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Contests
National and Regional Contests
Azerbaijan Contests
Azerbaijan Junior National Olympiad
2015 Azerbaijan National Olympiad
2015 Azerbaijan National Olympiad
Part of
Azerbaijan Junior National Olympiad
Subcontests
(5)
5
1
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Azerbaijan NMO 2015
In the convex quadrilateral
A
B
C
D
ABCD
A
BC
D
angle
∠
B
A
D
=
90
\angle{BAD}=90
∠
B
A
D
=
90
,
∠
B
A
C
=
2
⋅
∠
B
D
C
\angle{BAC}=2\cdot\angle{BDC}
∠
B
A
C
=
2
⋅
∠
B
D
C
and
∠
D
B
A
+
∠
D
C
B
=
180
\angle{DBA}+\angle{DCB}=180
∠
D
B
A
+
∠
D
CB
=
180
. Then find the angle
∠
D
B
A
\angle{DBA}
∠
D
B
A
3
1
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Azerbaijan NMO 2015
Find all polynomials
P
(
x
)
P(x)
P
(
x
)
with real coefficents such that
P
(
P
(
x
)
)
=
(
x
2
+
x
+
1
)
⋅
P
(
x
)
P(P(x))=(x^2+x+1)\cdot P(x)
P
(
P
(
x
))
=
(
x
2
+
x
+
1
)
⋅
P
(
x
)
where
x
∈
R
x \in \mathbb{R}
x
∈
R
2
1
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Azerbaijan NMO 2015
Let
a
,
b
a,b
a
,
b
and
c
c
c
be the length of sides of a triangle.Then prove that
S
≤
a
2
+
b
2
+
c
2
6
S\le\frac{a^2+b^2+c^2}{6}
S
≤
6
a
2
+
b
2
+
c
2
where
S
S
S
is the area of triangle.
1
1
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Azerbaijan NMO 2015
Let
a
,
b
a,b
a
,
b
and
c
c
c
be positive reals such that
a
b
c
=
1
8
abc=\frac{1}{8}
ab
c
=
8
1
.Then prove that
a
2
+
b
2
+
c
2
+
a
2
b
2
+
a
2
c
2
+
b
2
c
2
≥
15
16
a^2+b^2+c^2+a^2b^2+a^2c^2+b^2c^2\ge\frac{15}{16}
a
2
+
b
2
+
c
2
+
a
2
b
2
+
a
2
c
2
+
b
2
c
2
≥
16
15
4
1
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Azerbaijan NMO 2015
Natural number
M
M
M
has
6
6
6
divisors, such that sum of them are equal to
3500
3500
3500
.Find the all values of
M
M
M
.