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National and Regional Contests
Bosnia Herzegovina Contests
JBMO TST - Bosnia and Herzegovina
2013 Bosnia and Herzegovina Junior BMO TST
2013 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 2013 Problem 4
It is given polygon with
2013
2013
2013
sides
A
1
A
2
.
.
.
A
2013
A_{1}A_{2}...A_{2013}
A
1
A
2
...
A
2013
. His vertices are marked with numbers such that sum of numbers marked by any
9
9
9
consecutive vertices is constant and its value is
300
300
300
. If we know that
A
13
A_{13}
A
13
is marked with
13
13
13
and
A
20
A_{20}
A
20
is marked with
20
20
20
, determine with which number is marked
A
2013
A_{2013}
A
2013
3
1
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Bosnia and Herzegovina JBMO TST 2013 Problem 3
Let
M
M
M
and
N
N
N
be touching points of incircle with sides
A
B
AB
A
B
and
A
C
AC
A
C
of triangle
A
B
C
ABC
A
BC
, and
P
P
P
intersection point of line
M
N
MN
MN
and angle bisector of
∠
A
B
C
\angle ABC
∠
A
BC
. Prove that
∠
B
P
C
=
9
0
∘
\angle BPC =90 ^{\circ}
∠
BPC
=
9
0
∘
2
1
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Bosnia and Herzegovina JBMO TST 2013 Problem 2
Let
a
a
a
,
b
b
b
and
c
c
c
be positive real numbers such that
a
2
+
b
2
+
c
2
=
3
a^2+b^2+c^2=3
a
2
+
b
2
+
c
2
=
3
. Prove the following inequality:
a
3
c
(
a
2
−
a
b
+
b
2
)
+
b
3
a
(
b
2
−
b
c
+
c
2
)
+
c
3
b
(
c
2
−
c
a
+
a
2
)
≤
1
a
b
c
\frac{a}{3c(a^2-ab+b^2)} + \frac{b}{3a(b^2-bc+c^2)} + \frac{c}{3b(c^2-ca+a^2)} \leq \frac{1}{abc}
3
c
(
a
2
−
ab
+
b
2
)
a
+
3
a
(
b
2
−
b
c
+
c
2
)
b
+
3
b
(
c
2
−
c
a
+
a
2
)
c
≤
ab
c
1
1
1
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Bosnia and Herzegovina JBMO TST 2013 Problem 1
It is given
n
n
n
positive integers. Product of any one of them with sum of remaining numbers increased by
1
1
1
is divisible with sum of all
n
n
n
numbers. Prove that sum of squares of all
n
n
n
numbers is divisible with sum of all
n
n
n
numbers