MathDB
Problems
Contests
National and Regional Contests
Austria Contests
Austrian MO National Competition
2017 Federal Competition For Advanced Students, P2
2017 Federal Competition For Advanced Students, P2
Part of
Austrian MO National Competition
Subcontests
(6)
4
1
Hide problems
infinite triples of positive rational, 16xyz = (x + y)^2(x + z)^2, x+y+z =M, max
(a) Determine the maximum
M
M
M
of
x
+
y
+
z
x+y +z
x
+
y
+
z
where
x
,
y
x, y
x
,
y
and
z
z
z
are positive real numbers with
16
x
y
z
=
(
x
+
y
)
2
(
x
+
z
)
2
16xyz = (x + y)^2(x + z)^2
16
x
yz
=
(
x
+
y
)
2
(
x
+
z
)
2
. (b) Prove the existence of infinitely many triples
(
x
,
y
,
z
)
(x, y, z)
(
x
,
y
,
z
)
of positive rational numbers that satisfy
16
x
y
z
=
(
x
+
y
)
2
(
x
+
z
)
2
16xyz = (x + y)^2(x + z)^2
16
x
yz
=
(
x
+
y
)
2
(
x
+
z
)
2
and
x
+
y
+
z
=
M
x + y + z = M
x
+
y
+
z
=
M
.Proposed by Karl Czakler
5
1
Hide problems
another austrian midpoint, many perpendiculars
Let
A
B
C
ABC
A
BC
be an acute triangle. Let
H
H
H
denote its orthocenter and
D
,
E
D, E
D
,
E
and
F
F
F
the feet of its altitudes from
A
,
B
A, B
A
,
B
and
C
C
C
, respectively. Let the common point of
D
F
DF
D
F
and the altitude through
B
B
B
be
P
P
P
. The line perpendicular to
B
C
BC
BC
through
P
P
P
intersects
A
B
AB
A
B
in
Q
Q
Q
. Furthermore,
E
Q
EQ
EQ
intersects the altitude through
A
A
A
in
N
N
N
. Prove that
N
N
N
is the midpoint of
A
H
AH
A
H
.Proposed by Karl Czakler
6
1
Hide problems
maximal n, exist n distinct subsets of {1,2,...,2017} that no 2 have it as union
Let
S
=
{
1
,
2
,
.
.
.
,
2017
}
S = \{1,2,..., 2017\}
S
=
{
1
,
2
,
...
,
2017
}
. Find the maximal
n
n
n
with the property that there exist
n
n
n
distinct subsets of
S
S
S
such that for no two subsets their union equals
S
S
S
.Proposed by Gerhard Woeginger
3
1
Hide problems
a_{n+1} = a_n + 2/a_n , sequence of rationals without square of rational
Let
(
a
n
)
n
≥
0
(a_n)_{n\ge 0}
(
a
n
)
n
≥
0
be the sequence of rational numbers with
a
0
=
2016
a_0 = 2016
a
0
=
2016
and
a
n
+
1
=
a
n
+
2
a
n
a_{n+1} = a_n + \frac{2}{a_n}
a
n
+
1
=
a
n
+
a
n
2
for all
n
≥
0
n \ge 0
n
≥
0
. Show that the sequence does not contain a square of a rational number.Proposed by Theresia Eisenkölbl
2
1
Hide problems
necklace with 2016 pearls, black, green or blue
A necklace contains
2016
2016
2016
pearls, each of which has one of the colours black, green or blue. In each step we replace simultaneously each pearl with a new pearl, where the colour of the new pearl is determined as follows: If the two original neighbours were of the same colour, the new pearl has their colour. If the neighbours had two different colours, the new pearl has the third colour. (a) Is there such a necklace that can be transformed with such steps to a necklace of blue pearls if half of the pearls were black and half of the pearls were green at the start? (b) Is there such a necklace that can be transformed with such steps to a necklace of blue pearls if thousand of the pearls were black at the start and the rest green? (c) Is it possible to transform a necklace that contains exactly two adjacent black pearls and
2014
2014
2014
blue pearls to a necklace that contains one green pearl and
2015
2015
2015
blue pearls? Proposed byTheresia Eisenkölbl
1
1
Hide problems
Determine all f with f(f(x + y)f(x - y)) = x^2 + \alpha yf(y) for some alpha
Let
α
\alpha
α
be a fixed real number. Find all functions
f
:
R
→
R
f:\mathbb R \to \mathbb R
f
:
R
→
R
such that
f
(
f
(
x
+
y
)
f
(
x
−
y
)
)
=
x
2
+
α
y
f
(
y
)
f(f(x + y)f(x - y)) = x^2 + \alpha yf(y)
f
(
f
(
x
+
y
)
f
(
x
−
y
))
=
x
2
+
α
y
f
(
y
)
for all
x
,
y
∈
R
x,y \in \mathbb R
x
,
y
∈
R
.Proposed by Walther Janous