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Bosnia Herzegovina Contests
Bosnia Herzegovina Team Selection Test
2017 Bosnia Herzegovina Team Selection Test
2017 Bosnia Herzegovina Team Selection Test
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Bosnia Herzegovina Team Selection Test
Subcontests
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6
1
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NIce geometry
Given is an acute triangle
A
B
C
ABC
A
BC
.
M
M
M
is an arbitrary point at the side
A
B
AB
A
B
and
N
N
N
is midpoint of
A
C
AC
A
C
. The foots of the perpendiculars from
A
A
A
to
M
C
MC
MC
and
M
N
MN
MN
are points
P
P
P
and
Q
Q
Q
. Prove that center of the circumcircle of triangle
P
Q
N
PQN
PQN
lies on the fixed line for all points
M
M
M
from the side
A
B
AB
A
B
.
3
1
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Problem with sequences
Find all real constants c for which there exist strictly increasing sequence
a
a
a
of positive integers such that
(
a
2
n
−
1
+
a
2
n
)
/
a
n
=
c
(a_{2n-1}+a_{2n})/{a_n}=c
(
a
2
n
−
1
+
a
2
n
)
/
a
n
=
c
for all positive intеgers n.
4
1
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6n + 4 mathematicans take part in a conference
There are 6n \plus{} 4 mathematicians participating in a conference which includes 2n \plus{} 1 meetings. Each meeting has one round table that suits for
4
4
4
people and
n
n
n
round tables that each table suits for
6
6
6
people. We have known that two arbitrary people sit next to or have opposite places doesn't exceed one time. 1. Determine whether or not there is the case n \equal{} 1. 2. Determine whether or not there is the case
n
>
1
n > 1
n
>
1
.
1
1
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Circumcircle of ILK is tangent to ABC iff AB+AC=3BC
Incircle of triangle
A
B
C
ABC
A
BC
touches
A
B
,
A
C
AB,AC
A
B
,
A
C
at
P
,
Q
P,Q
P
,
Q
.
B
I
,
C
I
BI, CI
B
I
,
C
I
intersect with
P
Q
PQ
PQ
at
K
,
L
K,L
K
,
L
. Prove that circumcircle of
I
L
K
ILK
I
L
K
is tangent to incircle of
A
B
C
ABC
A
BC
if and only if AB\plus{}AC\equal{}3BC.