MathDB
Problems
Contests
National and Regional Contests
Bulgaria Contests
Bulgaria Team Selection Test
2008 Bulgaria Team Selection Test
2008 Bulgaria Team Selection Test
Part of
Bulgaria Team Selection Test
Subcontests
(3)
3
2
Hide problems
Find all real numbers
Let
R
+
\mathbb{R}^{+}
R
+
be the set of positive real numbers. Find all real numbers
a
a
a
for which there exists a function
f
:
R
+
→
R
+
f :\mathbb{R}^{+} \to \mathbb{R}^{+}
f
:
R
+
→
R
+
such that
3
(
f
(
x
)
)
2
=
2
f
(
f
(
x
)
)
+
a
x
4
3(f(x))^{2}=2f(f(x))+ax^{4}
3
(
f
(
x
)
)
2
=
2
f
(
f
(
x
))
+
a
x
4
, for all
x
∈
R
+
x \in \mathbb{R}^{+}
x
∈
R
+
.
Digraph with infinitely many vertices
Let
G
G
G
be a directed graph with infinitely many vertices. It is known that for each vertex the outdegree is greater than the indegree. Let
O
O
O
be a fixed vertex of
G
G
G
. For an arbitrary positive number
n
n
n
, let
V
n
V_{n}
V
n
be the number of vertices which can be reached from
O
O
O
passing through at most
n
n
n
edges (
O
O
O
counts). Find the smallest possible value of
V
n
V_{n}
V
n
.
2
2
Hide problems
Geometric inequality
The point
P
P
P
lies inside, or on the boundary of, the triangle
A
B
C
ABC
A
BC
. Denote by
d
a
d_{a}
d
a
,
d
b
d_{b}
d
b
and
d
c
d_{c}
d
c
the distances between
P
P
P
and
B
C
BC
BC
,
C
A
CA
C
A
, and
A
B
AB
A
B
, respectively. Prove that
max
{
A
P
,
B
P
,
C
P
}
≥
d
a
2
+
d
b
2
+
d
c
2
\max\{AP,BP,CP \} \ge \sqrt{d_{a}^{2}+d_{b}^{2}+d_{c}^{2}}
max
{
A
P
,
BP
,
CP
}
≥
d
a
2
+
d
b
2
+
d
c
2
. When does the equality holds?
Proving collinearity
In the triangle
A
B
C
ABC
A
BC
,
A
M
AM
A
M
is median,
M
∈
B
C
M \in BC
M
∈
BC
,
B
B
1
BB_{1}
B
B
1
and
C
C
1
CC_{1}
C
C
1
are altitudes,
C
1
∈
A
B
C_{1} \in AB
C
1
∈
A
B
,
B
1
∈
A
C
B_{1} \in AC
B
1
∈
A
C
. The line through
A
A
A
which is perpendicular to
A
M
AM
A
M
cuts the lines
B
B
1
BB_{1}
B
B
1
and
C
C
1
CC_{1}
C
C
1
at points
E
E
E
and
F
F
F
, respectively. Let
k
k
k
be the circumcircle of
△
E
F
M
\triangle EFM
△
EFM
. Suppose also that
k
1
k_{1}
k
1
and
k
2
k_{2}
k
2
are circles touching both
E
F
EF
EF
and the arc
E
F
EF
EF
of
k
k
k
which does not contain
M
M
M
. If
P
P
P
and
Q
Q
Q
are the points at which
k
1
k_{1}
k
1
intersects
k
2
k_{2}
k
2
, prove that
P
P
P
,
Q
Q
Q
, and
M
M
M
are collinear.
1
2
Hide problems
Square table and a pawn
Let
n
n
n
be a positive integer. There is a pawn in one of the cells of an
n
×
n
n\times n
n
×
n
table. The pawn moves from an arbitrary cell of the
k
k
k
th column,
k
∈
{
1
,
2
,
⋯
,
n
}
k \in \{1,2, \cdots, n \}
k
∈
{
1
,
2
,
⋯
,
n
}
, to an arbitrary cell in the
k
k
k
th row. Prove that there exists a sequence of
n
2
n^{2}
n
2
moves such that the pawn goes through every cell of the table and finishes in the starting cell.
Determine whether the number is rational
For each positive integer
n
n
n
, denote by
a
n
a_{n}
a
n
the first digit of
2
n
2^{n}
2
n
(base ten). Is the number
0.
a
1
a
2
a
3
⋯
0.a_{1}a_{2}a_{3}\cdots
0.
a
1
a
2
a
3
⋯
rational?