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Problems
Contests
National and Regional Contests
Azerbaijan Contests
Azerbaijan Team Selection Test
2016 Azerbaijan Team Selection Test
2016 Azerbaijan Team Selection Test
Part of
Azerbaijan Team Selection Test
Subcontests
(3)
1
1
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Tangent Circles
Tangents from the point
A
A
A
to the circle
Γ
\Gamma
Γ
touche this circle at
C
C
C
and
D
D
D
.Let
B
B
B
be a point on
Γ
\Gamma
Γ
,different from
C
C
C
and
D
D
D
. The circle
ω
\omega
ω
that passes through points
A
A
A
and
B
B
B
intersect with lines
A
C
AC
A
C
and
A
D
AD
A
D
at
F
F
F
and
E
E
E
,respectively.Prove that the circumcircles of triangles
A
B
C
ABC
A
BC
and
D
E
B
DEB
D
EB
are tangent if and only if the points
C
,
D
,
F
C,D,F
C
,
D
,
F
and
E
E
E
are cyclic.
3
2
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Functional Equation
Prove that there does not exist a function
f
:
R
+
→
R
+
f : \mathbb R^+\to\mathbb R^+
f
:
R
+
→
R
+
such that
f
(
f
(
x
)
+
y
)
=
f
(
x
)
+
3
x
+
y
f
(
y
)
f(f(x)+y)=f(x)+3x+yf(y)
f
(
f
(
x
)
+
y
)
=
f
(
x
)
+
3
x
+
y
f
(
y
)
for all positive reals
x
,
y
x,y
x
,
y
.
Maximum such k
During a day
2016
2016
2016
customers visited the store. Every customer has been only once at the store(a customer enters the store,spends some time, and leaves the store). Find the greatest integer
k
k
k
that makes the following statement always true.We can find
k
k
k
customers such that either all of them have been at the store at the same time, or any two of them have not been at the same store at the same time.
2
2
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Polynomial Equation
Find all polynomials
P
(
x
)
P(x)
P
(
x
)
with real coefficents, such that for all
x
,
y
,
z
x,y,z
x
,
y
,
z
satisfying
x
+
y
+
z
=
0
x+y+z=0
x
+
y
+
z
=
0
, the equation below is true:
P
(
x
+
y
)
3
+
P
(
y
+
z
)
3
+
P
(
z
+
x
)
3
=
3
P
(
(
x
+
y
)
(
y
+
z
)
(
z
+
x
)
)
P(x+y)^3+P(y+z)^3+P(z+x)^3=3P((x+y)(y+z)(z+x))
P
(
x
+
y
)
3
+
P
(
y
+
z
)
3
+
P
(
z
+
x
)
3
=
3
P
((
x
+
y
)
(
y
+
z
)
(
z
+
x
))
Interesting Polynomial
A positive interger
n
n
n
is called rising if its decimal representation
a
k
a
k
−
1
⋯
a
0
a_ka_{k-1}\cdots a_0
a
k
a
k
−
1
⋯
a
0
satisfies the condition
a
k
≤
a
k
−
1
≤
⋯
≤
a
0
a_k\le a_{k-1}\le\cdots \le a_0
a
k
≤
a
k
−
1
≤
⋯
≤
a
0
. Polynomial
P
P
P
with real coefficents is called interger-valued if for all integer numbers
n
n
n
,
P
(
n
)
P(n)
P
(
n
)
takes interger values.
P
(
n
)
P(n)
P
(
n
)
is called rising-valued if for all rising numbers
n
n
n
,
P
(
n
)
P(n)
P
(
n
)
takes integer values. Does it necessarily mean that, "every rising-valued
P
P
P
is also interger-valued
P
P
P
"?