MathDB
Problems
Contests
International Contests
Balkan MO
2011 Balkan MO
2011 Balkan MO
Part of
Balkan MO
Subcontests
(4)
3
1
Hide problems
Sets of of divisors
Let
S
S
S
be a finite set of positive integers which has the following property:if
x
x
x
is a member of
S
S
S
,then so are all positive divisors of
x
x
x
. A non-empty subset
T
T
T
of
S
S
S
is good if whenever
x
,
y
∈
T
x,y\in T
x
,
y
∈
T
and
x
<
y
x<y
x
<
y
, the ratio
y
/
x
y/x
y
/
x
is a power of a prime number. A non-empty subset
T
T
T
of
S
S
S
is bad if whenever
x
,
y
∈
T
x,y\in T
x
,
y
∈
T
and
x
<
y
x<y
x
<
y
, the ratio
y
/
x
y/x
y
/
x
is not a power of a prime number. A set of an element is considered both good and bad. Let
k
k
k
be the largest possible size of a good subset of
S
S
S
. Prove that
k
k
k
is also the smallest number of pairwise-disjoint bad subsets whose union is
S
S
S
.
2
1
Hide problems
Unusual inequality, x+y+z=0
Given real numbers
x
,
y
,
z
x,y,z
x
,
y
,
z
such that
x
+
y
+
z
=
0
x+y+z=0
x
+
y
+
z
=
0
, show that
x
(
x
+
2
)
2
x
2
+
1
+
y
(
y
+
2
)
2
y
2
+
1
+
z
(
z
+
2
)
2
z
2
+
1
≥
0
\dfrac{x(x+2)}{2x^2+1}+\dfrac{y(y+2)}{2y^2+1}+\dfrac{z(z+2)}{2z^2+1}\ge 0
2
x
2
+
1
x
(
x
+
2
)
+
2
y
2
+
1
y
(
y
+
2
)
+
2
z
2
+
1
z
(
z
+
2
)
≥
0
When does equality hold?
4
1
Hide problems
Inequality with areas in hexagon
Let
A
B
C
D
E
F
ABCDEF
A
BC
D
EF
be a convex hexagon of area
1
1
1
, whose opposite sides are parallel. The lines
A
B
AB
A
B
,
C
D
CD
C
D
and
E
F
EF
EF
meet in pairs to determine the vertices of a triangle. Similarly, the lines
B
C
BC
BC
,
D
E
DE
D
E
and
F
A
FA
F
A
meet in pairs to determine the vertices of another triangle. Show that the area of at least one of these two triangles is at least
3
/
2
3/2
3/2
.
1
1
Hide problems
Perpendicularity in a cyclic quad
Let
A
B
C
D
ABCD
A
BC
D
be a cyclic quadrilateral which is not a trapezoid and whose diagonals meet at
E
E
E
. The midpoints of
A
B
AB
A
B
and
C
D
CD
C
D
are
F
F
F
and
G
G
G
respectively, and
ℓ
\ell
ℓ
is the line through
G
G
G
parallel to
A
B
AB
A
B
. The feet of the perpendiculars from E onto the lines
ℓ
\ell
ℓ
and
C
D
CD
C
D
are
H
H
H
and
K
K
K
, respectively. Prove that the lines
E
F
EF
EF
and
H
K
HK
HK
are perpendicular.