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Problems
Contests
International Contests
Balkan MO
2006 Balkan MO
2006 Balkan MO
Part of
Balkan MO
Subcontests
(4)
2
1
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Pass through the same point but i think well known
Let
A
B
C
ABC
A
BC
be a triangle and
m
m
m
a line which intersects the sides
A
B
AB
A
B
and
A
C
AC
A
C
at interior points
D
D
D
and
F
F
F
, respectively, and intersects the line
B
C
BC
BC
at a point
E
E
E
such that
C
C
C
lies between
B
B
B
and
E
E
E
. The parallel lines from the points
A
A
A
,
B
B
B
,
C
C
C
to the line
m
m
m
intersect the circumcircle of triangle
A
B
C
ABC
A
BC
at the points
A
1
A_1
A
1
,
B
1
B_1
B
1
and
C
1
C_1
C
1
, respectively (apart from
A
A
A
,
B
B
B
,
C
C
C
). Prove that the lines
A
1
E
A_1E
A
1
E
,
B
1
F
B_1F
B
1
F
and
C
1
D
C_1D
C
1
D
pass through the same point. Greece
3
1
Hide problems
Nice result of divisibility
Find all triplets of positive rational numbers
(
m
,
n
,
p
)
(m,n,p)
(
m
,
n
,
p
)
such that the numbers
m
+
1
n
p
m+\frac 1{np}
m
+
n
p
1
,
n
+
1
p
m
n+\frac 1{pm}
n
+
p
m
1
,
p
+
1
m
n
p+\frac 1{mn}
p
+
mn
1
are integers. Valentin Vornicu, Romania
4
1
Hide problems
A nice cyclic sequence
Let
m
m
m
be a positive integer and
{
a
n
}
n
≥
0
\{a_n\}_{n\geq 0}
{
a
n
}
n
≥
0
be a sequence given by
a
0
=
a
∈
N
a_0 = a \in \mathbb N
a
0
=
a
∈
N
, and
a
n
+
1
=
{
a
n
2
if
a
n
≡
0
(
m
o
d
2
)
,
a
n
+
m
otherwise.
a_{n+1} = \begin{cases} \displaystyle \frac{a_n}2 & \textrm { if } a_n \equiv 0 \pmod 2, \\ a_n + m & \textrm{ otherwise. } \end{cases}
a
n
+
1
=
{
2
a
n
a
n
+
m
if
a
n
≡
0
(
mod
2
)
,
otherwise.
Find all values of
a
a
a
such that the sequence is periodical (starting from the beginning).
1
1
Hide problems
cyclic sum 1 / (a(b+1)) + ... >= 3 / (1+abc)
Let
a
a
a
,
b
b
b
,
c
c
c
be positive real numbers. Prove the inequality
1
a
(
b
+
1
)
+
1
b
(
c
+
1
)
+
1
c
(
a
+
1
)
≥
3
1
+
a
b
c
.
\frac{1}{a\left(b+1\right)}+\frac{1}{b\left(c+1\right)}+\frac{1}{c\left(a+1\right)}\geq \frac{3}{1+abc}.
a
(
b
+
1
)
1
+
b
(
c
+
1
)
1
+
c
(
a
+
1
)
1
≥
1
+
ab
c
3
.