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Contests
National and Regional Contests
Bulgaria Contests
Bulgaria National Olympiad
2008 Bulgaria National Olympiad
2008 Bulgaria National Olympiad
Part of
Bulgaria National Olympiad
Subcontests
(3)
3
2
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nice problem
Let
n
∈
N
n\in\mathbb{N}
n
∈
N
and
0
≤
a
1
≤
a
2
≤
…
≤
a
n
≤
π
0\leq a_1\leq a_2\leq\ldots\leq a_n\leq\pi
0
≤
a
1
≤
a
2
≤
…
≤
a
n
≤
π
and
b
1
,
b
2
,
…
,
b
n
b_1,b_2,\ldots ,b_n
b
1
,
b
2
,
…
,
b
n
are real numbers for which the following inequality is satisfied : \left|\sum_{i\equal{}1}^{n} b_i\cos(ka_i)\right|<\frac{1}{k} for all
k
∈
N
k\in\mathbb{N}
k
∈
N
. Prove that b_1\equal{}b_2\equal{}\ldots \equal{}b_n\equal{}0.
base subsets -- Bulgaria MO
Let
M
M
M
be the set of the integer numbers from the range
[
−
n
,
n
]
[-n, n]
[
−
n
,
n
]
. The subset
P
P
P
of
M
M
M
is called a base subset if every number from
M
M
M
can be expressed as a sum of some different numbers from
P
P
P
. Find the smallest natural number
k
k
k
such that every
k
k
k
numbers that belongs to
M
M
M
form a base subset.
2
2
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Find all natural numbers
Let
n
n
n
be a fixed natural number. Find all natural numbers
m
m
m
for which
1
a
n
+
1
b
n
≥
a
m
+
b
m
\frac{1}{a^n}+\frac{1}{b^n}\ge a^m+b^m
a
n
1
+
b
n
1
≥
a
m
+
b
m
is satisfied for every two positive numbers
a
a
a
and
b
b
b
with sum equal to
2
2
2
.
Union of arithmetical progressions
Is it possible to find
2008
2008
2008
infinite arithmetical progressions such that there exist finitely many positive integers not in any of these progressions, no two progressions intersect and each progression contains a prime number bigger than
2008
2008
2008
?
1
2
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Circles
Let
A
B
C
ABC
A
BC
be an acute triangle and
C
L
CL
C
L
be the angle bisector of
∠
A
C
B
\angle ACB
∠
A
CB
. The point
P
P
P
lies on the segment
C
L
CL
C
L
such that \angle APB\equal{}\pi\minus{}\frac{_1}{^2}\angle ACB. Let
k
1
k_1
k
1
and
k
2
k_2
k
2
be the circumcircles of the triangles
A
P
C
APC
A
PC
and
B
P
C
BPC
BPC
. BP\cap k_1\equal{}Q, AP\cap k_2\equal{}R. The tangents to
k
1
k_1
k
1
at
Q
Q
Q
and
k
2
k_2
k
2
at
B
B
B
intersect at
S
S
S
and the tangents to
k
1
k_1
k
1
at
A
A
A
and
k
2
k_2
k
2
at
R
R
R
intersect at
T
T
T
. Prove that AS\equal{}BT.
k-th power of natural number
Find the smallest natural number
k
k
k
for which there exists natural numbers
m
m
m
and
n
n
n
such that 1324 \plus{} 279m \plus{} 5^n is
k
k
k
-th power of some natural number.