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Problems
Contests
National and Regional Contests
Austria Contests
Austrian MO National Competition
1998 Federal Competition For Advanced Students, Part 2
1998 Federal Competition For Advanced Students, Part 2
Part of
Austrian MO National Competition
Subcontests
(3)
3
2
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Calculate the sum in the sequence
Let
a
n
a_n
a
n
be a sequence recursively de fined by
a
0
=
0
,
a
1
=
1
a_0 = 0, a_1 = 1
a
0
=
0
,
a
1
=
1
and
a
n
+
2
=
a
n
+
1
+
a
n
a_{n+2} = a_{n+1} + a_n
a
n
+
2
=
a
n
+
1
+
a
n
. Calculate the sum of
a
n
(
2
5
)
n
a_n\left( \frac 25\right)^n
a
n
(
5
2
)
n
for all positive integers
n
n
n
. For what value of the base
b
b
b
we get the sum
1
1
1
?
Relationship between the lengths of the diagonals AC and BD
In a parallelogram
A
B
C
D
ABCD
A
BC
D
with the side ratio
A
B
:
B
C
=
2
:
3
AB : BC = 2 : \sqrt 3
A
B
:
BC
=
2
:
3
the normal through
D
D
D
to
A
C
AC
A
C
and the normal through
C
C
C
to
A
B
AB
A
B
intersects in the point
E
E
E
on the line
A
B
AB
A
B
. What is the relationship between the lengths of the diagonals
A
C
AC
A
C
and
B
D
BD
B
D
?
2
2
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A problem about product of even squares and odd cubes
Let
Q
n
Q_n
Q
n
be the product of the squares of even numbers less than or equal to
n
n
n
and
K
n
K_n
K
n
equal to the product of cubes of odd numbers less than or equal to
n
n
n
. What is the highest power of
98
98
98
, that a)
Q
n
Q_n
Q
n
, b)
K
n
K_n
K
n
or c)
Q
n
K
n
Q_nK_n
Q
n
K
n
divides? If one divides
Q
98
K
98
Q_{98}K_{98}
Q
98
K
98
by the highest power of
98
98
98
, then one get a number
N
N
N
. By which power-of-two number is
N
N
N
still divisible?
An interestin polynomial problem with integer roots
Let
P
(
x
)
=
x
3
−
p
x
2
+
q
x
−
r
P(x) = x^3 - px^2 + qx - r
P
(
x
)
=
x
3
−
p
x
2
+
q
x
−
r
be a cubic polynomial with integer roots
a
,
b
,
c
a, b, c
a
,
b
,
c
.(a) Show that the greatest common divisor of
p
,
q
,
r
p, q, r
p
,
q
,
r
is equal to
1
1
1
if the greatest common divisor of
a
,
b
,
c
a, b, c
a
,
b
,
c
is equal to
1
1
1
.(b) What are the roots of polynomial
Q
(
x
)
=
x
3
−
98
x
2
+
98
s
x
−
98
t
Q(x) = x^3-98x^2+98sx-98t
Q
(
x
)
=
x
3
−
98
x
2
+
98
s
x
−
98
t
with
s
,
t
s, t
s
,
t
positive integers.
1
2
Hide problems
Find all rational numbers x
Let
a
≥
0
a \geq 0
a
≥
0
be a natural number. Determine all rational
x
x
x
, so that
1
+
(
a
−
1
)
x
3
=
1
+
(
a
−
1
)
x
\sqrt{1+(a-1)\sqrt[3]x}=\sqrt{1+(a-1)\sqrt x}
1
+
(
a
−
1
)
3
x
=
1
+
(
a
−
1
)
x
All occurring square roots, are not negative.Note. It seems the set of natural numbers =
N
=
{
0
,
1
,
2
,
…
}
\mathbb N = \{0,1,2,\ldots\}
N
=
{
0
,
1
,
2
,
…
}
in this problem.
How many chains are there?
Let
M
M
M
be the set of the vertices of a regular hexagon, our Olympiad symbol. How many chains
∅
⊂
A
⊂
B
⊂
C
⊂
D
⊂
M
\emptyset \subset A \subset B \subset C \subset D \subset M
∅
⊂
A
⊂
B
⊂
C
⊂
D
⊂
M
of six different set, beginning with the empty set and ending with the
M
M
M
, are there?