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Contests
National and Regional Contests
Greece Contests
Greece Junior Math Olympiad
2004 Greece Junior Math Olympiad
2004 Greece Junior Math Olympiad
Part of
Greece Junior Math Olympiad
Subcontests
(4)
4
1
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Algebra Inequality
Determine the rational number
a
b
\frac{a}{b}
b
a
, where
a
,
b
a,b
a
,
b
are positive integers, with minimal denominator, which is such that
52
303
<
a
b
<
16
91
\frac{52}{303} < \frac{a}{b}< \frac{16}{91}
303
52
<
b
a
<
91
16
1
1
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Number Theory
The numbers
203
203
203
and
298
298
298
divided with the positive integer
x
x
x
give both remainder
13
13
13
. Which are the possible values of
x
x
x
?
2
1
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isosceles trapezoid and areas, computational (Greece Junior 2004)
Let
A
B
C
D
ABCD
A
BC
D
be a rectangle. Let
K
,
L
K,L
K
,
L
be the midpoints of
B
C
,
A
D
BC, AD
BC
,
A
D
respectively. From point
B
B
B
the perpendicular line on
A
K
AK
A
K
, intersects
A
K
AK
A
K
at point
E
E
E
and
C
L
CL
C
L
at point
Z
Z
Z
. a) Prove that the quadrilateral
A
K
Z
L
AKZL
A
K
Z
L
is an isosceles trapezoid b) Prove that
2
S
A
B
K
Z
=
S
A
B
C
D
2S_{ABKZ}=S_{ABCD}
2
S
A
B
K
Z
=
S
A
BC
D
c) If quadrilateral
A
B
C
D
ABCD
A
BC
D
is a square of side
a
a
a
, calculate the area of the isosceles trapezoid
A
K
Z
L
AKZL
A
K
Z
L
in terms of side
B
C
=
a
BC=a
BC
=
a
3
1
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An ineq easy
x,y,z positive real numbers such that
x
2
+
y
2
+
z
2
=
25
x^2+y^2+z^2=25
x
2
+
y
2
+
z
2
=
25
Find the min price of
A
=
x
y
z
+
y
z
x
+
z
x
y
A=\frac{xy}{z}+\frac{yz}{x}+\frac{zx}{y}
A
=
z
x
y
+
x
yz
+
y
z
x