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Problems
Contests
National and Regional Contests
Austria Contests
Austrian MO National Competition
1997 Federal Competition For Advanced Students, P2
1997 Federal Competition For Advanced Students, P2
Part of
Austrian MO National Competition
Subcontests
(6)
6
1
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polynomial (maybe posted before)
For every natural number
n
n
n
, find all polynomials x^2\plus{}ax\plus{}b, where
a
2
≥
4
b
a^2 \ge 4b
a
2
≥
4
b
, that divide x^{2n}\plus{}ax^n\plus{}b.
5
1
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operations (maybe posted before)
We define the following operation which will be applied to a row of bars being situated side-by-side on positions
1
,
2
,
.
.
.
,
N
1,2,...,N
1
,
2
,
...
,
N
. Each bar situated at an odd numbered position is left as is, while each bar at an even numbered position is replaced by two bars. After that, all bars will be put side-by-side in such a way that all bars form a new row and are situated on positions
1
,
.
.
.
,
M
.
1,...,M.
1
,
...
,
M
.
From an initial number
a
0
>
0
a_0>0
a
0
>
0
of bars there originates a sequence
(
a
n
)
n
≥
0
,
(a_n)_{n \ge 0},
(
a
n
)
n
≥
0
,
where
a
n
a_n
a
n
is the number of bars after having applied the operation
n
n
n
times.
(
a
)
(a)
(
a
)
Prove that for no
n
>
0
n>0
n
>
0
can we have a_n\equal{}1997.
(
b
)
(b)
(
b
)
Determine all natural numbers that can only occur as
a
0
a_0
a
0
or
a
1
a_1
a
1
.
4
1
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quadruples (maybe posted before)
Determine all quadruples
(
a
,
b
,
c
,
d
)
(a,b,c,d)
(
a
,
b
,
c
,
d
)
of real numbers satisfying the equation: 256a^3 b^3 c^3 d^3\equal{}(a^6\plus{}b^2\plus{}c^2\plus{}d^2)(a^2\plus{}b^6\plus{}c^2\plus{}d^2)(a^2\plus{}b^2\plus{}c^6\plus{}d^2)(a^2\plus{}b^2\plus{}c^2\plus{}d^6).
3
1
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cyclic quadrilateral (maybe posted before)
Let be given a triangle
A
B
C
ABC
A
BC
. Points
P
P
P
on side
A
C
AC
A
C
and
Y
Y
Y
on the production of
C
B
CB
CB
beyond
B
B
B
are chosen so that
Y
Y
Y
subtends equal angles with
A
P
AP
A
P
and
P
C
PC
PC
. Similarly,
Q
Q
Q
on side
B
C
BC
BC
and
X
X
X
on the production of
A
C
AC
A
C
beyond
C
C
C
are such that
X
X
X
subtends equal angles with
B
Q
BQ
BQ
and
Q
C
QC
QC
. Lines
Y
P
YP
Y
P
and
X
B
XB
XB
meet at
R
R
R
,
X
Q
XQ
XQ
and
Y
A
YA
Y
A
meet at
S
S
S
, and
X
B
XB
XB
and
Y
A
YA
Y
A
meet at
D
D
D
. Prove that
P
Q
R
S
PQRS
PQRS
is a parallelogram if and only if
A
C
B
D
ACBD
A
CB
D
is a cyclic quadrilateral.
2
1
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sequence
A positive integer
K
K
K
is given. Define the sequence
(
a
n
)
(a_n)
(
a
n
)
by a_1\equal{}1 and
a
n
a_n
a
n
is the
n
n
n
-th natural number greater than a_{n\minus{}1} which is congruent to
n
n
n
modulo
K
K
K
.
(
a
)
(a)
(
a
)
Find an explicit formula for
a
n
a_n
a
n
.
(
b
)
(b)
(
b
)
What is the result if K\equal{}2?
1
1
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system
Let
a
a
a
be a fixed integer. Find all integer solutions
x
,
y
,
z
x,y,z
x
,
y
,
z
of the system: 5x\plus{}(a\plus{}2)y\plus{}(a\plus{}2)z\equal{}a, (2a\plus{}4)x\plus{}(a^2\plus{}3)y\plus{}(2a\plus{}2)z\equal{}3a\minus{}1, (2a\plus{}4)x\plus{}(2a\plus{}2)y\plus{}(a^2\plus{}3)z\equal{}a\plus{}1.