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Problems
Contests
National and Regional Contests
Greece Contests
Greece Junior Math Olympiad
2006 Greece Junior Math Olympiad
2006 Greece Junior Math Olympiad
Part of
Greece Junior Math Olympiad
Subcontests
(4)
4
1
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Simple but rather nice !
If
x
,
y
x , y
x
,
y
are real numbers such that
x
2
+
x
y
+
y
2
=
1
x^2 + xy + y^2 = 1
x
2
+
x
y
+
y
2
=
1
, find the least and the greatest value( minimum and maximum) of the expression
K
=
x
3
y
+
x
y
3
K = x^3y + xy^3
K
=
x
3
y
+
x
y
3
Babis Sorry !!! I forgot to write that these 4 problems( 4 topics) were JUNIOR LEVEL
3
1
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Pigeon holle- one more !
Prove that between every
27
27
27
different positive integers , less than
100
100
100
, there exist some two which are NOT relative prime. babis
2
1
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Dioph. equation
Find all positive integers
x
,
y
x , y
x
,
y
which are roots of the equation
2
x
y
−
y
=
2005
2 x^y-y= 2005
2
x
y
−
y
=
2005
Babis
1
1
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Geometry - i think simple !
Let
P
P
P
an interior point of an equilateral triangle
A
B
C
ABC
A
BC
. Prove that there exists triangle with sides
P
A
,
P
B
,
P
C
PA , PB , PC
P
A
,
PB
,
PC
. Babis