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National and Regional Contests
Austria Contests
Austrian MO National Competition
2015 Federal Competition For Advanced Students
2015 Federal Competition For Advanced Students
Part of
Austrian MO National Competition
Subcontests
(4)
4
1
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Problem 4 — Emergency NT Problem
A police emergency number is a positive integer that ends with the digits
133
133
133
in decimal representation. Prove that every police emergency number has a prime factor larger than
7
7
7
.(In Austria,
133
133
133
is the emergency number of the police.)(Robert Geretschläger)
3
1
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Problem 3 — Cut the (Pearl) Rope
Alice and Bob play a game with a string of
2015
2015
2015
pearls.In each move, one player cuts the string between two pearls and the other player chooses one of the resulting parts of the string while the other part is discarded.In the first move, Alice cuts the string, thereafter, the players take turns.A player loses if he or she obtains a string with a single pearl such that no more cut is possible.Who of the two players does have a winning strategy?(Theresia Eisenkölbl)
2
1
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Problem 2 — Angle from Triangle Lines Must Be Right
Let
A
B
C
ABC
A
BC
be an acute-angled triangle with
A
C
<
A
B
AC < AB
A
C
<
A
B
and circumradius
R
R
R
. Furthermore, let
D
D
D
be the foot ofthe altitude from
A
A
A
on
B
C
BC
BC
and let
T
T
T
denote the point on the line
A
D
AD
A
D
such that
A
T
=
2
R
AT = 2R
A
T
=
2
R
holds with
D
D
D
lying between
A
A
A
and
T
T
T
. Finally, let
S
S
S
denote the mid-point of the arc
B
C
BC
BC
on the circumcircle that does not include
A
A
A
.Prove:
∠
A
S
T
=
9
0
∘
\angle AST = 90^\circ
∠
A
ST
=
9
0
∘
.(Karl Czakler)
1
1
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Problem 1 — Symmetric Squares, Symmetric Products
Let
a
a
a
,
b
b
b
,
c
c
c
,
d
d
d
be positive numbers. Prove that
(
a
2
+
b
2
+
c
2
+
d
2
)
2
≥
(
a
+
b
)
(
b
+
c
)
(
c
+
d
)
(
d
+
a
)
(a^2 + b^2 + c^2 + d^2)^2 \ge (a+b)(b+c)(c+d)(d+a)
(
a
2
+
b
2
+
c
2
+
d
2
)
2
≥
(
a
+
b
)
(
b
+
c
)
(
c
+
d
)
(
d
+
a
)
When does equality hold?(Georg Anegg)