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Bosnia Herzegovina Contests
Bosnia And Herzegovina - Regional Olympiad
2012 Bosnia And Herzegovina - Regional Olympiad
2012 Bosnia And Herzegovina - Regional Olympiad
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Bosnia And Herzegovina - Regional Olympiad
Subcontests
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4
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Regional Olympiad - FBH 2012 Grade 10 Problem 3
Quadrilateral
A
B
C
D
ABCD
A
BC
D
is cyclic. Line through point
D
D
D
parallel with line
B
C
BC
BC
intersects
C
A
CA
C
A
in point
P
P
P
, line
A
B
AB
A
B
in point
Q
Q
Q
and circumcircle of
A
B
C
D
ABCD
A
BC
D
in point
R
R
R
. Line through point
D
D
D
parallel with line
A
B
AB
A
B
intersects
A
C
AC
A
C
in point
S
S
S
, line
B
C
BC
BC
in point
T
T
T
and circumcircle of
A
B
C
D
ABCD
A
BC
D
in point
U
U
U
. If
P
Q
=
Q
R
PQ=QR
PQ
=
QR
, prove that
S
T
=
T
U
ST=TU
ST
=
T
U
Regional Olympiad - FBH 2012 Grade 9 Problem 3
Find remainder when dividing upon
2012
2012
2012
number
A
=
1
⋅
2
+
2
⋅
3
+
3
⋅
4
+
.
.
.
+
2009
⋅
2010
+
2010
⋅
2011
A=1\cdot2+2\cdot3+3\cdot4+...+2009\cdot2010+2010\cdot2011
A
=
1
⋅
2
+
2
⋅
3
+
3
⋅
4
+
...
+
2009
⋅
2010
+
2010
⋅
2011
Regional Olympiad - FBH 2012 Grade 11 Problem 3
Prove tha number
19
⋅
8
n
+
17
19 \cdot 8^n +17
19
⋅
8
n
+
17
is composite for every positive integer
n
n
n
2
4
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1
3
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Regional Olympiad - FBH 2012 Grade 9 Problem 1
Find all possible values of
1
a
(
1
b
+
1
c
+
1
b
+
c
)
+
1
b
(
1
c
+
1
a
+
1
c
+
a
)
+
1
c
(
1
a
+
1
b
+
1
a
+
b
)
−
1
a
+
b
+
c
(
1
a
+
1
b
+
1
c
+
1
a
+
b
+
1
b
+
c
+
1
c
+
a
)
+
1
a
2
+
1
b
2
+
1
c
2
\frac{1}{a}\left(\frac{1}{b}+\frac{1}{c}+\frac{1}{b+c}\right)+\frac{1}{b}\left(\frac{1}{c}+\frac{1}{a}+\frac{1}{c+a}\right)+\frac{1}{c}\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{a+b}\right)-\frac{1}{a+b+c}\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{a+b}+\frac{1}{b+c}+\frac{1}{c+a}\right)+\frac{1}{a^2}+\frac{1}{b^2}+\frac{1}{c^2}
a
1
(
b
1
+
c
1
+
b
+
c
1
)
+
b
1
(
c
1
+
a
1
+
c
+
a
1
)
+
c
1
(
a
1
+
b
1
+
a
+
b
1
)
−
a
+
b
+
c
1
(
a
1
+
b
1
+
c
1
+
a
+
b
1
+
b
+
c
1
+
c
+
a
1
)
+
a
2
1
+
b
2
1
+
c
2
1
where
a
a
a
,
b
b
b
and
c
c
c
are positive real numbers such that
a
b
+
b
c
+
c
a
=
a
b
c
ab+bc+ca=abc
ab
+
b
c
+
c
a
=
ab
c
Regional Olympiad - FBH 2012 Grade 10 Problem 1
Solve equation
x
2
−
a
−
x
=
a
x^2-\sqrt{a-x}=a
x
2
−
a
−
x
=
a
where
x
x
x
is real number and
a
a
a
is real parameter
Regional Olympiad - FBH 2012 Grade 11 Problem 1
For which real numbers
x
x
x
and
α
\alpha
α
inequality holds:
log
2
x
+
log
x
2
+
2
cos
α
≤
0
\log _2 {x}+\log _x {2}+2\cos{\alpha} \leq 0
lo
g
2
x
+
lo
g
x
2
+
2
cos
α
≤
0