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National and Regional Contests
Bosnia Herzegovina Contests
Junior Regional - Federation of Bosnia Herzegovina
2016 Junior Regional Olympiad - FBH
2016 Junior Regional Olympiad - FBH
Part of
Junior Regional - Federation of Bosnia Herzegovina
Subcontests
(5)
5
3
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Junior Regional Olympiad - FBH 2016 Grade 6 Problem 5
Pages of some book are numerated with numbers
1
1
1
to
100
100
100
. From the book several double pages were ripped out and sum of enumerations of that pages is equal to
4949
4949
4949
. How many double pages were ripped out?
Junior Regional Olympiad - FBH 2016 Grade 7 Problem 5
In table https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvZC9hLzBjNjFlZWFjM2ZlOTQzMTk2YTdkMzQ2MjJiYzYyMWFlN2Y0ZGZlLnBuZw==&rn=dGFibGljYWEucG5n
10
10
10
numbers are circled, in every row and every column exactly one. Prove that among them, there are at least two equal
Junior Regional Olympiad - FBH 2016 Grade 8 Problem 5
605
605
605
spheres of same radius are divided in two parts. From one part, upright "pyramid" is made with square base. From the other part, upright "pyramid" is made with equilateral triangle base. Both "pyramids" are put together from equal numbers of sphere rows. Find number of spheres in every "pyramid"
4
3
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Junior Regional Olympiad - FBH 2016 Grade 6 Problem 4
Let
C
C
C
and
D
D
D
be points inside angle
∠
A
O
B
\angle AOB
∠
A
OB
such that
5
∠
C
O
D
=
4
∠
A
O
C
5\angle COD = 4\angle AOC
5∠
CO
D
=
4∠
A
OC
and
3
∠
C
O
D
=
2
∠
D
O
B
3\angle COD = 2\angle DOB
3∠
CO
D
=
2∠
D
OB
. If
∠
A
O
B
=
10
5
∘
\angle AOB = 105^{\circ}
∠
A
OB
=
10
5
∘
, find
∠
C
O
D
\angle COD
∠
CO
D
Junior Regional Olympiad - FBH 2016 Grade 7 Problem 4
In right angled triangle
A
B
C
ABC
A
BC
point
D
D
D
is midpoint of hypotenuse, and
E
E
E
and
F
F
F
are points on shorter sides
A
C
AC
A
C
and
B
C
BC
BC
, respectively, such that
D
E
⊥
D
F
DE \perp DF
D
E
⊥
D
F
. Prove that
E
F
2
=
A
E
2
+
B
F
2
EF^2=AE^2+BF^2
E
F
2
=
A
E
2
+
B
F
2
Junior Regional Olympiad - FBH 2016 Grade 8 Problem 4
In set of positive integers solve the equation
x
3
+
x
2
y
+
x
y
2
+
y
3
=
8
(
x
2
+
x
y
+
y
2
+
1
)
x^3+x^2y+xy^2+y^3=8(x^2+xy+y^2+1)
x
3
+
x
2
y
+
x
y
2
+
y
3
=
8
(
x
2
+
x
y
+
y
2
+
1
)
3
3
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Junior Regional Olympiad - FBH 2016 Grade 6 Problem 3
From three boys and three girls, every boy knows exactly two girls and every girl knows exactly two boys. Prove that we can arrange boys and girls in pairs such that in every pair people know each other
Junior Regional Olympiad - FBH 2016 Grade 7 Problem 3
Prove that when dividing a prime number with
30
30
30
, remainder is always not a composite number
Junior Regional Olympiad - FBH 2016 Grade 8 Problem 3
In trapezoid
A
B
C
D
ABCD
A
BC
D
holds
A
D
∣
∣
B
C
AD \mid \mid BC
A
D
∣∣
BC
,
∠
A
B
C
=
3
0
∘
\angle ABC = 30^{\circ}
∠
A
BC
=
3
0
∘
,
∠
B
C
D
=
6
0
∘
\angle BCD = 60^{\circ}
∠
BC
D
=
6
0
∘
and
B
C
=
7
BC=7
BC
=
7
. Let
E
E
E
,
M
M
M
,
F
F
F
and
N
N
N
be midpoints of sides
A
B
AB
A
B
,
B
C
BC
BC
,
C
D
CD
C
D
and
D
A
DA
D
A
, respectively. If
M
N
=
3
MN=3
MN
=
3
, find
E
F
EF
EF
2
3
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Junior Regional Olympiad - FBH 2016 Grade 6 Problem 2
Which fraction is bigger:
5553
5557
\frac{5553}{5557}
5557
5553
or
6664
6669
\frac{6664}{6669}
6669
6664
?
Junior Regional Olympiad - FBH 2016 Grade 7 Problem 2
If
w
=
1
+
−
3
+
2
3
−
1
−
−
3
+
2
3
w=\sqrt{1+\sqrt{-3+2\sqrt{3}}}-\sqrt{1-\sqrt{-3+2\sqrt{3}}}
w
=
1
+
−
3
+
2
3
−
1
−
−
3
+
2
3
prove that
w
=
3
−
1
w=\sqrt{3}-1
w
=
3
−
1
Junior Regional Olympiad - FBH 2016 Grade 8 Problem 2
Find set of positive integers divisible with
8
8
8
which sum of digits is
7
7
7
and product is
6
6
6
1
3
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Junior Regional Olympiad - FBH 2016 Grade 6 Problem 1
Find unknown digits
a
a
a
and
b
b
b
such that number
a
783
b
‾
\overline{a783b}
a
783
b
is divisible with
56
56
56
Junior Regional Olympiad - FBH 2016 Grade 7 Problem 1
One company from Tesanj has last year produced profit for
112
%
112 \%
112%
of expected one . Determine how many percents expected profit is from produced one
Junior Regional Olympiad - FBH 2016 Grade 8 Problem 1
If
a
>
b
>
c
a>b>c
a
>
b
>
c
are real numbers prove that
1
a
−
b
+
1
b
−
c
>
2
a
−
c
\frac{1}{a-b}+\frac{1}{b-c}>\frac{2}{a-c}
a
−
b
1
+
b
−
c
1
>
a
−
c
2