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International Contests
Junior Balkan MO
2009 Junior Balkan MO
2009 Junior Balkan MO
Part of
Junior Balkan MO
Subcontests
(4)
4
1
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Points and circles in the plane...
Each one of 2009 distinct points in the plane is coloured in blue or red, so that on every blue-centered unit circle there are exactly two red points. Find the gratest possible number of blue points.
3
1
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xyz = (1 - x)(1 - y)(1 - z)
Let
x
x
x
,
y
y
y
,
z
z
z
be real numbers such that
0
<
x
,
y
,
z
<
1
0 < x,y,z < 1
0
<
x
,
y
,
z
<
1
and xyz \equal{} (1 \minus{} x)(1 \minus{} y)(1 \minus{} z). Show that at least one of the numbers (1 \minus{} x)y,(1 \minus{} y)z,(1 \minus{} z)x is greater than or equal to
1
4
\frac {1}{4}
4
1
2
1
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2^a 3^b + 9 = c^2
Solve in non-negative integers the equation 2^{a}3^{b} \plus{} 9 \equal{} c^{2}
1
1
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Parallel lines in pentagon
Let
A
B
C
D
E
ABCDE
A
BC
D
E
be a convex pentagon such that AB\plus{}CD\equal{}BC\plus{}DE and
k
k
k
a circle with center on side
A
E
AE
A
E
that touches the sides
A
B
AB
A
B
,
B
C
BC
BC
,
C
D
CD
C
D
and
D
E
DE
D
E
at points
P
P
P
,
Q
Q
Q
,
R
R
R
and
S
S
S
(different from vertices of the pentagon) respectively. Prove that lines
P
S
PS
PS
and
A
E
AE
A
E
are parallel.