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Contests
National and Regional Contests
Austria Contests
Austrian MO Regional Competition
2015 Regional Competition For Advanced Students
2015 Regional Competition For Advanced Students
Part of
Austrian MO Regional Competition
Subcontests
(4)
4
1
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Problem 4 -- Problematic Parallel and Perpendicular Proofs
Let
A
B
C
ABC
A
BC
be an isosceles triangle with
A
C
=
B
C
AC = BC
A
C
=
BC
and
∠
A
C
B
<
6
0
∘
\angle ACB < 60^\circ
∠
A
CB
<
6
0
∘
. We denote the incenter and circumcenter by
I
I
I
and
O
O
O
, respectively. The circumcircle of triangle
B
I
O
BIO
B
I
O
intersects the leg
B
C
BC
BC
also at point
D
≠
B
D \ne B
D
=
B
.(a) Prove that the lines
A
C
AC
A
C
and
D
I
DI
D
I
are parallel. (b) Prove that the lines
O
D
OD
O
D
and
I
B
IB
I
B
are mutually perpendicular.(Walther Janous)
3
1
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Problem 3 -- Whiteboard Wipeout
Let
n
≥
3
n \ge 3
n
≥
3
be a fixed integer. The numbers
1
,
2
,
3
,
⋯
,
n
1,2,3, \cdots , n
1
,
2
,
3
,
⋯
,
n
are written on a board. In every move one chooses two numbers and replaces them by their arithmetic mean. This is done until only a single number remains on the board.Determine the least integer that can be reached at the end by an appropriate sequence of moves.(Theresia Eisenkölbl)
2
1
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Problem 2 -- One Not More Than One
Let
x
x
x
,
y
y
y
, and
z
z
z
be positive real numbers with
x
+
y
+
z
=
3
x+y+z = 3
x
+
y
+
z
=
3
. Prove that at least one of the three numbers
x
(
x
+
y
−
z
)
x(x+y-z)
x
(
x
+
y
−
z
)
y
(
y
+
z
−
x
)
y(y+z-x)
y
(
y
+
z
−
x
)
z
(
z
+
x
−
y
)
z(z+x-y)
z
(
z
+
x
−
y
)
is less or equal
1
1
1
.(Karl Czakler)
1
1
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Problem 1 -- Groovy GCDs
Determine all triples
(
a
,
b
,
c
)
(a,b,c)
(
a
,
b
,
c
)
of positive integers satisfying the conditions
gcd
(
a
,
20
)
=
b
\gcd(a,20) = b
g
cd
(
a
,
20
)
=
b
gcd
(
b
,
15
)
=
c
\gcd(b,15) = c
g
cd
(
b
,
15
)
=
c
gcd
(
a
,
c
)
=
5
\gcd(a,c) = 5
g
cd
(
a
,
c
)
=
5
(Richard Henner)