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National and Regional Contests
Bosnia Herzegovina Contests
Bosnia Herzegovina Team Selection Test
2016 Bosnia and Herzegovina Team Selection Test
2016 Bosnia and Herzegovina Team Selection Test
Part of
Bosnia Herzegovina Team Selection Test
Subcontests
(5)
5
1
Hide problems
Collinear points THC
Let
k
k
k
be a circumcircle of triangle
A
B
C
ABC
A
BC
(
A
C
<
B
C
)
(AC<BC)
(
A
C
<
BC
)
. Also, let
C
L
CL
C
L
be an angle bisector of angle
A
C
B
ACB
A
CB
(
L
∈
A
B
)
(L \in AB)
(
L
∈
A
B
)
,
M
M
M
be a midpoint of arc
A
B
AB
A
B
of circle
k
k
k
containing the point
C
C
C
, and let
I
I
I
be an incenter of a triangle
A
B
C
ABC
A
BC
. Circle
k
k
k
cuts line
M
I
MI
M
I
at point
K
K
K
and circle with diameter
C
I
CI
C
I
at
H
H
H
. If the circumcircle of triangle
C
L
K
CLK
C
L
K
intersects
A
B
AB
A
B
again at
T
T
T
, prove that
T
T
T
,
H
H
H
and
C
C
C
are collinear. .
4
1
Hide problems
Biggest integer which is not a sum
Determine the largest positive integer
n
n
n
which cannot be written as the sum of three numbers bigger than
1
1
1
which are pairwise coprime.
3
1
Hide problems
Multiplies of p in increasing sequence
For an infinite sequence
a
1
<
a
2
<
a
3
<
.
.
.
a_1<a_2<a_3<...
a
1
<
a
2
<
a
3
<
...
of positive integers we say that it is nice if for every positive integer
n
n
n
holds
a
2
n
=
2
a
n
a_{2n}=2a_n
a
2
n
=
2
a
n
. Prove the following statements:
a
)
a)
a
)
If there is given a nice sequence and prime number
p
>
a
1
p>a_1
p
>
a
1
, there exist some term of the sequence which is divisible by
p
p
p
.
b
)
b)
b
)
For every prime number
p
>
2
p>2
p
>
2
, there exist a nice sequence such that no terms of the sequence are divisible by
p
p
p
.
2
1
Hide problems
Guessing the sum on a table
Let
n
n
n
be a positive integer and let
t
t
t
be an integer.
n
n
n
distinct integers are written on a table. Bob, sitting in a room nearby, wants to know whether there exist some of these numbers such that their sum is equal to
t
t
t
. Alice is standing in front of the table and she wants to help him. At the beginning, she tells him only the initial sum of all numbers on the table. After that, in every move he says one of the
4
4
4
sentences:
i
.
i.
i
.
Is there a number on the table equal to
k
k
k
?
i
i
.
ii.
ii
.
If a number
k
k
k
exists on the table, erase him.
i
i
i
.
iii.
iii
.
If a number
k
k
k
does not exist on the table, add him.
i
v
.
iv.
i
v
.
Do the numbers written on the table can be arranged in two sets with equal sum of elements? On these questions Alice answers yes or no, and the operations he says to her she does (if it is possible) and does not tell him did she do it. Prove that in less than
3
n
3n
3
n
moves, Bob can find out whether there exist numbers initially written on the board such that their sum is equal to
t
t
t
1
1
Hide problems
Parallel lines and tangents
Let
A
B
C
D
ABCD
A
BC
D
be a quadrilateral inscribed in circle
k
k
k
. Lines
A
B
AB
A
B
and
C
D
CD
C
D
intersect at point
E
E
E
such that
A
B
=
B
E
AB=BE
A
B
=
BE
. Let
F
F
F
be the intersection point of tangents on circle
k
k
k
in points
B
B
B
and
D
D
D
, respectively. If the lines
A
B
AB
A
B
and
D
F
DF
D
F
are parallel, prove that
A
A
A
,
C
C
C
and
F
F
F
are collinear.