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National and Regional Contests
Bosnia Herzegovina Contests
Junior Regional - Federation of Bosnia Herzegovina
2015 Junior Regional Olympiad - FBH
2015 Junior Regional Olympiad - FBH
Part of
Junior Regional - Federation of Bosnia Herzegovina
Subcontests
(5)
5
3
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Junior Regional Olympiad - FBH 2015 Grade 6 Problem 5
In how many ways you can pay
2015
$
2015\$
2015$
using bills of
1
$
1\$
1$
,
10
$
10\$
10$
,
100
$
100\$
100$
and
200
$
200\$
200$
Junior Regional Olympiad - FBH 2015 Grade 7 Problem 5
It is given
2015
2015
2015
numbers such that every one of them when gets replaced with sum of the rest, we get same
2015
2015
2015
same numbers. Prove that product of all numbers is
0
0
0
Junior Regional Olympiad - FBH 2015 Grade 8 Problem 5
Prove that for every parititon of set
X
=
{
1
,
2
,
.
.
.
,
9
}
X=\{1,2,...,9\}
X
=
{
1
,
2
,
...
,
9
}
on two disjoint sets at least one of them contains three elements such that sum of some two of them is equal to third
4
3
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Junior Regional Olympiad - FBH 2015 Grade 6 Problem 4
Which number we need to substract from numerator and add to denominator of
28
a
3
‾
7276
\frac{\overline{28a3}}{7276}
7276
28
a
3
such that we get fraction equal to
2
7
\frac{2}{7}
7
2
Junior Regional Olympiad - FBH 2015 Grade 7 Problem 4
On the market one seller is selling watermelons, melons and young corn cobs. Total number of watermelons, melons and corn cobs is
239
239
239
. One buyer bought
2
3
\frac{2}{3}
3
2
of all watermelons,
3
5
\frac{3}{5}
5
3
of all melons and
5
7
\frac{5}{7}
7
5
of all corn cobs. Other buyer bought
1
13
\frac{1}{13}
13
1
of all watermelons,
1
4
\frac{1}{4}
4
1
of all melons and
1
5
\frac{1}{5}
5
1
of all corn cobs. How many pieces in total bought second buyer and how many seller had at the beggining of each watermelons, melons and corn cobs?
Junior Regional Olympiad - FBH 2015 Grade 8 Problem 4
Let
n
n
n
be a positive integer and
a
=
2
n
⋅
7
n
+
1
+
11
a=2^n\cdot 7^{n+1}+11
a
=
2
n
⋅
7
n
+
1
+
11
and
b
=
2
n
+
1
⋅
7
n
+
3
b=2^{n+1}\cdot 7^n+3
b
=
2
n
+
1
⋅
7
n
+
3
.
a
)
a)
a
)
Prove that fraction
a
b
\frac{a}{b}
b
a
is irreducible
b
)
b)
b
)
Prove that number
a
+
b
−
7
a+b-7
a
+
b
−
7
is not a perfect square for any positive integer
n
n
n
3
3
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Junior Regional Olympiad - FBH 2015 Grade 6 Problem 3
Let
A
D
AD
A
D
be a median of
A
B
C
ABC
A
BC
and
S
S
S
its midpoint. Let
E
E
E
be a intersection point of
A
B
AB
A
B
and
C
S
CS
CS
. Prove that
B
E
=
2
A
E
BE=2AE
BE
=
2
A
E
Junior Regional Olympiad - FBH 2015 Grade 7 Problem 3
Find the area of quadrilateral
A
B
C
D
ABCD
A
BC
D
if: two opposite angles are right;two sides which form right angle are of equal length and sum of lengths of other two sides is
10
10
10
Junior Regional Olympiad - FBH 2015 Grade 8 Problem 3
Let
D
D
D
be a midpoint of
B
C
BC
BC
of triangle
A
B
C
ABC
A
BC
. On side
A
B
AB
A
B
is given point
E
E
E
, and on side
A
C
AC
A
C
is given point
F
F
F
such that
∠
E
D
F
=
9
0
∘
\angle EDF = 90^{\circ}
∠
E
D
F
=
9
0
∘
. Prove that
B
E
+
C
F
>
E
F
BE+CF>EF
BE
+
CF
>
EF
2
3
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Junior Regional Olympiad - FBH 2015 Grade 6 Problem 2
One day students in school organised a exchange between them such that :
11
11
11
strawberries change for
14
14
14
raspberries,
22
22
22
cherries change for
21
21
21
raspberries,
10
10
10
cherries change for
3
3
3
bananas and
5
5
5
pears for
2
2
2
bananas. How many pears has Amila to give to get
7
7
7
strawberries
Junior Regional Olympiad - FBH 2015 Grade 7 Problem 2
Seller reduced price of one shirt for
20
%
20\%
20%
,and they raised it by
10
%
10\%
10%
. Does he needs to reduce or raise the price and how many, so that price of shirt will be reduced by
10
%
10\%
10%
from the original price
Junior Regional Olympiad - FBH 2015 Grade 8 Problem 2
Show tha value
A
=
(
b
−
c
)
2
(
a
−
b
)
(
a
−
c
)
+
(
c
−
a
)
2
(
b
−
c
)
(
b
−
a
)
+
(
a
−
b
)
2
(
c
−
a
)
(
c
−
b
)
A=\frac{(b-c)^2}{(a-b)(a-c)}+\frac{(c-a)^2}{(b-c)(b-a)}+\frac{(a-b)^2}{(c-a)(c-b)}
A
=
(
a
−
b
)
(
a
−
c
)
(
b
−
c
)
2
+
(
b
−
c
)
(
b
−
a
)
(
c
−
a
)
2
+
(
c
−
a
)
(
c
−
b
)
(
a
−
b
)
2
does not depend on values of
a
a
a
,
b
b
b
and
c
c
c
1
3
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Junior Regional Olympiad - FBH 2015 Grade 6 Problem 1
Find two angles which add to
18
0
∘
180^{\circ}
18
0
∘
which difference is
1
′
1^{'}
1
′
Junior Regional Olympiad - FBH 2015 Grade 7 Problem 1
Every one of the six trucks of construction company drove for
8
8
8
hours and they all together spent
720
720
720
litres of oil. How many litres should
9
9
9
trucks spend, if every one of them drives for
6
6
6
hours?
Junior Regional Olympiad - FBH 2015 Grade 8 Problem 1
Father is
42
42
42
years old, and son has
14
14
14
years. In how many years father will be twice as old as his son?