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Problems
Contests
National and Regional Contests
Austria Contests
Austrian MO Regional Competition
2020 Regional Competition For Advanced Students
2020 Regional Competition For Advanced Students
Part of
Austrian MO Regional Competition
Subcontests
(4)
3
1
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concyclic wanted, incenter, perp. bisector, angle bisectors
Let a triangle
A
B
C
ABC
A
BC
be given with
A
B
<
A
C
AB <AC
A
B
<
A
C
. Let the inscribed center of the triangle be
I
I
I
. The perpendicular bisector of side
B
C
BC
BC
intersects the angle bisector of
B
A
C
BAC
B
A
C
at point
S
S
S
and the angle bisector of
C
B
A
CBA
CB
A
at point
T
T
T
. Prove that the points
C
,
I
,
S
C, I, S
C
,
I
,
S
and
T
T
T
lie on a circle.(Karl Czakler)
2
1
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7-digits numbers with digits 1, 3, 4, 6, 7, 8,9 each once
The set
M
M
M
consists of all
7
7
7
-digit positive integer numbers that contain (in decimal notation) each of the digits
1
,
3
,
4
,
6
,
7
,
8
1, 3, 4, 6, 7, 8
1
,
3
,
4
,
6
,
7
,
8
and
9
9
9
exactly once. (a) Find the smallest positive difference
d
d
d
of two numbers from
M
M
M
. (b) How many pairs
(
x
,
y
)
(x, y)
(
x
,
y
)
with
x
x
x
and
y
y
y
from M are there for which
x
−
y
=
d
x - y = d
x
−
y
=
d
?(Gerhard Kirchner)
4
1
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p^2 = q^2 + r^n , diophantine with 3 primes
Find all quadruples
(
p
,
q
,
r
,
n
)
(p, q, r, n)
(
p
,
q
,
r
,
n
)
of prime numbers
p
,
q
,
r
p, q, r
p
,
q
,
r
and positive integer numbers
n
n
n
, such that
p
2
=
q
2
+
r
n
p^2 = q^2 + r^n
p
2
=
q
2
+
r
n
(Walther Janous)
1
1
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Determine all solutions, equation
Let
a
a
a
be a positive integer. Determine all
a
a
a
such that the equation
(
1
+
1
x
)
⋅
(
1
+
1
x
+
1
)
⋯
(
1
+
1
x
+
a
)
=
a
−
x
\biggl( 1+\frac{1}{x} \biggr) \cdot \biggl( 1+\frac{1}{x+1} \biggr) \cdots \biggl( 1+\frac{1}{x+a} \biggr)=a-x
(
1
+
x
1
)
⋅
(
1
+
x
+
1
1
)
⋯
(
1
+
x
+
a
1
)
=
a
−
x
has at least one integer solution for
x
x
x
. For every such
a
a
a
state the respective solutions.(Richard Henner)