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Contests
National and Regional Contests
Bosnia Herzegovina Contests
Bosnia Herzegovina Team Selection Test
2015 Bosnia Herzegovina Team Selection Test
2015 Bosnia Herzegovina Team Selection Test
Part of
Bosnia Herzegovina Team Selection Test
Subcontests
(6)
6
1
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Incircle geometry
Let
D
D
D
,
E
E
E
and
F
F
F
be points in which incircle of triangle
A
B
C
ABC
A
BC
touches sides
B
C
BC
BC
,
C
A
CA
C
A
and
A
B
AB
A
B
, respectively, and let
I
I
I
be a center of that circle.Furthermore, let
P
P
P
be a foot of perpendicular from point
I
I
I
to line
A
D
AD
A
D
, and let
M
M
M
be midpoint of
D
E
DE
D
E
. If
{
N
}
=
P
M
∩
A
C
\{N\}=PM\cap{AC}
{
N
}
=
PM
∩
A
C
, prove that
D
N
∥
E
F
DN \parallel EF
D
N
∥
EF
5
1
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Weights in sets
Let
N
N
N
be a positive integer. It is given set of weights which satisfies following conditions:
i
)
i)
i
)
Every weight from set has some weight from
1
,
2
,
.
.
.
,
N
1,2,...,N
1
,
2
,
...
,
N
;
i
i
)
ii)
ii
)
For every
i
∈
1
,
2
,
.
.
.
,
N
i\in {1,2,...,N}
i
∈
1
,
2
,
...
,
N
in given set there exists weight
i
i
i
;
i
i
i
)
iii)
iii
)
Sum of all weights from given set is even positive integer. Prove that set can be partitioned into two disjoint sets which have equal weight
4
1
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Set of 8 conescutive positive integers
Let
X
X
X
be a set which consists from
8
8
8
consecutive positive integers. Set
X
X
X
is divided on two disjoint subsets
A
A
A
and
B
B
B
with equal number of elements. If sum of squares of elements from set
A
A
A
is equal to sum of squares of elements from set
B
B
B
, prove that sum of elements of set
A
A
A
is equal to sum of elements of set
B
B
B
.
3
1
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Prove that there exist infinitely many
Prove that there exist infinitely many composite positive integers
n
n
n
such that
n
n
n
divides
3
n
−
1
−
2
n
−
1
3^{n-1}-2^{n-1}
3
n
−
1
−
2
n
−
1
.
2
1
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Lots of circles and lines
Let
D
D
D
be an arbitrary point on side
A
B
AB
A
B
of triangle
A
B
C
ABC
A
BC
. Circumcircles of triangles
B
C
D
BCD
BC
D
and
A
C
D
ACD
A
C
D
intersect sides
A
C
AC
A
C
and
B
C
BC
BC
at points
E
E
E
and
F
F
F
, respectively. Perpendicular bisector of
E
F
EF
EF
cuts
A
B
AB
A
B
at point
M
M
M
, and line perpendicular to
A
B
AB
A
B
at
D
D
D
at point
N
N
N
. Lines
A
B
AB
A
B
and
E
F
EF
EF
intersect at point
T
T
T
, and the second point of intersection of circumcircle of triangle
C
M
D
CMD
CM
D
and line
T
C
TC
TC
is
U
U
U
. Prove that
N
C
=
N
U
NC=NU
NC
=
N
U
1
1
Hide problems
Find minimum value
Determine the minimum value of the expression
a
+
1
a
(
a
+
2
)
+
b
+
1
b
(
b
+
2
)
+
c
+
1
c
(
c
+
2
)
\frac {a+1}{a(a+2)}+ \frac {b+1}{b(b+2)}+\frac {c+1}{c(c+2)}
a
(
a
+
2
)
a
+
1
+
b
(
b
+
2
)
b
+
1
+
c
(
c
+
2
)
c
+
1
for positive real numbers
a
,
b
,
c
a,b,c
a
,
b
,
c
such that
a
+
b
+
c
≤
3
a+b+c \leq 3
a
+
b
+
c
≤
3
.