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Problems
Contests
International Contests
Balkan MO
2010 Balkan MO
2010 Balkan MO
Part of
Balkan MO
Subcontests
(4)
4
1
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f(n+p) != f(n) for any n and prime p
For each integer
n
n
n
(
n
≥
2
n \ge 2
n
≥
2
), let
f
(
n
)
f(n)
f
(
n
)
denote the sum of all positive integers that are at most
n
n
n
and not relatively prime to
n
n
n
. Prove that
f
(
n
+
p
)
≠
f
(
n
)
f(n+p) \neq f(n)
f
(
n
+
p
)
=
f
(
n
)
for each such
n
n
n
and every prime
p
p
p
.
3
1
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Covering any 3 points with 1-strip => cover all with 2-strip
A strip of width
w
w
w
is the set of all points which lie on, or between, two parallel lines distance
w
w
w
apart. Let
S
S
S
be a set of
n
n
n
(
n
≥
3
n \ge 3
n
≥
3
) points on the plane such that any three different points of
S
S
S
can be covered by a strip of width
1
1
1
. Prove that
S
S
S
can be covered by a strip of width
2
2
2
.
2
1
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Eight-point cicle
Let
A
B
C
ABC
A
BC
be an acute triangle with orthocentre
H
H
H
, and let
M
M
M
be the midpoint of
A
C
AC
A
C
. The point
C
1
C_1
C
1
on
A
B
AB
A
B
is such that
C
C
1
CC_1
C
C
1
is an altitude of the triangle
A
B
C
ABC
A
BC
. Let
H
1
H_1
H
1
be the reflection of
H
H
H
in
A
B
AB
A
B
. The orthogonal projections of
C
1
C_1
C
1
onto the lines
A
H
1
AH_1
A
H
1
,
A
C
AC
A
C
and
B
C
BC
BC
are
P
P
P
,
Q
Q
Q
and
R
R
R
, respectively. Let
M
1
M_1
M
1
be the point such that the circumcentre of triangle
P
Q
R
PQR
PQR
is the midpoint of the segment
M
M
1
MM_1
M
M
1
. Prove that
M
1
M_1
M
1
lies on the segment
B
H
1
BH_1
B
H
1
.
1
1
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Cyclic sum a^2b(b-c) / (a+b) >= 0
Let
a
,
b
a,b
a
,
b
and
c
c
c
be positive real numbers. Prove that
a
2
b
(
b
−
c
)
a
+
b
+
b
2
c
(
c
−
a
)
b
+
c
+
c
2
a
(
a
−
b
)
c
+
a
≥
0.
\frac{a^2b(b-c)}{a+b}+\frac{b^2c(c-a)}{b+c}+\frac{c^2a(a-b)}{c+a} \ge 0.
a
+
b
a
2
b
(
b
−
c
)
+
b
+
c
b
2
c
(
c
−
a
)
+
c
+
a
c
2
a
(
a
−
b
)
≥
0.