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National and Regional Contests
Bosnia Herzegovina Contests
JBMO TST - Bosnia and Herzegovina
2009 Bosnia and Herzegovina Junior BMO TST
2009 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 2009 Problem 4
On circle there are
2009
2009
2009
positive integers which sum is
7036
7036
7036
. Show that it is possible to find two pairs of neighboring numbers such that sum of both pairs is greater or equal to
8
8
8
3
1
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Bosnia and Herzegovina JBMO TST 2009 Problem 3
Let
p
p
p
be a prime number,
p
≠
3
p\neq 3
p
=
3
and let
a
a
a
and
b
b
b
be positive integers such that
p
∣
a
+
b
p \mid a+b
p
∣
a
+
b
and
p
2
∣
a
3
+
b
3
p^2\mid a^3+b^3
p
2
∣
a
3
+
b
3
. Show that
p
2
∣
a
+
b
p^2 \mid a+b
p
2
∣
a
+
b
or
p
3
∣
a
3
+
b
3
p^3 \mid a^3+b^3
p
3
∣
a
3
+
b
3
2
1
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Bosnia and Herzegovina JBMO TST 2009 Problem 2
Let
a
a
a
,
b
b
b
,
c
c
c
and
d
d
d
be positive real numbers such that
a
+
b
+
c
+
d
=
8
a+b+c+d=8
a
+
b
+
c
+
d
=
8
. Prove that
1
a
+
1
b
+
4
c
+
16
d
≥
8
\frac{1}{a}+\frac{1}{b}+\frac{4}{c}+\frac{16}{d}\geq8
a
1
+
b
1
+
c
4
+
d
16
≥
8
1
1
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Bosnia and Herzegovina JBMO TST 2009 Problem 1
Lengths of sides of triangle
A
B
C
ABC
A
BC
are positive integers, and smallest side is equal to
2
2
2
. Determine the area of triangle
P
P
P
if
v
c
=
v
a
+
v
b
v_c = v_a + v_b
v
c
=
v
a
+
v
b
, where
v
a
v_a
v
a
,
v
b
v_b
v
b
and
v
c
v_c
v
c
are lengths of altitudes in triangle
A
B
C
ABC
A
BC
from vertices
A
A
A
,
B
B
B
and
C
C
C
, respectively.