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National and Regional Contests
Bosnia Herzegovina Contests
Junior Regional - Federation of Bosnia Herzegovina
2014 Junior Regional Olympiad - FBH
2014 Junior Regional Olympiad - FBH
Part of
Junior Regional - Federation of Bosnia Herzegovina
Subcontests
(5)
5
3
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Junior Regional Olympiad - FBH 2014 Grade 6 Problem 5
Let
A
B
C
D
E
F
ABCDEF
A
BC
D
EF
be a hexagon. Sides and diagonals of hexagon are colored in two colors: blue and yellow. Prove that there exist a triangle with vertices from set
{
A
,
B
,
C
,
D
,
E
,
F
}
\{A,B,C,D,E,F\}
{
A
,
B
,
C
,
D
,
E
,
F
}
which sides are all same colour
Junior Regional Olympiad - FBH 2014 Grade 7 Problem 5
From digits
0
0
0
,
1
1
1
,
3
3
3
,
4
4
4
,
7
7
7
and
9
9
9
were written
5
5
5
digit numbers which all digits are different. How many numbers from them are divisible with
5
5
5
Junior Regional Olympiad - FBH 2014 Grade 8 Problem 5
How many are there
4
4
4
digit numbers such that they have two odd digits and two even digits
4
3
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Junior Regional Olympiad - FBH 2014 Grade 6 Problem 4
Find all prime numbers
p
p
p
and
q
q
q
such that
3
p
2
q
+
2
p
q
2
=
483
3p^2q+2pq^2=483
3
p
2
q
+
2
p
q
2
=
483
Junior Regional Olympiad - FBH 2014 Grade 7 Problem 4
Positive integer
n
n
n
when divided with number
3
3
3
gives remainder
a
a
a
, when divided with
5
5
5
has remainder
b
b
b
and when divided with
7
7
7
gives remainder
c
c
c
. Find remainder when dividing number
n
n
n
with
105
105
105
if
4
a
+
3
b
+
2
c
=
30
4a+3b+2c=30
4
a
+
3
b
+
2
c
=
30
Junior Regional Olympiad - FBH 2014 Grade 8 Problem 4
Find all prime numbers
p
p
p
and
q
q
q
such that
(
2
p
−
q
)
2
=
17
p
−
10
q
(2p-q)^2=17p-10q
(
2
p
−
q
)
2
=
17
p
−
10
q
3
3
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Junior Regional Olympiad - FBH 2014 Grade 6 Problem 3
Let
A
B
C
ABC
A
BC
be a right angled triangle. Prove that angle bisector of right angle is simultaneously an angle bisector of angle between median and altitude to hypotenuse.
Junior Regional Olympiad - FBH 2014 Grade 7 Problem 3
Let
A
B
C
D
ABCD
A
BC
D
be a trapezoid with base sides
A
B
AB
A
B
and
C
D
CD
C
D
and let
A
B
=
a
AB=a
A
B
=
a
,
B
C
=
b
BC=b
BC
=
b
,
C
D
=
c
CD=c
C
D
=
c
,
D
A
=
d
DA=d
D
A
=
d
,
A
C
=
m
AC=m
A
C
=
m
and
B
D
=
n
BD=n
B
D
=
n
. We know that
m
2
+
n
2
=
(
a
+
c
)
2
m^2+n^2=(a+c)^2
m
2
+
n
2
=
(
a
+
c
)
2
a
)
a)
a
)
Prove that lines
A
C
AC
A
C
and
B
D
BD
B
D
are perpendicular
b
)
b)
b
)
Prove that
a
c
<
b
d
ac<bd
a
c
<
b
d
Junior Regional Olympiad - FBH 2014 Grade 8 Problem 3
If
B
K
BK
B
K
is an angle bisector of
∠
A
B
C
\angle ABC
∠
A
BC
in triangle
A
B
C
ABC
A
BC
. Find angles of triangle
A
B
C
ABC
A
BC
if
B
K
=
K
C
=
2
A
K
BK=KC=2AK
B
K
=
K
C
=
2
A
K
2
3
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Junior Regional Olympiad - FBH 2014 Grade 6 Problem 2
In one class in the school, number of abscent students is
1
6
\frac{1}{6}
6
1
of number of students who were present. When teacher sent one student to bring chalk, number of abscent students was
1
5
\frac{1}{5}
5
1
of number of students who were present. How many students are in that class?
Junior Regional Olympiad - FBH 2014 Grade 7 Problem 2
We know that raw wheat has
70
%
70\%
70%
moisture and dry wheat has
10
%
10\%
10%
moisture. One miller bought
3
3
3
tons of raw wheat with price of
0.4
$
0.4 \$
0.4$
per kilo. At which price miller has to sell dry wheat, so he gets
80
%
80\%
80%
profit?
Junior Regional Olympiad - FBH 2014 Grade 8 Problem 2
Find value of
1
1
+
x
+
x
y
+
1
1
+
y
+
y
z
+
1
1
+
z
+
z
x
\frac{1}{1+x+xy}+\frac{1}{1+y+yz}+\frac{1}{1+z+zx}
1
+
x
+
x
y
1
+
1
+
y
+
yz
1
+
1
+
z
+
z
x
1
if
x
x
x
,
y
y
y
and
z
z
z
are real numbers usch that
x
y
z
=
1
xyz=1
x
yz
=
1
1
3
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Junior Regional Olympiad - FBH 2014 Grade 6 Problem 1
If
a
a
a
and
b
b
b
are digits, how many are there
4
4
4
digit numbers
3
a
b
4
‾
\overline{3ab4}
3
ab
4
divisible with
9
9
9
. Which numbers are they (
4
4
4
digit numbers)?
Junior Regional Olympiad - FBH 2014 Grade 7 Problem 1
If for numbers
a
a
a
,
b
b
b
and
c
c
c
holds
a
:
b
=
4
:
3
a : b=4:3
a
:
b
=
4
:
3
and
b
:
c
=
2
:
5
b : c=2:5
b
:
c
=
2
:
5
, find the value
(
3
a
−
2
b
)
:
(
b
+
2
c
)
(3a-2b):(b+2c)
(
3
a
−
2
b
)
:
(
b
+
2
c
)
Junior Regional Olympiad - FBH 2014 Grade 8 Problem 1
Compare numbers
A
=
5
+
2
5
A=5+2\sqrt{5}
A
=
5
+
2
5
and
B
=
45
+
20
5
B=\sqrt{45+20\sqrt{5}}
B
=
45
+
20
5