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National and Regional Contests
Bosnia Herzegovina Contests
JBMO TST - Bosnia and Herzegovina
2010 Bosnia and Herzegovina Junior BMO TST
2010 Bosnia and Herzegovina Junior BMO TST
Part of
JBMO TST - Bosnia and Herzegovina
Subcontests
(4)
4
1
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Bosnia and Herzegovina JBMO TST 2010 Problem 4
On circle in clockwise order are written positive integers from
1
1
1
to
2010
2010
2010
. Let us cross out number
1
1
1
, then number
10
10
10
, then number
19
19
19
, and so on every
9
9
9
th number in that direction. Which number will be first crossed out twice? How many numbers at that moment are not crossed out?
3
1
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Bosnia and Herzegovina JBMO TST 2010 Problem 3
Points
M
M
M
and
N
N
N
are given on sides
A
D
AD
A
D
and
B
C
BC
BC
of rhombus
A
B
C
D
ABCD
A
BC
D
, respectively. Line
M
C
MC
MC
intersects line
B
D
BD
B
D
in point
T
T
T
, line
M
N
MN
MN
intersects line
B
D
BD
B
D
in point
U
U
U
, line
C
U
CU
C
U
intersects line
A
B
AB
A
B
in point
Q
Q
Q
and line
Q
T
QT
QT
intersects line
C
D
CD
C
D
in
P
P
P
. Prove that triangles
Q
C
P
QCP
QCP
and
M
C
N
MCN
MCN
have equal area
2
1
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Bosnia and Herzegovina JBMO TST 2010 Problem 2
Let us consider every third degree polynomial
P
(
x
)
P(x)
P
(
x
)
with coefficients as nonnegative positive integers such that
P
(
1
)
=
20
P(1)=20
P
(
1
)
=
20
. Among them determine polynomial for which is:
a
)
a)
a
)
Minimal value of
P
(
4
)
P(4)
P
(
4
)
b
)
b)
b
)
Maximal value of
P
(
3
)
/
P
(
2
)
P(3)/P(2)
P
(
3
)
/
P
(
2
)
1
1
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Bosnia and Herzegovina JBMO TST 2010 Problem 1
Prove that number
2
2008
⋅
2
2010
+
5
2012
2^{2008}\cdot2^{2010}+5^{2012}
2
2008
⋅
2
2010
+
5
2012
is not prime