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Contests
National and Regional Contests
Albania Contests
Albania-Balkan MO TST
2010 BMO TST
2010 BMO TST
Part of
Albania-Balkan MO TST
Subcontests
(4)
3
1
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Trapezoid inscribed in a cirlce
Let
K
K
K
be the circumscribed circle of the trapezoid
A
B
C
D
ABCD
A
BC
D
. In this trapezoid the diagonals
A
C
AC
A
C
and
B
D
BD
B
D
are perpendicular. The parallel sides AB\equal{}a and CD\equal{}c are diameters of the circles
K
a
K_{a}
K
a
and
K
b
K_{b}
K
b
respectively. Find the perimeter and the area of the part inside the circle
K
K
K
, that is outside circles
K
a
K_{a}
K
a
and
K
b
K_{b}
K
b
.
2
1
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Sequence built on the roots of the equation x^2-ax+1=0, a>=2
Let
a
≥
2
a\geq 2
a
≥
2
be a real number; with the roots
x
1
x_{1}
x
1
and
x
2
x_{2}
x
2
of the equation x^2\minus{}ax\plus{}1\equal{}0 we build the sequence with S_{n}\equal{}x_{1}^n \plus{} x_{2}^n. a)Prove that the sequence \frac{S_{n}}{S_{n\plus{}1}}, where
n
n
n
takes value from
1
1
1
up to infinity, is strictly non increasing. b)Find all value of
a
a
a
for the which this inequality hold for all natural values of
n
n
n
\frac{S_{1}}{S_{2}}\plus{}\cdots \plus{}\frac{S_{n}}{S_{n\plus{}1}}>n\minus{}1
4
1
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a, b,c are the sides of a triangle
Let's consider the inequality a^3\plus{}b^3\plus{}c^3
a
,
b
,
c
a,b,c
a
,
b
,
c
are the sides of a triangle and
k
k
k
a real number. a) Prove the inequality for k\equal{}1. b) Find the smallest value of
k
k
k
such that the inequality holds for all triangles.
1
1
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About the divisors of 1111...11(2011 ones)
a) Is the number
1111
⋯
11
1111\cdots11
1111
⋯
11
(with
2010
2010
2010
ones) a prime number? b) Prove that every prime factor of
1111
⋯
11
1111\cdots11
1111
⋯
11
(with
2011
2011
2011
ones) is of the form 4022j\plus{}1 where
j
j
j
is a natural number.