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National and Regional Contests
Albania Contests
Albania-Balkan MO TST
2009 BMO TST
2009 BMO TST
Part of
Albania-Balkan MO TST
Subcontests
(4)
2
1
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Albanian BMO TST 2009 Question 2
Let
C
1
C_{1}
C
1
and
C
2
C_{2}
C
2
be concentric circles, with
C
2
C_{2}
C
2
in the interior of
C
1
C_{1}
C
1
. From a point
A
A
A
on
C
1
C_{1}
C
1
, draw the tangent
A
B
AB
A
B
to
C
2
C_{2}
C
2
(
B
∈
C
2
)
(B \in C_{2})
(
B
∈
C
2
)
. Let
C
C
C
be the second point of intersection of
A
B
AB
A
B
and
C
1
C_{1}
C
1
,and let
D
D
D
be the midpoint of
A
B
AB
A
B
. A line passing through
A
A
A
intersects
C
2
C_{2}
C
2
at
E
E
E
and
F
F
F
in such a way that the perpendicular bisectors of
D
E
DE
D
E
and
C
F
CF
CF
intersect at a point
M
M
M
on
A
B
AB
A
B
. Find, with proof, the ratio
A
M
/
M
C
AM/MC
A
M
/
MC
.This question is taken from Mathematical Olympiad Challenges , the 9-th exercise in 1.3 Power of a Point.
4
1
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Albanian BMO TST 2009 Question 4
Find all the polynomials
P
(
x
)
P(x)
P
(
x
)
of a degree
≤
n
\leq n
≤
n
with real non-negative coefficients such that
P
(
x
)
⋅
P
(
1
x
)
≤
[
P
(
1
)
]
2
P(x) \cdot P(\frac{1}{x}) \leq [P(1)]^2
P
(
x
)
⋅
P
(
x
1
)
≤
[
P
(
1
)
]
2
,
∀
x
>
0
\forall x>0
∀
x
>
0
.
3
1
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Albanian BMO TST 2009 Question 3
For the give functions in
N
\mathbb{N}
N
: (a) Euler's
ϕ
\phi
ϕ
function (
ϕ
(
n
)
\phi(n)
ϕ
(
n
)
- the number of natural numbers smaller than
n
n
n
and coprime with
n
n
n
); (b) the
σ
\sigma
σ
function such that the
σ
(
n
)
\sigma(n)
σ
(
n
)
is the sum of natural divisors of
n
n
n
. solve the equation
ϕ
(
σ
(
2
x
)
)
=
2
x
\phi(\sigma(2^x))=2^x
ϕ
(
σ
(
2
x
))
=
2
x
.
1
1
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Albanian BMO TST 2009 Question 1
Given the equation
x
4
−
x
3
−
1
=
0
x^4-x^3-1=0
x
4
−
x
3
−
1
=
0
(a) Find the number of its real roots. (b) We denote by
S
S
S
the sum of the real roots and by
P
P
P
their product. Prove that
P
<
−
11
10
P< - \frac{11}{10}
P
<
−
10
11
and
S
>
6
11
S> \frac {6}{11}
S
>
11
6
.