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Bosnia Herzegovina Contests
Bosnia And Herzegovina - Regional Olympiad
2011 Bosnia And Herzegovina - Regional Olympiad
2011 Bosnia And Herzegovina - Regional Olympiad
Part of
Bosnia And Herzegovina - Regional Olympiad
Subcontests
(4)
4
3
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Regional Olympiad - FBH 2011 Grade 9 Problem 4
For positive integer
n
n
n
, prove that at least one of the numbers
A
=
2
n
−
1
,
B
=
5
n
−
1
,
C
=
13
n
−
1
A=2n-1 , B=5n-1, C=13n-1
A
=
2
n
−
1
,
B
=
5
n
−
1
,
C
=
13
n
−
1
is not perfect square
Regional Olympiad - FBH 2011 Grade 10 Problem 4
Let
n
n
n
be a positive integer and set
S
=
{
n
,
n
+
1
,
n
+
2
,
.
.
.
,
5
n
}
S=\{n,n+1,n+2,...,5n\}
S
=
{
n
,
n
+
1
,
n
+
2
,
...
,
5
n
}
a
)
a)
a
)
If set
S
S
S
is divided into two disjoint sets , prove that there exist three numbers
x
x
x
,
y
y
y
and
z
z
z
(possibly equal) which belong to same subset of
S
S
S
and
x
+
y
=
z
x+y=z
x
+
y
=
z
b
)
b)
b
)
Does
a
)
a)
a
)
hold for set
S
=
{
n
,
n
+
1
,
n
+
2
,
.
.
.
,
5
n
−
1
}
S=\{n,n+1,n+2,...,5n-1\}
S
=
{
n
,
n
+
1
,
n
+
2
,
...
,
5
n
−
1
}
Regional Olympiad - FBH 2011 Grade 11 Problem 4
Prove that among any
6
6
6
irrational numbers you can pick three numbers
a
a
a
,
b
b
b
and
c
c
c
such that numbers
a
+
b
a+b
a
+
b
,
b
+
c
b+c
b
+
c
and
c
+
a
c+a
c
+
a
are irrational
3
4
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2
4
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1
3
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Regional Olympiad - FBH 2011 Grade 9 Problem 1
Factorise
(
a
+
2
b
−
3
c
)
3
+
(
b
+
2
c
−
3
a
)
3
+
(
c
+
2
a
−
3
b
)
3
(a+2b-3c)^3+(b+2c-3a)^3+(c+2a-3b)^3
(
a
+
2
b
−
3
c
)
3
+
(
b
+
2
c
−
3
a
)
3
+
(
c
+
2
a
−
3
b
)
3
Regional Olympiad - FBH 2011 Grade 10 Problem 1
Find the real number coefficient
c
c
c
of polynomial
x
2
+
x
+
c
x^2+x+c
x
2
+
x
+
c
, if his roots
x
1
x_1
x
1
and
x
2
x_2
x
2
satisfy following:
2
x
1
3
2
+
x
2
+
2
x
2
3
2
+
x
1
=
−
1
\frac{2x_1^3}{2+x_2}+\frac{2x_2^3}{2+x_1}=-1
2
+
x
2
2
x
1
3
+
2
+
x
1
2
x
2
3
=
−
1
Regional Olympiad - FBH 2011 Grade 11 Problem 1
Determine value of real parameter
λ
\lambda
λ
such that equation
1
sin
x
+
1
cos
x
=
λ
\frac{1}{\sin{x}} + \frac{1}{\cos{x}} = \lambda
sin
x
1
+
cos
x
1
=
λ
has root in interval
(
0
,
π
2
)
\left(0,\frac{\pi}{2}\right)
(
0
,
2
π
)