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National and Regional Contests
Austria Contests
Austrian MO Regional Competition
2005 Regional Competition For Advanced Students
2005 Regional Competition For Advanced Students
Part of
Austrian MO Regional Competition
Subcontests
(4)
4
1
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36th Austrian Mathematical Olympiad 2005
Prove: if an infinte arithmetic sequence ( a_n\equal{}a_0\plus{}nd) of positive real numbers contains two different powers of an integer
a
>
1
a>1
a
>
1
, then the sequence contains an infinite geometric sequence ( b_n\equal{}b_0q^n) of real numbers.
3
1
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36th Austrian Mathematical Olympiad 2005
For which values of
k
k
k
and
d
d
d
has the system x^3\plus{}y^3\equal{}2 and y\equal{}kx\plus{}d no real solutions
(
x
,
y
)
(x,y)
(
x
,
y
)
?
2
1
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36th Austrian Mathematical Olympiad 2005
Construct the semicircle
h
h
h
with the diameter
A
B
AB
A
B
and the midpoint
M
M
M
. Now construct the semicircle
k
k
k
with the diameter
M
B
MB
MB
on the same side as
h
h
h
. Let
X
X
X
and
Y
Y
Y
be points on
k
k
k
, such that the arc
B
X
BX
BX
is
3
2
\frac{3}{2}
2
3
times the arc
B
Y
BY
B
Y
. The line
M
Y
MY
M
Y
intersects the line
B
X
BX
BX
in
D
D
D
and the semicircle
h
h
h
in
C
C
C
. Show that
Y
Y
Y
ist he midpoint of
C
D
CD
C
D
.
1
1
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36th Austrian Mathematical Olympiad 2005
Show for all integers
n
≥
2005
n \ge 2005
n
≥
2005
the following chaine of inequalities: (n\plus{}830)^{2005}