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Bosnia And Herzegovina - Regional Olympiad
2018 Bosnia And Herzegovina - Regional Olympiad
2018 Bosnia And Herzegovina - Regional Olympiad
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Bosnia And Herzegovina - Regional Olympiad
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Regional Olympiad - FBH 2018 Grade 9 Problem 5
Let
H
H
H
be an orhocenter of an acute triangle
A
B
C
ABC
A
BC
and
M
M
M
midpoint of side
B
C
BC
BC
. If
D
D
D
and
E
E
E
are foots of perpendicular of
H
H
H
on internal and external angle bisector of angle
∠
B
A
C
\angle BAC
∠
B
A
C
, prove that
M
M
M
,
D
D
D
and
E
E
E
are collinear
Regional Olympiad - FBH 2018 Grade 10 Problem 5
Board with dimesions
2018
×
2018
2018 \times 2018
2018
×
2018
is divided in unit cells
1
×
1
1 \times 1
1
×
1
. In some cells of board are placed black chips and in some white chips (in every cell maximum is one chip). Firstly we remove all black chips from columns which contain white chips, and then we remove all white chips from rows which contain black chips. If
W
W
W
is number of remaining white chips, and
B
B
B
number of remaining black chips on board and
A
=
m
i
n
{
W
,
B
}
A=min\{W,B\}
A
=
min
{
W
,
B
}
, determine maximum of
A
A
A
Regional Olympiad - FBH 2018 Grade 11 Problem 5
It is given
2018
2018
2018
points in plane. Prove that it is possible to cover them with circles such that:
i
)
i)
i
)
sum of lengths of all diameters of all circles is not greater than
2018
2018
2018
i
i
)
ii)
ii
)
distance between any two circles is greater than
1
1
1
4
4
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3
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Regional Olympiad - FBH 2018 Grade 9 Problem 2
Determine all triplets
(
a
,
b
,
c
)
(a,b,c)
(
a
,
b
,
c
)
of real numbers such that sets
{
a
2
−
4
c
,
b
2
−
2
a
,
c
2
−
2
b
}
\{a^2-4c, b^2-2a, c^2-2b \}
{
a
2
−
4
c
,
b
2
−
2
a
,
c
2
−
2
b
}
and
{
a
−
c
,
b
−
4
c
,
a
+
b
}
\{a-c,b-4c,a+b\}
{
a
−
c
,
b
−
4
c
,
a
+
b
}
are equal and
2
a
+
2
b
+
6
=
5
c
2a+2b+6=5c
2
a
+
2
b
+
6
=
5
c
. In every set all elements are pairwise distinct
Regional Olympiad - FBH 2018 Grade 10 Problem 2
Find all positive integers
n
n
n
such that number
n
4
−
4
n
3
+
22
n
2
−
36
n
+
18
n^4-4n^3+22n^2-36n+18
n
4
−
4
n
3
+
22
n
2
−
36
n
+
18
is perfect square of positive integer
Regional Olympiad - FBH 2018 Grade 11 Problem 2
Let
a
1
,
a
2
,
.
.
.
,
a
2018
a_1, a_2,...,a_{2018}
a
1
,
a
2
,
...
,
a
2018
be a sequence of numbers such that all its elements are elements of a set
{
−
1
,
1
}
\{-1,1\}
{
−
1
,
1
}
. Sum
S
=
∑
1
≤
i
<
j
≤
2018
a
i
a
j
S=\sum \limits_{1 \leq i < j \leq 2018} a_i a_j
S
=
1
≤
i
<
j
≤
2018
∑
a
i
a
j
can be negative and can also be positive. Find the minimal value of this sum
1
4
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