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National and Regional Contests
Austria Contests
Austrian MO Regional Competition
2004 Regional Competition For Advanced Students
2004 Regional Competition For Advanced Students
Part of
Austrian MO Regional Competition
Subcontests
(4)
4
1
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Infinitely prime numbers in a sequence
The sequence
<
x
n
>
< x_n >
<
x
n
>
is defined through: x_{n \plus{} 1} \equal{} \left(\frac {n}{2004} \plus{} \frac {1}{n}\right)x_n^2 \minus{} \frac {n^3}{2004} \plus{} 1 for
n
>
0
n > 0
n
>
0
Let
x
1
x_1
x
1
be a non-negative integer smaller than
204
204
204
so that all members of the sequence are non-negative integers. Show that there exist infinitely many prime numbers in this sequence.
3
1
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35th Austrian Mathematical Olympiad 2004
Given is a convex quadrilateral
A
B
C
D
ABCD
A
BC
D
with \angle ADC\equal{}\angle BCD>90^{\circ}. Let
E
E
E
be the point of intersection of the line
A
C
AC
A
C
with the parallel line to
A
D
AD
A
D
through
B
B
B
and
F
F
F
be the point of intersection of the line
B
D
BD
B
D
with the parallel line to
B
C
BC
BC
through
A
A
A
. Show that
E
F
EF
EF
is parallel to
C
D
CD
C
D
1
1
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35th Austrian Mathematical Olympiad 2004
Determine all integers
a
a
a
and
b
b
b
, so that (a^3\plus{}b)(a\plus{}b^3)\equal{}(a\plus{}b)^4
2
1
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35th Austrian Mathematical Olympiad 2004
Solve the following equation for real numbers: \sqrt{4\minus{}x\sqrt{4\minus{}(x\minus{}2)\sqrt{1\plus{}(x\minus{}5)(x\minus{}7)}}}\equal{}\frac{5x\minus{}6\minus{}x^2}{2} (all square roots are non negative)