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National and Regional Contests
Bosnia Herzegovina Contests
Junior Regional - Federation of Bosnia Herzegovina
2017 Junior Regional Olympiad - FBH
2017 Junior Regional Olympiad - FBH
Part of
Junior Regional - Federation of Bosnia Herzegovina
Subcontests
(5)
5
3
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Junior Regional Olympiad - FBH 2017 Grade 6 Problem 5
Fathers childhood lasted for one sixth part of his life, and he married one
8
8
8
th after that and he immediately left to army. When one
12
12
12
th of his life passed, father returned from the army and
5
5
5
years after he got a son. Son who lived for one half of fathers years, died
4
4
4
years before his father. How many years lived his father, and how many years he had when his son was born?
Junior Regional Olympiad - FBH 2017 Grade 7 Problem 5
Find all positive integers
a
a
a
and
b
b
b
such that number
p
=
2
+
a
3
+
b
p=\frac{\sqrt{2}+\sqrt{a}}{\sqrt{3}+\sqrt{b}}
p
=
3
+
b
2
+
a
is rational number
Junior Regional Olympiad - FBH 2017 Grade 8 Problem 5
Points
K
K
K
and
L
L
L
are on side
A
B
AB
A
B
of triangle
A
B
C
ABC
A
BC
such that
K
L
=
B
C
KL=BC
K
L
=
BC
and
A
K
=
L
B
AK=LB
A
K
=
L
B
. Let
M
M
M
be a midpoint of
A
C
AC
A
C
. Prove that
∠
K
M
L
=
9
0
∘
\angle KML = 90^{\circ}
∠
K
M
L
=
9
0
∘
4
3
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Junior Regional Olympiad - FBH 2017 Grade 7 Problem 4
Group of
27
27
27
climbers shared among themself
13
13
13
breads. Every man had
2
2
2
breads, every woman half of a bread, and every child
1
3
\frac{1}{3}
3
1
of a bread. How many men, women and children where there ?
Junior Regional Olympiad - FBH 2017 Grade 6 Problem 4
If we divide number
19250
19250
19250
with one number, we get remainder
11
11
11
. If we divide number
20302
20302
20302
with the same number, we get the reamainder
3
3
3
. Which number is that?
Junior Regional Olympiad - FBH 2017 Grade 8 Problem 4
Let
n
n
n
and
k
k
k
be positive integers for which we have
4
4
4
statements:
i
)
i)
i
)
n
+
1
n+1
n
+
1
is divisible with
k
k
k
i
i
)
ii)
ii
)
n
=
2
k
+
5
n=2k+5
n
=
2
k
+
5
i
i
i
)
iii)
iii
)
n
+
k
n+k
n
+
k
is divisible with
3
3
3
i
v
)
iv)
i
v
)
n
+
7
k
n+7k
n
+
7
k
is prime Determine all possible values for
n
n
n
and
k
k
k
, if out of the
4
4
4
statements, three of them are true and one is false
3
3
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Junior Regional Olympiad - FBH 2017 Grade 7 Problem 3
On blackboard there are
10
10
10
different positive integers which sum is equal to
62
62
62
. Prove that product of those numbers is divisible with
60
60
60
Junior Regional Olympiad - FBH 2017 Grade 6 Problem 3
In acute triangle
A
B
C
ABC
A
BC
holds
∠
B
A
C
=
8
0
∘
\angle BAC=80^{\circ}
∠
B
A
C
=
8
0
∘
, and altitudes
h
a
h_a
h
a
and
h
b
h_b
h
b
intersect in point
H
H
H
. if
∠
A
H
B
=
12
6
∘
\angle AHB = 126^{\circ}
∠
A
H
B
=
12
6
∘
, which side is the smallest, and which is the biggest in
A
B
C
ABC
A
BC
Junior Regional Olympiad - FBH 2017 Grade 8 Problem 3
Find all real numbers
x
x
x
such that:
x
−
7
2015
+
x
−
6
2016
+
x
−
5
2017
=
x
−
2015
7
+
x
−
2016
6
+
x
−
2017
5
\sqrt{\frac{x-7}{2015}}+\sqrt{\frac{x-6}{2016}}+\sqrt{\frac{x-5}{2017}}=\sqrt{\frac{x-2015}{7}}+\sqrt{\frac{x-2016}{6}}+\sqrt{\frac{x-2017}{5}}
2015
x
−
7
+
2016
x
−
6
+
2017
x
−
5
=
7
x
−
2015
+
6
x
−
2016
+
5
x
−
2017
2
3
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Junior Regional Olympiad - FBH 2017 Grade 6 Problem 2
In three cisterns of milk lies
780
780
780
litres of milk. When we pour off from first cistern quarter of milk, from second cistern fifth of milk and from third cistern
3
7
\frac{3}{7}
7
3
of milk, in all cisterns remain same amount of milk. How many milk is in cisterns?
