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Austrian MO Regional Competition
2003 Regional Competition For Advanced Students
2003 Regional Competition For Advanced Students
Part of
Austrian MO Regional Competition
Subcontests
(4)
4
1
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34th Austrian Mathematical Olympiad 2003
For every real number
b
b
b
determine all real numbers
x
x
x
satisfying x\minus{}b\equal{} \sum_{k\equal{}0}^{\infty}x^k.
3
1
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34th Austrian Mathematical Olympiad 2003
Given are two parallel lines
g
g
g
and
h
h
h
and a point
P
P
P
, that lies outside of the corridor bounded by
g
g
g
and
h
h
h
. Construct three lines
g
1
g_1
g
1
,
g
2
g_2
g
2
and
g
3
g_3
g
3
through the point
P
P
P
. These lines intersect
g
g
g
in
A
1
,
A
2
,
A
3
A_1,A_2, A_3
A
1
,
A
2
,
A
3
and
h
h
h
in
B
1
,
B
2
,
B
3
B_1, B_2, B_3
B
1
,
B
2
,
B
3
respectively. Let
C
1
C_1
C
1
be the intersection of the lines
A
1
B
2
A_1B_2
A
1
B
2
and
A
2
B
1
A_2B_1
A
2
B
1
,
C
2
C_2
C
2
be the intersection of the lines
A
1
B
3
A_1B_3
A
1
B
3
and
A
3
B
1
A_3B_1
A
3
B
1
and let
C
3
C_3
C
3
be the intersection of the lines
A
2
B
3
A_2B_3
A
2
B
3
and
A
3
B
2
A_3B_2
A
3
B
2
. Show that there exists exactly one line
n
n
n
, that contains the points
C
1
,
C
2
,
C
3
C_1,C_2,C_3
C
1
,
C
2
,
C
3
and that
n
n
n
is parallel to
g
g
g
and
h
h
h
.
2
1
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34th Austrian Mathematical Olympiad 2003
Find all prime numbers
p
p
p
with 5^p\plus{}4p^4 is the square of an integer.
1
1
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34th Austrian Mathematical Olympiad 2003
Find the minimum value of the expression \frac{a\plus{}1}{a(a\plus{}2)}\plus{}\frac{b\plus{}1}{b(b\plus{}2)}\plus{}\frac{c\plus{}1}{c(c\plus{}2)}, where
a
,
b
,
c
a,b,c
a
,
b
,
c
are positive real numbers with a\plus{}b\plus{}c \le 3.