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Problems
Contests
International Contests
Balkan MO
1992 Balkan MO
1992 Balkan MO
Part of
Balkan MO
Subcontests
(4)
4
1
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A set in which three elements pairwise coprime exist
For each integer
n
≥
3
n\geq 3
n
≥
3
, find the least natural number
f
(
n
)
f(n)
f
(
n
)
having the property
⋆
\star
⋆
For every
A
⊂
{
1
,
2
,
…
,
n
}
A \subset \{1, 2, \ldots, n\}
A
⊂
{
1
,
2
,
…
,
n
}
with
f
(
n
)
f(n)
f
(
n
)
elements, there exist elements
x
,
y
,
z
∈
A
x, y, z \in A
x
,
y
,
z
∈
A
that are pairwise coprime.
3
1
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Three points on the sides of triangle ABC imply inequality
Let
D
D
D
,
E
E
E
,
F
F
F
be points on the sides
B
C
BC
BC
,
C
A
CA
C
A
,
A
B
AB
A
B
respectively of a triangle
A
B
C
ABC
A
BC
(distinct from the vertices). If the quadrilateral
A
F
D
E
AFDE
A
F
D
E
is cyclic, prove that
4
A
[
D
E
F
]
A
[
A
B
C
]
≤
(
E
F
A
D
)
2
.
\frac{ 4 \mathcal A[DEF] }{\mathcal A[ABC] } \leq \left( \frac{EF}{AD} \right)^2 .
A
[
A
BC
]
4
A
[
D
EF
]
≤
(
A
D
EF
)
2
.
Greece
2
1
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An inequality for all positive integers n
Prove that for all positive integers
n
n
n
the following inequality takes place
(
2
n
2
+
3
n
+
1
)
n
≥
6
n
(
n
!
)
2
.
(2n^2+3n+1)^n \geq 6^n (n!)^2 .
(
2
n
2
+
3
n
+
1
)
n
≥
6
n
(
n
!
)
2
.
Cyprus
1
1
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Divisibility question
For all positive integers
m
,
n
m,n
m
,
n
define
f
(
m
,
n
)
=
m
3
4
n
+
6
−
m
3
4
n
+
4
−
m
5
+
m
3
f(m,n) = m^{3^{4n}+6} - m^{3^{4n}+4} - m^5 + m^3
f
(
m
,
n
)
=
m
3
4
n
+
6
−
m
3
4
n
+
4
−
m
5
+
m
3
. Find all numbers
n
n
n
with the property that
f
(
m
,
n
)
f(m, n)
f
(
m
,
n
)
is divisible by 1992 for every
m
m
m
. Bulgaria