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National and Regional Contests
Bosnia Herzegovina Contests
JBMO TST - Bosnia and Herzegovina
2016 Bosnia and Herzegovina Junior BMO TST
2016 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 2016 Problem 4
Let
x
x
x
,
y
y
y
and
z
z
z
be positive real numbers such that
x
y
+
y
z
+
z
x
=
3
\sqrt{xy} + \sqrt{yz} + \sqrt{zx} = 3
x
y
+
yz
+
z
x
=
3
. Prove that
x
3
+
x
+
y
3
+
y
+
z
3
+
z
≥
6
(
x
+
y
+
z
)
\sqrt{x^3+x} + \sqrt{y^3+y} + \sqrt{z^3+z} \geq \sqrt{6(x+y+z)}
x
3
+
x
+
y
3
+
y
+
z
3
+
z
≥
6
(
x
+
y
+
z
)
3
1
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Bosnia and Herzegovina JBMO TST 2016 Problem 3
Let
O
O
O
be a center of circle which passes through vertices of quadrilateral
A
B
C
D
ABCD
A
BC
D
, which has perpendicular diagonals. Prove that sum of distances of point
O
O
O
to sides of quadrilateral
A
B
C
D
ABCD
A
BC
D
is equal to half of perimeter of
A
B
C
D
ABCD
A
BC
D
.
2
1
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Bosnia and Herzegovina JBMO TST 2016 Problem 2
We color numbers
1
,
2
,
3
,
.
.
.
,
20
1,2,3,...,20
1
,
2
,
3
,
...
,
20
in two colors, blue and yellow, such that both colors are used (not all numbers are colored in one color). Determine number of ways we can color those numbers, such that product of all blue numbers and product of all yellow numbers have greatest common divisor
1
1
1
.
1
1
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Bosnia and Herzegovina JBMO TST 2016 Problem 1
Prove that it is not possible that numbers
(
n
+
1
)
⋅
2
n
(n+1)\cdot 2^n
(
n
+
1
)
⋅
2
n
and
(
n
+
3
)
⋅
2
n
+
2
(n+3)\cdot 2^{n+2}
(
n
+
3
)
⋅
2
n
+
2
are perfect squares, where
n
n
n
is positive integer.