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Contests
National and Regional Contests
Bulgaria Contests
Bulgaria Team Selection Test
2005 Bulgaria Team Selection Test
2005 Bulgaria Team Selection Test
Part of
Bulgaria Team Selection Test
Subcontests
(6)
6
1
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Partition of a Group of Nine Persons
In a group of nine persons it is not possible to choose four persons such that every one knows the three others. Prove that this group of nine persons can be partitioned into four groups such that nobody knows anyone from his or her group.
5
1
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Orthocenter and Incenter
Let
A
B
C
ABC
A
BC
,
A
C
≠
B
C
AC \not= BC
A
C
=
BC
, be an acute triangle with orthocenter
H
H
H
and incenter
I
I
I
. The lines
C
H
CH
C
H
and
C
I
CI
C
I
meet the circumcircle of
△
A
B
C
\bigtriangleup ABC
△
A
BC
at points
D
D
D
and
L
L
L
, respectively. Prove that
∠
C
I
H
=
9
0
∘
\angle CIH = 90^{\circ}
∠
C
I
H
=
9
0
∘
if and only if
∠
I
D
L
=
9
0
∘
\angle IDL = 90^{\circ}
∠
I
D
L
=
9
0
∘
4
1
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Maximum Possible Number of Positive Numbers
Let
a
i
a_{i}
a
i
and
b
i
b_{i}
b
i
, where
i
∈
{
1
,
2
,
…
,
2005
}
i \in \{1,2, \dots, 2005 \}
i
∈
{
1
,
2
,
…
,
2005
}
, be real numbers such that the inequality
(
a
i
x
−
b
i
)
2
≥
∑
j
=
1
,
j
≠
i
2005
(
a
j
x
−
b
j
)
(a_{i}x-b_{i})^{2} \ge \sum_{j=1, j \not= i}^{2005} (a_{j}x-b_{j})
(
a
i
x
−
b
i
)
2
≥
∑
j
=
1
,
j
=
i
2005
(
a
j
x
−
b
j
)
holds for all
x
∈
R
x \in \mathbb{R}
x
∈
R
and all
i
∈
{
1
,
2
,
…
,
2005
}
i \in \{1,2, \dots, 2005 \}
i
∈
{
1
,
2
,
…
,
2005
}
. Find the maximum possible number of positive numbers amongst
a
i
a_{i}
a
i
and
b
i
b_{i}
b
i
,
i
∈
{
1
,
2
,
…
,
2005
}
i \in \{1,2, \dots, 2005 \}
i
∈
{
1
,
2
,
…
,
2005
}
.
3
1
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Functional Equation
Let
R
∗
\mathbb{R}^{*}
R
∗
be the set of non-zero real numbers. Find all functions
f
:
R
∗
→
R
∗
f : \mathbb{R}^{*} \to \mathbb{R}^{*}
f
:
R
∗
→
R
∗
such that
f
(
x
2
+
y
)
=
(
f
(
x
)
)
2
+
f
(
x
y
)
f
(
x
)
f(x^{2}+y) = (f(x))^{2} + \frac{f(xy)}{f(x)}
f
(
x
2
+
y
)
=
(
f
(
x
)
)
2
+
f
(
x
)
f
(
x
y
)
, for all
x
,
y
∈
R
∗
x,y \in \mathbb{R}^{*}
x
,
y
∈
R
∗
and
−
x
2
≠
y
-x^{2} \not= y
−
x
2
=
y
.
2
1
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The Number of Subsets
Find the number of the subsets
B
B
B
of the set
{
1
,
2
,
⋯
,
2005
}
\{1,2,\cdots, 2005 \}
{
1
,
2
,
⋯
,
2005
}
such that the sum of the elements of
B
B
B
is congruent to
2006
2006
2006
modulo
2048
2048
2048
1
1
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Let ABC be an acute triangle...
Let
A
B
C
ABC
A
BC
be an acute triangle. Find the locus of the points
M
M
M
, in the interior of
△
A
B
C
\bigtriangleup ABC
△
A
BC
, such that
A
B
−
F
G
=
M
F
.
A
G
+
M
G
.
B
F
C
M
AB-FG= \frac{MF.AG+MG.BF}{CM}
A
B
−
FG
=
CM
MF
.
A
G
+
MG
.
BF
, where
F
F
F
and
G
G
G
are the feet of the perpendiculars from
M
M
M
to the lines
B
C
BC
BC
and
A
C
AC
A
C
, respectively.