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Contests
National and Regional Contests
Bosnia Herzegovina Contests
JBMO TST - Bosnia and Herzegovina
2007 Bosnia and Herzegovina Junior BMO TST
2007 Bosnia and Herzegovina Junior BMO TST
Part of
JBMO TST - Bosnia and Herzegovina
Subcontests
(4)
3
1
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circles inside a square side length 1 with sum of radii 2007
Is it possible to place some circles inside a square side length
1
1
1
, such that no two circles intersect and the sum of their radii is
2007
2007
2007
?
2
1
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x^2(x + y) is divisible by y^2(y - x)^2
Find all pairs of relatively prime numbers (
x
,
y
x, y
x
,
y
) such that
x
2
(
x
+
y
)
x^2(x + y)
x
2
(
x
+
y
)
is divisible by
y
2
(
y
−
x
)
2
y^2(y - x)^2
y
2
(
y
−
x
)
2
. .
1
1
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ways to write 1000 as the sum of at least two consecutive positive integers
Write the number
1000
1000
1000
as the sum of at least two consecutive positive integers. How many (different) ways are there to write it?
4
1
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< IMA = <INB, incenter and arc midpoint of circumcircle related
Let
I
I
I
be the incenter of the triangle
A
B
C
ABC
A
BC
(
A
B
<
B
C
AB < BC
A
B
<
BC
). Let
M
M
M
be the midpoint of
A
C
AC
A
C
, and let
N
N
N
be the midpoint of the arc
A
C
AC
A
C
of the circumcircle of
A
B
C
ABC
A
BC
which contains
B
B
B
. Prove that
∠
I
M
A
=
∠
I
N
B
\angle IMA = \angle INB
∠
I
M
A
=
∠
I
NB
.