MathDB
Problems
Contests
National and Regional Contests
Bulgaria Contests
Bulgaria National Olympiad
2005 Bulgaria National Olympiad
2005 Bulgaria National Olympiad
Part of
Bulgaria National Olympiad
Subcontests
(6)
3
1
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Bulguaria 3
Let
M
=
(
0
,
1
)
∩
Q
M=(0,1)\cap \mathbb Q
M
=
(
0
,
1
)
∩
Q
. Determine, with proof, whether there exists a subset
A
⊂
M
A\subset M
A
⊂
M
with the property that every number in
M
M
M
can be uniquely written as the sum of finitely many distinct elements of
A
A
A
.
4
1
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Bulgaria 4
Let
A
B
C
ABC
A
BC
be a triangle with
A
C
≠
B
C
AC\neq BC
A
C
=
BC
, and let
A
′
B
′
C
A^{\prime }B^{\prime }C
A
′
B
′
C
be a triangle obtained from
A
B
C
ABC
A
BC
after some rotation centered at
C
C
C
. Let
M
,
E
,
F
M,E,F
M
,
E
,
F
be the midpoints of the segments
B
A
′
,
A
C
BA^{\prime },AC
B
A
′
,
A
C
and
C
B
′
CB^{\prime }
C
B
′
respectively. If
E
M
=
F
M
EM=FM
EM
=
FM
, find
E
M
F
^
\widehat{EMF}
EMF
.
6
1
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Bulgaria 6
Let
a
,
b
a,b
a
,
b
and
c
c
c
be positive integers such that
a
b
ab
ab
divides
c
(
c
2
−
c
+
1
)
c(c^{2}-c+1)
c
(
c
2
−
c
+
1
)
and
a
+
b
a+b
a
+
b
is divisible by
c
2
+
1
c^{2}+1
c
2
+
1
. Prove that the sets
{
a
,
b
}
\{a,b\}
{
a
,
b
}
and
{
c
,
c
2
−
c
+
1
}
\{c,c^{2}-c+1\}
{
c
,
c
2
−
c
+
1
}
coincide.
2
1
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Bulgaria 2
Consider two circles
k
1
,
k
2
k_{1},k_{2}
k
1
,
k
2
touching externally at point
T
T
T
. a line touches
k
2
k_{2}
k
2
at point
X
X
X
and intersects
k
1
k_{1}
k
1
at points
A
A
A
and
B
B
B
. Let
S
S
S
be the second intersection point of
k
1
k_{1}
k
1
with the line
X
T
XT
XT
. On the arc
T
S
^
\widehat{TS}
TS
not containing
A
A
A
and
B
B
B
is chosen a point
C
C
C
. Let
C
Y
\ CY
C
Y
be the tangent line to
k
2
k_{2}
k
2
with
Y
∈
k
2
Y\in k_{2}
Y
∈
k
2
, such that the segment
C
Y
CY
C
Y
does not intersect the segment
S
T
ST
ST
. If
I
=
X
Y
∩
S
C
I=XY\cap SC
I
=
X
Y
∩
SC
. Prove that : (a) the points
C
,
T
,
Y
,
I
C,T,Y,I
C
,
T
,
Y
,
I
are concyclic. (b)
I
I
I
is the excenter of triangle
A
B
C
ABC
A
BC
with respect to the side
B
C
BC
BC
.
5
1
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Bulgaria 5
For positive integers
t
,
a
,
b
,
t,a,b,
t
,
a
,
b
,
a
(
t
,
a
,
b
)
(t,a,b)
(
t
,
a
,
b
)
-game is a two player game defined by the following rules. Initially, the number
t
t
t
is written on a blackboard. At his first move, the 1st player replaces
t
t
t
with either
t
−
a
t-a
t
−
a
or
t
−
b
t-b
t
−
b
. Then, the 2nd player subtracts either
a
a
a
or
b
b
b
from this number, and writes the result on the blackboard, erasing the old number. After this, the first player once again erases either
a
a
a
or
b
b
b
from the number written on the blackboard, and so on. The player who first reaches a negative number loses the game. Prove that there exist infinitely many values of
t
t
t
for which the first player has a winning strategy for all pairs
(
a
,
b
)
(a,b)
(
a
,
b
)
with
a
+
b
=
2005
a+b=2005
a
+
b
=
2005
.
1
1
Hide problems
Bulguaria 1
Determine all triples
(
x
,
y
,
z
)
\left( x,y,z\right)
(
x
,
y
,
z
)
of positive integers for which the number
2005
x
+
y
+
2005
y
+
z
+
2005
z
+
x
\sqrt{\frac{2005}{x+y}}+\sqrt{\frac{2005}{y+z}}+\sqrt{\frac{2005}{z+x}}
x
+
y
2005
+
y
+
z
2005
+
z
+
x
2005
is an integer .