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National and Regional Contests
Austria Contests
Austrian MO Regional Competition
2009 Regional Competition For Advanced Students
2009 Regional Competition For Advanced Students
Part of
Austrian MO Regional Competition
Subcontests
(4)
4
1
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Considerable different arithmetic progressions
Two infinite arithmetic progressions are called considerable different if the do not only differ by the absence of finitely many members at the beginning of one of the sequences. How many pairwise considerable different non-constant arithmetic progressions of positive integers that contain an infinite non-constant geometric progression
(
b
n
)
n
≥
0
(b_n)_{n\ge0}
(
b
n
)
n
≥
0
with b_2\equal{}40 \cdot 2009 are there?
3
1
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Show that three lines don't intersect in one point
Let
D
D
D
,
E
E
E
,
F
F
F
be the feet of the altitudes wrt sides
B
C
BC
BC
,
C
A
CA
C
A
,
A
B
AB
A
B
of acute-angled triangle
△
A
B
C
\triangle ABC
△
A
BC
, respectively. In triangle
△
C
F
B
\triangle CFB
△
CFB
, let
P
P
P
be the foot of the altitude wrt side
B
C
BC
BC
. Define
Q
Q
Q
and
R
R
R
wrt triangles
△
A
D
C
\triangle ADC
△
A
D
C
and
△
B
E
A
\triangle BEA
△
BE
A
analogously. Prove that lines
A
P
AP
A
P
,
B
Q
BQ
BQ
,
C
R
CR
CR
don't intersect in one common point.
2
1
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Number of solutions of equation
How many integer solutions
(
x
0
(x_0
(
x
0
,
x
1
x_1
x
1
,
x
2
x_2
x
2
,
x
3
x_3
x
3
,
x
4
x_4
x
4
,
x
5
x_5
x
5
,
x
6
)
x_6)
x
6
)
does the equation 2x_0^2\plus{}x_1^2\plus{}x_2^2\plus{}x_3^2\plus{}x_4^2\plus{}x_5^2\plus{}x_6^2\equal{}9 have?
1
1
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Find largest interval
Find the largest interval M \subseteq \mathbb{R^ \plus{} }, such that for all
a
a
a
,
b
b
b
,
c
c
c
,
d
∈
M
d \in M
d
∈
M
the inequality \sqrt {ab} \plus{} \sqrt {cd} \ge \sqrt {a \plus{} b} \plus{} \sqrt {c \plus{} d} holds. Does the inequality \sqrt {ab} \plus{} \sqrt {cd} \ge \sqrt {a \plus{} c} \plus{} \sqrt {b \plus{} d} hold too for all
a
a
a
,
b
b
b
,
c
c
c
,
d
∈
M
d \in M
d
∈
M
? ( \mathbb{R^ \plus{} } denotes the set of positive reals.)