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National and Regional Contests
Austria Contests
Austrian MO Regional Competition
2006 Regional Competition For Advanced Students
2006 Regional Competition For Advanced Students
Part of
Austrian MO Regional Competition
Subcontests
(4)
4
1
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37th Austrian Mathematical Olympiad
Let
<
h
n
>
<h_n>
<
h
n
>
n
∈
N
n\in\mathbb N
n
∈
N
a harmonic sequence of positive real numbers (that means that every
h
n
h_n
h
n
is the harmonic mean of its two neighbours h_{n\minus{}1} and h_{n\plus{}1} : h_n\equal{}\frac{2h_{n\minus{}1}h_{n\plus{}1}}{h_{n\minus{}1}\plus{}h_{n\plus{}1}}) Show that: if the sequence includes a member
h
j
h_j
h
j
, which is the square of a rational number, it includes infinitely many members
h
k
h_k
h
k
, which are squares of rational numbers.
3
1
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37th Austrian Mathematical Olympiad 2006
In a non isosceles triangle
A
B
C
ABC
A
BC
let
w
w
w
be the angle bisector of the exterior angle at
C
C
C
. Let
D
D
D
be the point of intersection of
w
w
w
with the extension of
A
B
AB
A
B
. Let
k
A
k_A
k
A
be the circumcircle of the triangle
A
D
C
ADC
A
D
C
and analogy
k
B
k_B
k
B
the circumcircle of the triangle
B
D
C
BDC
B
D
C
. Let
t
A
t_A
t
A
be the tangent line to
k
A
k_A
k
A
in A and
t
B
t_B
t
B
the tangent line to
k
B
k_B
k
B
in B. Let
P
P
P
be the point of intersection of
t
A
t_A
t
A
and
t
B
t_B
t
B
. Given are the points
A
A
A
and
B
B
B
. Determine the set of points P\equal{}P(C ) over all points
C
C
C
, so that
A
B
C
ABC
A
BC
is a non isosceles, acute-angled triangle.
2
1
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37th Austrian Mathematical Olympiad 2006
Let
n
>
1
n>1
n
>
1
be a positive integer an
a
a
a
a real number. Determine all real solutions
(
x
1
,
x
2
,
…
,
x
n
)
(x_1,x_2,\dots,x_n)
(
x
1
,
x
2
,
…
,
x
n
)
to following system of equations: x_1\plus{}ax_2\equal{}0 x_2\plus{}a^2x_3\equal{}0 … x_k\plus{}a^kx_{k\plus{}1}\equal{}0 … x_n\plus{}a^nx_1\equal{}0
1
1
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37th Austrian Mathematical Olympiad 2006
Let
0
<
x
<
y
0 < x <y
0
<
x
<
y
be real numbers. Let H\equal{}\frac{2xy}{x\plus{}y} , G\equal{}\sqrt{xy} , A\equal{}\frac{x\plus{}y}{2} , Q\equal{}\sqrt{\frac{x^2\plus{}y^2}{2}} be the harmonic, geometric, arithmetic and root mean square (quadratic mean) of
x
x
x
and
y
y
y
. As generally known
H
<
G
<
A
<
Q
H<G<A<Q
H
<
G
<
A
<
Q
. Arrange the intervals
[
H
,
G
]
[H,G]
[
H
,
G
]
,
[
G
,
A
]
[G,A]
[
G
,
A
]
and
[
A
,
Q
]
[A,Q]
[
A
,
Q
]
in ascending order by their length.