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Austrian MO Regional Competition
2024 Austrian MO Regional Competition
2024 Austrian MO Regional Competition
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Austrian MO Regional Competition
Subcontests
(4)
1
1
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sum ab/(c-1) >= 12 if a,b,c>1
Let
a
a
a
,
b
b
b
and
c
c
c
be real numbers larger than
1
1
1
. Prove the inequality
a
b
c
−
1
+
b
c
a
−
1
+
c
a
b
−
1
≥
12.
\frac{ab}{c-1}+\frac{bc}{a - 1}+\frac{ca}{b -1} \ge 12.
c
−
1
ab
+
a
−
1
b
c
+
b
−
1
c
a
≥
12.
When does equality hold?(Karl Czakler)
4
1
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n^5 +5^n is divisible by 11 iff n^5 x 5^n +1 is divisible by 11
Let
n
n
n
be a positive integer. Prove that
a
(
n
)
=
n
5
+
5
n
a(n) = n^5 +5^n
a
(
n
)
=
n
5
+
5
n
is divisible by
11
11
11
if and only if
b
(
n
)
=
n
5
⋅
5
n
+
1
b(n) = n^5 · 5^n +1
b
(
n
)
=
n
5
⋅
5
n
+
1
is divisible by
11
11
11
.(Walther Janous)
3
1
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ten thousand matches and a bowl, 2 player game
On a table, we have ten thousand matches, two of which are inside a bowl. Anna and Bernd play the following game: They alternate taking turns and Anna begins. A turn consists of counting the matches in the bowl, choosing a proper divisor
d
d
d
of this number and adding
d
d
d
matches to the bowl. The game ends when more than
2024
2024
2024
matches are in the bowl. The person who played the last turn wins. Prove that Anna can win independently of how Bernd plays.(Richard Henner)
2
1
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orthocenter of ABC is circumcenter of APQ
Let
A
B
C
ABC
A
BC
be an acute triangle with orthocenter
H
H
H
. The circumcircle of the triangle
B
H
C
BHC
B
H
C
intersects
A
C
AC
A
C
a second time in point
P
P
P
and
A
B
AB
A
B
a second time in point
Q
Q
Q
. Prove that
H
H
H
is the circumcenter of the triangle
A
P
Q
APQ
A
PQ
. (Karl Czakler)