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Problems
Contests
National and Regional Contests
Austria Contests
Austrian MO Regional Competition
2021 Austrian MO Regional Competition
2021 Austrian MO Regional Competition
Part of
Austrian MO Regional Competition
Subcontests
(4)
4
1
Hide problems
x | (y + 1), y | (z + 1) and z | (x + 1)
Determine all triples
(
x
,
y
,
z
)
(x, y, z)
(
x
,
y
,
z
)
of positive integers satisfying
x
∣
(
y
+
1
)
x | (y + 1)
x
∣
(
y
+
1
)
,
y
∣
(
z
+
1
)
y | (z + 1)
y
∣
(
z
+
1
)
and
z
∣
(
x
+
1
)
z | (x + 1)
z
∣
(
x
+
1
)
.(Walther Janous)
3
1
Hide problems
repeat operation with numbers 1-2021 until one remains
The numbers
1
,
2
,
.
.
.
,
2020
1, 2, ..., 2020
1
,
2
,
...
,
2020
and
2021
2021
2021
are written on a blackboard. The following operation is executed: Two numbers are chosen, both are erased and replaced by the absolute value of their difference. This operation is repeated until there is only one number left on the blackboard. (a) Show that
2021
2021
2021
can be the final number on the blackboard. (b) Show that
2020
2020
2020
cannot be the final number on the blackboard.(Karl Czakler)
2
1
Hide problems
BC is tangent to circumcircle of BDE, isosceles ABC
Let
A
B
C
ABC
A
BC
be an isosceles triangle with
A
C
=
B
C
AC = BC
A
C
=
BC
and circumcircle
k
k
k
. The point
D
D
D
lies on the shorter arc of
k
k
k
over the chord
B
C
BC
BC
and is different from
B
B
B
and
C
C
C
. Let
E
E
E
denote the intersection of
C
D
CD
C
D
and
A
B
AB
A
B
. Prove that the line through
B
B
B
and
C
C
C
is a tangent of the circumcircle of the triangle
B
D
E
BDE
B
D
E
.(Karl Czakler)
1
1
Hide problems
c >= 1 if (a + 1)/(b + c)= b/a , for a,b \in N, c>0
Let
a
a
a
and
b
b
b
be positive integers and
c
c
c
be a positive real number satisfying
a
+
1
b
+
c
=
b
a
.
\frac{a + 1}{b + c}=\frac{b}{a}.
b
+
c
a
+
1
=
a
b
.
Prove that
c
≥
1
c \ge 1
c
≥
1
holds.(Karl Czakler)