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Problems
Contests
National and Regional Contests
Austria Contests
Austrian MO National Competition
2008 Federal Competition For Advanced Students, Part 2
2008 Federal Competition For Advanced Students, Part 2
Part of
Austrian MO National Competition
Subcontests
(3)
2
2
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Polynomial with coefficients in integers
(a) Does there exist a polynomial
P
(
x
)
P(x)
P
(
x
)
with coefficients in integers, such that P(d) \equal{} \frac{2008}{d} holds for all positive divisors of
2008
2008
2008
? (b) For which positive integers
n
n
n
does a polynomial
P
(
x
)
P(x)
P
(
x
)
with coefficients in integers exists, such that P(d) \equal{} \frac{n}{d} holds for all positive divisors of
n
n
n
?
Find the missing positive integers
Which positive integers are missing in the sequence
{
a
n
}
\left\{a_n\right\}
{
a
n
}
, with a_n \equal{} n \plus{} \left[\sqrt n\right] \plus{}\left[\sqrt [3]n\right] for all
n
≥
1
n \ge 1
n
≥
1
? (
[
x
]
\left[x\right]
[
x
]
denotes the largest integer less than or equal to
x
x
x
, i.e.
g
g
g
with g \le x < g \plus{} 1.)
1
2
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Inequality with condition a + b + c = 1
Prove the inequality \sqrt {a^{1 \minus{} a}b^{1 \minus{} b}c^{1 \minus{} c}} \le \frac {1}{3} holds for all positive real numbers
a
a
a
,
b
b
b
and
c
c
c
with a \plus{} b \plus{} c \equal{} 1.
Determine all functions
Determine all functions
f
f
f
mapping the set of positive integers to the set of non-negative integers satisfying the following conditions: (1) f(mn) \equal{} f(m)\plus{}f(n), (2) f(2008) \equal{} 0, and (3) f(n) \equal{} 0 for all
n
≡
39
(
m
o
d
2008
)
n \equiv 39\pmod {2008}
n
≡
39
(
mod
2008
)
.
3
2
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Construct a square with a straightedge and compass
We are given a line
g
g
g
with four successive points
P
P
P
,
Q
Q
Q
,
R
R
R
,
S
S
S
, reading from left to right. Describe a straightedge and compass construction yielding a square
A
B
C
D
ABCD
A
BC
D
such that
P
P
P
lies on the line
A
D
AD
A
D
,
Q
Q
Q
on the line
B
C
BC
BC
,
R
R
R
on the line
A
B
AB
A
B
and
S
S
S
on the line
C
D
CD
C
D
.
Determine the set of all acceptable points
We are given a square
A
B
C
D
ABCD
A
BC
D
. Let
P
P
P
be a point not equal to a corner of the square or to its center
M
M
M
. For any such
P
P
P
, we let
E
E
E
denote the common point of the lines
P
D
PD
P
D
and
A
C
AC
A
C
, if such a point exists. Furthermore, we let
F
F
F
denote the common point of the lines
P
C
PC
PC
and
B
D
BD
B
D
, if such a point exists. All such points
P
P
P
, for which
E
E
E
and
F
F
F
exist are called acceptable points. Determine the set of all acceptable points, for which the line
E
F
EF
EF
is parallel to
A
D
AD
A
D
.