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Azerbaijan Contests
Azerbaijan Team Selection Test
Azerbaijan IMO TST 2016 First Round
Azerbaijan IMO TST 2016 First Round
Part of
Azerbaijan Team Selection Test
Subcontests
(5)
5
1
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Azerbaijan Team Selection Test 2016 First Round
The largest side of the triangle
A
B
C
ABC
A
BC
is equal to
1
1
1
unit. Prove that , the circles centred at
A
,
B
A,B
A
,
B
and
C
C
C
wit radiuses
1
3
\frac{1}{\sqrt{3}}
3
1
can compeletely cover the triangle
A
B
C
ABC
A
BC
.
4
1
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Azerbaijan Team Selection Test 2016 First Round
Find the solution of the functional equation
P
(
x
)
+
P
(
1
−
x
)
=
1
P(x)+P(1-x)=1
P
(
x
)
+
P
(
1
−
x
)
=
1
with power
2015
2015
2015
P.S:
P
(
y
)
=
y
2015
P(y)=y^{2015}
P
(
y
)
=
y
2015
is also a function with power
2015
2015
2015
2
1
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Azerbaijan Team Selection Test 2016 First Round
A
B
C
ABC
A
BC
be atriangle with sides
A
B
=
20
AB=20
A
B
=
20
,
A
C
=
21
AC=21
A
C
=
21
and
B
C
=
29
BC=29
BC
=
29
. Let
D
D
D
and
E
E
E
be points on the side
B
C
BC
BC
such that
B
D
=
8
BD=8
B
D
=
8
and
E
C
=
9
EC=9
EC
=
9
. Find the angle
∠
D
A
E
\angle DAE
∠
D
A
E
.
1
1
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Azerbaijan Team Selection Test 2016 First Round
Find the maximum value of natural components of number
96
96
96
that we can seperate such that all of them must be relatively prime number withh each other.
3
1
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Azerbaijan IMO TST 2016 First Round
Find the solution of the equation
8
x
(
2
x
2
−
1
)
(
8
x
4
−
8
x
2
+
1
)
=
1
8x(2x^2-1)(8x^4-8x^2+1)=1
8
x
(
2
x
2
−
1
)
(
8
x
4
−
8
x
2
+
1
)
=
1
in the interval
[
0
,
1
]
[0,1]
[
0
,
1
]
?