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National and Regional Contests
Bosnia Herzegovina Contests
Bosnia And Herzegovina - Regional Olympiad
2009 Bosnia And Herzegovina - Regional Olympiad
2009 Bosnia And Herzegovina - Regional Olympiad
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Bosnia And Herzegovina - Regional Olympiad
Subcontests
(4)
4
3
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Regional Olympiad - FBH 2009 Grade 9 Problem 4
Let
C
C
C
be a circle with center
O
O
O
and radius
R
R
R
. From point
A
A
A
of circle
C
C
C
we construct a tangent
t
t
t
on circle
C
C
C
. We construct line
d
d
d
through point
O
O
O
whch intersects tangent
t
t
t
in point
M
M
M
and circle
C
C
C
in points
B
B
B
and
D
D
D
(
B
B
B
lies between points
O
O
O
and
M
M
M
). If
A
M
=
R
3
AM=R\sqrt{3}
A
M
=
R
3
, prove:
a
)
a)
a
)
Triangle
A
M
D
AMD
A
M
D
is isosceles
b
)
b)
b
)
Circumcenter of
A
M
D
AMD
A
M
D
lies on circle
C
C
C
Regional Olympiad - FBH 2009 Grade 10 Problem 4
Let
x
x
x
and
y
y
y
be positive integers such that
x
2
−
1
y
+
1
+
y
2
−
1
x
+
1
\frac{x^2-1}{y+1}+\frac{y^2-1}{x+1}
y
+
1
x
2
−
1
+
x
+
1
y
2
−
1
is integer. Prove that numbers
x
2
−
1
y
+
1
\frac{x^2-1}{y+1}
y
+
1
x
2
−
1
and
y
2
−
1
x
+
1
\frac{y^2-1}{x+1}
x
+
1
y
2
−
1
are integers
Regional Olympiad - FBH 2009 Grade 11 Problem 4
What is the minimal value of
2
x
+
1
+
3
y
+
1
+
4
z
+
1
\sqrt{2x+1}+\sqrt{3y+1}+\sqrt{4z+1}
2
x
+
1
+
3
y
+
1
+
4
z
+
1
, if
x
x
x
,
y
y
y
and
z
z
z
are nonnegative real numbers such that
x
+
y
+
z
=
4
x+y+z=4
x
+
y
+
z
=
4
3
3
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Regional Olympiad - FBH 2009 Grade 9 Problem 3
Is it possible in a plane mark
10
10
10
red,
10
10
10
blue and
10
10
10
green points (all distinct) such that three conditions hold:
i
)
i)
i
)
For every red point
A
A
A
there exists a blue point closer to point
A
A
A
than any other green point
i
i
)
ii)
ii
)
For every blue point
B
B
B
there exists a green point closer to point
B
B
B
than any other red point
i
i
i
)
iii)
iii
)
For every green point
C
C
C
there exists a red point closer to point
C
C
C
than any other blue point
Regional Olympiad - FBH 2009 Grade 10 Problem 3
Decomposition of number
n
n
n
is showing
n
n
n
as a sum of positive integers (not neccessarily distinct). Order of addends is important. For every positive integer
n
n
n
show that number of decompositions is
2
n
−
1
2^{n-1}
2
n
−
1
Regional Olympiad - FBH 2009 Grade 11 Problem 3
There are
n
n
n
positive integers on the board. We can add only positive integers
c
=
a
+
b
a
−
b
c=\frac{a+b}{a-b}
c
=
a
−
b
a
+
b
, where
a
a
a
and
b
b
b
are numbers already writted on the board.
a
)
a)
a
)
Find minimal value of
n
n
n
, such that with adding numbers with described method, we can get any positive integer number written on the board
b
)
b)
b
)
For such
n
n
n
, find numbers written on the board at the beginning
2
4
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1
4
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