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National and Regional Contests
Austria Contests
Austrian MO Regional Competition
2007 Regional Competition For Advanced Students
2007 Regional Competition For Advanced Students
Part of
Austrian MO Regional Competition
Subcontests
(4)
4
1
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38th Austrian Mathematical Competition 2007
Let
M
M
M
be the intersection of the diagonals of a convex quadrilateral
A
B
C
D
ABCD
A
BC
D
. Determine all such quadrilaterals for which there exists a line
g
g
g
that passes through
M
M
M
and intersects the side
A
B
AB
A
B
in
P
P
P
and the side
C
D
CD
C
D
in
Q
Q
Q
, such that the four triangles
A
P
M
APM
A
PM
,
B
P
M
BPM
BPM
,
C
Q
M
CQM
CQM
,
D
Q
M
DQM
D
QM
are similar.
3
1
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38th Austrian Mathematical Competition 2007
Let
a
a
a
be a positive real number and
n
n
n
a non-negative integer. Determine S\minus{}T, where S\equal{} \sum_{k\equal{}\minus{}2n}^{2n\plus{}1} \frac{(k\minus{}1)^2}{a^{| \lfloor \frac{k}{2} \rfloor |}} and T\equal{} \sum_{k\equal{}\minus{}2n}^{2n\plus{}1} \frac{k^2}{a^{| \lfloor \frac{k}{2} \rfloor |}}
2
1
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38th Austrian Mathematical Competition 2007
Find all tuples
(
x
1
,
x
2
,
x
3
,
x
4
,
x
5
)
(x_1,x_2,x_3,x_4,x_5)
(
x
1
,
x
2
,
x
3
,
x
4
,
x
5
)
of positive integers with
x
1
>
x
2
>
x
3
>
x
4
>
x
5
>
0
x_1>x_2>x_3>x_4>x_5>0
x
1
>
x
2
>
x
3
>
x
4
>
x
5
>
0
and
⌊
x
1
+
x
2
3
⌋
2
+
⌊
x
2
+
x
3
3
⌋
2
+
⌊
x
3
+
x
4
3
⌋
2
+
⌊
x
4
+
x
5
3
⌋
2
=
38.
{\left \lfloor \frac{x_1+x_2}{3} \right \rfloor }^2 + {\left \lfloor \frac{x_2+x_3}{3} \right \rfloor }^2 + {\left \lfloor \frac{x_3+x_4}{3} \right \rfloor }^2 + {\left \lfloor \frac{x_4+x_5}{3} \right \rfloor }^2 = 38.
⌊
3
x
1
+
x
2
⌋
2
+
⌊
3
x
2
+
x
3
⌋
2
+
⌊
3
x
3
+
x
4
⌋
2
+
⌊
3
x
4
+
x
5
⌋
2
=
38.
1
1
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38th Austrian Mathematical Competition 2007
Let
0
<
x
0
,
x
1
,
…
,
x
669
<
1
0<x_0,x_1, \dots , x_{669}<1
0
<
x
0
,
x
1
,
…
,
x
669
<
1
be pairwise distinct real numbers. Show that there exists a pair
(
x
i
,
x
j
)
(x_i,x_j)
(
x
i
,
x
j
)
with 0