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National and Regional Contests
Greece Contests
Greece Junior Math Olympiad
2008 Greece Junior Math Olympiad
2008 Greece Junior Math Olympiad
Part of
Greece Junior Math Olympiad
Subcontests
(4)
2
1
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Algebra Inequality
If
x
,
y
,
z
x,y,z
x
,
y
,
z
are positive real numbers with
x
2
+
y
2
+
z
2
=
3
x^2+y^2+z^2=3
x
2
+
y
2
+
z
2
=
3
, prove that
3
2
<
1
+
y
2
x
+
2
+
1
+
z
2
y
+
2
+
1
+
x
2
z
+
2
<
3
\frac32<\frac{1+y^2}{x+2}+\frac{1+z^2}{y+2}+\frac{1+x^2}{z+2}<3
2
3
<
x
+
2
1
+
y
2
+
y
+
2
1
+
z
2
+
z
+
2
1
+
x
2
<
3
1
1
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Easiest Number Theory
Let
p
,
q
p,q
p
,
q
denote distinct prime numbers and
k
,
l
k,l
k
,
l
positive integers. Find all positive divisors of the numbers: (a)
A
=
p
k
A = p^k
A
=
p
k
(b)
B
=
p
k
q
l
B=p^kq^l
B
=
p
k
q
l
(c)
1944
1944
1944
4
1
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trapezoid with 2 right angles, concurrency wanted (Greece Junior 2008)
Let
A
B
C
D
ABCD
A
BC
D
be a trapezoid with
A
D
=
a
,
A
B
=
2
a
,
B
C
=
3
a
AD=a, AB=2a, BC=3a
A
D
=
a
,
A
B
=
2
a
,
BC
=
3
a
and
∠
A
=
∠
B
=
9
0
o
\angle A=\angle B =90 ^o
∠
A
=
∠
B
=
9
0
o
. Let
E
,
Z
E,Z
E
,
Z
be the midpoints of the sides
A
B
,
C
D
AB ,CD
A
B
,
C
D
respectively and
I
I
I
be the foot of the perpendicular from point
Z
Z
Z
on
B
C
BC
BC
. Prove that : i) triangle
B
D
Z
BDZ
B
D
Z
is isosceles ii) midpoint
O
O
O
of
E
Z
EZ
EZ
is the centroid of triangle
B
D
Z
BDZ
B
D
Z
iii) lines
A
Z
AZ
A
Z
and
D
I
DI
D
I
intersect at a point lying on line
B
O
BO
BO
3
1
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Perfect Square
Find the greatest value of positive integer
x
x
x
, such that the number A\equal{} 2^{182} \plus{} 4^x \plus{} 8^{700} is a perfect square .