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Problems
Contests
National and Regional Contests
Austria Contests
Austrian MO Beginners' Competition
2013 Austria Beginners' Competition
2013 Austria Beginners' Competition
Part of
Austrian MO Beginners' Competition
Subcontests
(4)
2
1
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numbers of paths on a grid
The following figure is given: https://cdn.artofproblemsolving.com/attachments/9/b/97a30e248fcd6f098a900c89721a2e1b3b3f0e.png Determine the number of paths from the starting square
A
A
A
to the target square
Z
Z
Z
, where a path consists of steps from a square to its top or right neighbor square .(W. Janous, WRG Ursulinen, Innsbruck)
3
1
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a/(b + 1)+b/(a + 1)<=1 if 0<=a, b<= 1
Let
a
a
a
and
b
b
b
be real numbers with
0
≤
a
,
b
≤
1
0\le a, b\le 1
0
≤
a
,
b
≤
1
. Prove that
a
b
+
1
+
b
a
+
1
≤
1
\frac{a}{b + 1}+\frac{b}{a + 1}\le 1
b
+
1
a
+
a
+
1
b
≤
1
When does equality holds?(K. Czakler, GRG 21, Vienna)
4
1
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rectangle wanted, 4 midpoints, point on altitude
Let
A
B
C
ABC
A
BC
be an acute-angled triangle and
D
D
D
a point on the altitude through
C
C
C
. Let
E
E
E
,
F
F
F
,
G
G
G
and
H
H
H
be the midpoints of the segments
A
D
AD
A
D
,
B
D
BD
B
D
,
B
C
BC
BC
and
A
C
AC
A
C
. Show that
E
E
E
,
F
F
F
,
G
G
G
, and
H
H
H
form a rectangle.(G. Anegg, Innsbruck)
1
1
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n + its second largest divisor = 2013
Find all natural numbers
n
>
1
n> 1
n
>
1
for which the following applies: The sum of the number
n
n
n
and its second largest divisor is
2013
2013
2013
.(R. Henner, Vienna)