k=1,2,3,...,n
k
=
1
,
2
,
3
,
...
,
n
.Find the greatest value of
n
n
n
.
4
1
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Kazakhstan MO 2018 inequality
Prove that for all reas
a
,
b
,
c
,
d
∈
(
0
,
1
)
a,b,c,d\in(0,1)
a
,
b
,
c
,
d
∈
(
0
,
1
)
we have
(
a
b
−
c
d
)
(
a
c
+
b
d
)
(
a
d
−
b
c
)
+
min
(
a
,
b
,
c
,
d
)
<
1.
\left(ab-cd\right)\left(ac+bd\right)\left(ad-bc\right)+\min{\left(a,b,c,d\right)} < 1.
(
ab
−
c
d
)
(
a
c
+
b
d
)
(
a
d
−
b
c
)
+
min
(
a
,
b
,
c
,
d
)
<
1.
3
1
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Kazakhstan MO 2018 function
Is there exist a function
f
:
N
→
N
f:\mathbb {N}\to \mathbb {N}
f
:
N
→
N
with for
∀
m
,
n
∈
N
\forall m,n \in \mathbb {N}
∀
m
,
n
∈
N
f
(
m
f
(
n
)
)
=
f
(
m
)
f
(
m
+
n
)
+
n
?
f\left(mf\left(n\right)\right)=f\left(m\right)f\left(m+n\right)+n ?
f
(
m
f
(
n
)
)
=
f
(
m
)
f
(
m
+
n
)
+
n
?
2
1
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Kazakhstan MO 2018 the concave sequence
The natural number
m
≥
2
m\geq 2
m
≥
2
is given.Sequence of natural numbers
(
b
0
,
b
1
,
…
,
b
m
)
(b_0,b_1,\ldots,b_m)
(
b
0
,
b
1
,
…
,
b
m
)
is called concave if
b
k
+
b
k
−
2
≤
2
b
k
−
1
b_k+b_{k-2}\le2b_{k-1}
b
k
+
b
k
−
2
≤
2
b
k
−
1
for all
2
≤
k
≤
m
.
2\le k\le m.
2
≤
k
≤
m
.
Prove that there exist not greater than
2
m
2^m
2
m
concave sequences starting with
b
0
=
1
b_0 =1
b
0
=
1
or
b
0
=
2
b_0 =2
b
0
=
2
1
1
Hide problems
Kazakhstan MO 2018
In an equilateral trapezoid, the point
O
O
O
is the midpoint of the base
A
D
AD
A
D
. A circle with a center at a point
O
O
O
and a radius
B
O
BO
BO
is tangent to a straight line
A
B
AB
A
B
. Let the segment
A
C
AC
A
C
intersect this circle at point
K
(
K
≠
C
)
K(K \ne C)
K
(
K
=
C
)
, and let
M
M
M
is a point such that
A
B
C
M
ABCM
A
BCM
is a parallelogram. The circumscribed circle of a triangle
C
M
D
CMD
CM
D
intersects the segment
A
C
AC
A
C
at a point
L
(
L
≠
C
)
L(L\ne C)
L
(
L
=
C
)
. Prove that
A
K
=
C
L
AK=CL
A
K
=
C
L
.