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Contests
National and Regional Contests
Austria Contests
Austrian MO Regional Competition
2023 Austrian MO Regional Competition
2023 Austrian MO Regional Competition
Part of
Austrian MO Regional Competition
Subcontests
(4)
3
1
Hide problems
(a_1 + b_1, a_2 +b_2,..., a_n + b_n) consecutive naturals for permutations
Determine all natural numbers
n
≥
2
n \ge 2
n
≥
2
with the property that there are two permutations
(
a
1
,
a
2
,
.
.
.
,
a
n
)
(a_1, a_2,... , a_n)
(
a
1
,
a
2
,
...
,
a
n
)
and
(
b
1
,
b
2
,
.
.
.
,
b
n
)
(b_1, b_2,... , b_n)
(
b
1
,
b
2
,
...
,
b
n
)
of the numbers
1
,
2
,
.
.
.
,
n
1, 2,..., n
1
,
2
,
...
,
n
such that
(
a
1
+
b
1
,
a
2
+
b
2
,
.
.
.
,
a
n
+
b
n
)
(a_1 + b_1, a_2 +b_2,..., a_n + b_n)
(
a
1
+
b
1
,
a
2
+
b
2
,
...
,
a
n
+
b
n
)
are consecutive natural numbers.(Walther Janous)
4
1
Hide problems
xyd = x + y + d^2 where d = gcd(x, y)
Determine all pairs
(
x
,
y
)
(x, y)
(
x
,
y
)
of positive integers such that for
d
=
g
c
d
(
x
,
y
)
d = gcd(x, y)
d
=
g
c
d
(
x
,
y
)
the equation
x
y
d
=
x
+
y
+
d
2
xyd = x + y + d^2
x
y
d
=
x
+
y
+
d
2
holds.(Walther Janous)
2
1
Hide problems
concyclic wanted, starting with a rhombusq
Let
A
B
C
D
ABCD
A
BC
D
be a rhombus with
∠
B
A
D
<
9
0
o
\angle BAD < 90^o
∠
B
A
D
<
9
0
o
. The circle passing through
D
D
D
with center
A
A
A
intersects the line
C
D
CD
C
D
a second time in point
E
E
E
. Let
S
S
S
be the intersection of the lines
B
E
BE
BE
and
A
C
AC
A
C
. Prove that the points
A
A
A
,
S
S
S
,
D
D
D
and
E
E
E
lie on a circle.(Karl Czakler)
1
1
Hide problems
(a - b)(b - c)(a- c)<= 2 if 0 <=a, b, c <= 2.
Let
a
a
a
,
b
b
b
and
c
c
c
be real numbers with
0
≤
a
,
b
,
c
≤
2
0 \le a, b, c \le 2
0
≤
a
,
b
,
c
≤
2
. Prove that
(
a
−
b
)
(
b
−
c
)
(
a
−
c
)
≤
2.
(a - b)(b - c)(a- c) \le 2.
(
a
−
b
)
(
b
−
c
)
(
a
−
c
)
≤
2.
When does equality hold?(Karl Czakler)