Junior Regional Olympiad - FBH 2017 Grade 8 Problem 2
Square table
5
×
5
5 \times 5
5
×
5
is filled with numbers in a following way. https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvYi8zLzQ0Y2M1NjdiNjQ3NjhlYTAwMWQ0MTg2ZjIwZWE4NzkwYzcwYWFkLnBuZw==&rn=dGFiZWxpY2EucG5n We can change the table in a way we take two arbitrary numbers from the table and we decrease both of them with value of smaller of those two. Can we get to the table with all zeros?
Junior Regional Olympiad - FBH 2017 Grade 7 Problem 2
In quadrilateral
A
B
C
D
ABCD
A
BC
D
holds
A
B
=
6
AB=6
A
B
=
6
,
A
D
=
4
AD=4
A
D
=
4
,
∠
D
A
B
=
∠
A
B
C
=
6
0
∘
\angle DAB=\angle ABC = 60^{\circ}
∠
D
A
B
=
∠
A
BC
=
6
0
∘
and
∠
A
D
C
=
9
0
∘
\angle ADC = 90^{\circ}
∠
A
D
C
=
9
0
∘
. Find length of diagonals and area of the quadrilateral
1
3
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Junior Regional Olympiad - FBH 2017 Grade 6 Problem 1
Lamija and Faris are playing the following game. Cards, which are numerated from
1
1
1
to
100
100
100
, are placed one next to other, starting from
1
1
1
to
100
100
100
. Now Faris picks every
7
7
7
th card, and after that every card which contains number
7
7
7
. After that Lamija picks from remaining cards ones divisible with
5
5
5
, and after that cards which contain number
5
5
5
. Who will have more cards and how many ? How would game end, if Lamija started with "
5
5
5
rule" and Faris continues with "
7
7
7
rule"?
Junior Regional Olympiad - FBH 2017 Grade 7 Problem 1
Price of the book increased by
20
%
20\%
20%
, and then decreased by
10
%
10\%
10%
. How many percents should we decrease current price so we get a price which is
54
%
54\%
54%
percent of an original one?
Junior Regional Olympiad - FBH 2017 Grade 8 Problem 1
It is given function
f
(
x
)
=
3
x
−
2
f(x)=3x-2
f
(
x
)
=
3
x
−
2
a
)
a)
a
)
Find
g
(
x
)
g(x)
g
(
x
)
if
f
(
2
x
−
g
(
x
)
)
=
−
3
(
1
+
2
m
)
x
+
34
f(2x-g(x))=-3(1+2m)x+34
f
(
2
x
−
g
(
x
))
=
−
3
(
1
+
2
m
)
x
+
34
b
)
b)
b
)
Solve the equation:
g
(
x
)
=
4
(
m
−
1
)
x
−
4
(
m
+
1
)
g(x)=4(m-1)x-4(m+1)
g
(
x
)
=
4
(
m
−
1
)
x
−
4
(
m
+
1
)
,
m
∈
R
m \in \mathbb{R}
m
∈
R