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National and Regional Contests
Bosnia Herzegovina Contests
Junior Regional - Federation of Bosnia Herzegovina
2018 Junior Regional Olympiad - FBH
2018 Junior Regional Olympiad - FBH
Part of
Junior Regional - Federation of Bosnia Herzegovina
Subcontests
(5)
5
3
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Junior Regional Olympiad - FBH 2018 Grade 6 Problem 5
In triangle
A
B
C
ABC
A
BC
length of altitude
C
H
CH
C
H
, with
H
∈
A
B
H \in AB
H
∈
A
B
, is equal to half of side
A
B
AB
A
B
. If
∠
B
A
C
=
4
5
∘
\angle BAC = 45^{\circ}
∠
B
A
C
=
4
5
∘
find
∠
A
B
C
\angle ABC
∠
A
BC
Junior Regional Olympiad - FBH 2018 Grade 7 Problem 5
Find all integers
x
x
x
and
y
y
y
such that
2
x
+
1
=
y
2
2^x+1=y^2
2
x
+
1
=
y
2
Junior Regional Olympiad - FBH 2018 Grade 8 Problem 5
It is given square
A
B
C
D
ABCD
A
BC
D
which is circumscribed by circle
k
k
k
. Let us construct a new square so vertices
E
E
E
and
F
F
F
lie on side
A
B
C
D
ABCD
A
BC
D
and vertices
G
G
G
and
H
H
H
on arc
A
B
AB
A
B
of circumcircle. Find out the ratio of area of squares
4
3
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Junior Regional Olympiad - FBH 2018 Grade 6 Problem 4
Determine the last digit of number
1
8
1
+
1
8
2
+
.
.
.
+
1
8
19
+
1
8
20
18^1+18^2+...+18^{19}+18^{20}
1
8
1
+
1
8
2
+
...
+
1
8
19
+
1
8
20
Junior Regional Olympiad - FBH 2018 Grade 7 Problem 4
It is given
4
4
4
circles in a plane and every one of them touches the other three as shown: https://services.artofproblemsolving.com/download.php?id=YXR0YWNobWVudHMvZC82L2FkYWQ5NThhMWRiMjAwZjYxOWFhYmE1M2YzZDU5YWI2N2IyYzk2LnBuZw==&rn=a3J1Z292aS5wbmc= Biggest circle has radius
2
2
2
, and every one of the medium has
1
1
1
. Find out the radius of fourth circle.
Junior Regional Olympiad - FBH 2018 Grade 8 Problem 4
Let
a
a
a
,
b
b
b
and
c
c
c
be positive real numbers such that
a
≥
b
≥
c
a \geq b \geq c
a
≥
b
≥
c
. Prove the inequality:
a
b
+
b
c
+
c
a
≤
b
a
+
c
b
+
a
c
\frac{a}{b}+\frac{b}{c}+\frac{c}{a} \leq \frac{b}{a}+\frac{c}{b}+\frac{a}{c}
b
a
+
c
b
+
a
c
≤
a
b
+
b
c
+
c
a
3
3
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Junior Regional Olympiad - FBH 2018 Grade 7 Problem 3
Let
a
a
a
,
b
b
b
and
m
m
m
be three positive real numbers and
a
>
b
a>b
a
>
b
. Which of the numbers
A
=
a
+
m
−
a
A=\sqrt{a+m}-\sqrt{a}
A
=
a
+
m
−
a
and
B
=
b
+
m
−
b
B=\sqrt{b+m}-\sqrt{b}
B
=
b
+
m
−
b
is bigger:
Junior Regional Olympiad - FBH 2018 Grade 6 Problem 3
In some primary school there were
94
94
94
students in
7
7
7
th grade. Some students are involved in extracurricular activities: spanish and german language and sports. Spanish language studies
40
40
40
students outside school program, german
27
27
27
students and
60
60
60
students do sports. Out of the students doing sports,
24
24
24
of them also goes to spanish language.
10
10
10
students who study spanish also study german.
12
12
12
students who study german also do sports. Only
4
4
4
students go to all three activities. How many of them does only one of the activities, and how much of them do not go to any activity?
Junior Regional Olympiad - FBH 2018 Grade 8 Problem 3
Find all
4
4
4
digit number
a
b
c
d
‾
\overline{abcd}
ab
c
d
such that
4
⋅
a
b
c
d
‾
+
30
=
d
c
b
a
‾
4\cdot \overline{abcd}+30=\overline{dcba}
4
⋅
ab
c
d
+
30
=
d
c
ba
2
3
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Junior Regional Olympiad - FBH 2018 Grade 6 Problem 2
Find all integers
n
n
n
such that
n
+
4
3
n
−
2
\frac{n+4}{3n-2}
3
n
−
2
n
+
4
is integer
Junior Regional Olympiad - FBH 2018 Grade 7 Problem 2
On blackboard is written
3
3
3
digit number so all three digits are distinct than zero. Out of it, we made three
2
2
2
digit numbers by crossing out first digit of original number, crossing out second digit of original number and crossing out third digit of original number. Sum of those three numbers is
293
293
293
. Which number is written on blackboard?
Junior Regional Olympiad - FBH 2018 Grade 8 Problem 2
Find all integers
m
m
m
such that
2
m
2
+
7
m
−
9
m
2
+
m
+
1
\frac{2m^2+7m-9}{m^2+m+1}
m
2
+
m
+
1
2
m
2
+
7
m
−
9
is integer
1
3
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Junior Regional Olympiad - FBH 2018 Grade 6 Problem 1
Four buddies bought a ball. First one paid half of the ball price. Second one gave one third of money that other three gave. Third one paid a quarter of sum paid by other three. Fourth paid
5
$
5\$
5$
. How much did the ball cost?
Junior Regional Olympiad - FBH 2018 Grade 7 Problem 1
Price of some item has decreased by
5
%
5\%
5%
. Then price increased by
40
%
40\%
40%
and now it is
1352.06
$
1352.06\$
1352.06$
cheaper than doubled original price. How much did the item originally cost?
Junior Regional Olympiad - FBH 2018 Grade 8 Problem 1
When askes: "What time is it?", father said to a son: "Quarter of time that passed and half of the remaining time gives the exact time". What time was it?