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Problems
Contests
National and Regional Contests
Austria Contests
Austrian MO National Competition
1997 Federal Competition For Advanced Students, Part 2
1997 Federal Competition For Advanced Students, Part 2
Part of
Austrian MO National Competition
Subcontests
(3)
3
2
Hide problems
PQRS is a parallelogram if and only if ACBD is cyclic
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.
Find all p(x) = x^2+ax+b such that p(x) | x^{2n} + ax^n + b
For every natural number
n
n
n
, find all polynomials
x
2
+
a
x
+
b
x^2+ax+b
x
2
+
a
x
+
b
, where
a
2
≥
4
b
a^2 \geq 4b
a
2
≥
4
b
, that divide
x
2
n
+
a
x
n
+
b
x^{2n} + ax^n + b
x
2
n
+
a
x
n
+
b
.
2
2
Hide problems
An operation which will be applied to a row of bars
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, \ldots ,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, \ldots,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 \geq 0}
(
a
n
)
n
≥
0
, where an is the number of bars after having applied the operation
n
n
n
times.(a) Prove that for no
n
>
0
n > 0
n
>
0
can we have
a
n
=
1997
a_n = 1997
a
n
=
1997
.(b) Determine all natural numbers that can only occur as
a
0
a_0
a
0
or
a
1
a_1
a
1
.
The explicit formula for the sequence
A positive integer
K
K
K
is given. Define the sequence
(
a
n
)
(a_n)
(
a
n
)
by
a
1
=
1
a_1 = 1
a
1
=
1
and
a
n
a_n
a
n
is the
n
n
n
-th positive integer greater than
a
n
−
1
a_{n-1}
a
n
−
1
which is congruent to
n
n
n
modulo
K
K
K
.(a) Find an explicit formula for
a
n
a_n
a
n
.(b) What is the result if
K
=
2
K = 2
K
=
2
?
1
2
Hide problems
Find all integers x,y for a fixed integer a
Let
a
a
a
be a fixed integer. Find all integer solutions
x
,
y
,
z
x, y, z
x
,
y
,
z
of the system
5
x
+
(
a
+
2
)
y
+
(
a
+
2
)
z
=
a
,
5x + (a + 2)y + (a + 2)z = a,
5
x
+
(
a
+
2
)
y
+
(
a
+
2
)
z
=
a
,
(
2
a
+
4
)
x
+
(
a
2
+
3
)
y
+
(
2
a
+
2
)
z
=
3
a
−
1
,
(2a + 4)x + (a^2 + 3)y + (2a + 2)z = 3a - 1,
(
2
a
+
4
)
x
+
(
a
2
+
3
)
y
+
(
2
a
+
2
)
z
=
3
a
−
1
,
(
2
a
+
4
)
x
+
(
2
a
+
2
)
y
+
(
a
2
+
3
)
z
=
a
+
1.
(2a + 4)x + (2a + 2)y + (a^2 + 3)z = a + 1.
(
2
a
+
4
)
x
+
(
2
a
+
2
)
y
+
(
a
2
+
3
)
z
=
a
+
1.
Determine all quadruples (a, b, c, d) of real numbers
Determine all quadruples
(
a
,
b
,
c
,
d
)
(a, b, c, d)
(
a
,
b
,
c
,
d
)
of real numbers satisfying the equation
256
a
3
b
3
c
3
d
3
=
(
a
6
+
b
2
+
c
2
+
d
2
)
(
a
2
+
b
6
+
c
2
+
d
2
)
(
a
2
+
b
2
+
c
6
+
d
2
)
(
a
2
+
b
2
+
c
2
+
d
6
)
.
256a^3b^3c^3d^3 = (a^6+b^2+c^2+d^2)(a^2+b^6+c^2+d^2)(a^2+b^2+c^6+d^2)(a^2+b^2+c^2+d^6).
256
a
3
b
3
c
3
d
3
=
(
a
6
+
b
2
+
c
2
+
d
2
)
(
a
2
+
b
6
+
c
2
+
d
2
)
(
a
2
+
b
2
+
c
6
+
d
2
)
(
a
2
+
b
2
+
c
2
+
d
6
)
.