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Problems
Contests
National and Regional Contests
Austria Contests
Austrian MO Beginners' Competition
2018 Austria Beginners' Competition
2018 Austria Beginners' Competition
Part of
Austrian MO Beginners' Competition
Subcontests
(4)
2
1
Hide problems
AM = AF iff <BAC = 60^o
Let
A
B
C
ABC
A
BC
be an acute-angled triangle,
M
M
M
the midpoint of the side
A
C
AC
A
C
and
F
F
F
the foot on
A
B
AB
A
B
of the altitude through the vertex
C
C
C
. Prove that
A
M
=
A
F
AM = AF
A
M
=
A
F
holds if and only if
∠
B
A
C
=
6
0
o
\angle BAC = 60^o
∠
B
A
C
=
6
0
o
.(Karl Czakler)
1
1
Hide problems
a/c+c/b \ge 4a/(a + b) for a,b,c>0
Let
a
,
b
a, b
a
,
b
and
c
c
c
denote positive real numbers. Prove that
a
c
+
c
b
≥
4
a
a
+
b
\frac{a}{c}+\frac{c}{b}\ge \frac{4a}{a + b}
c
a
+
b
c
≥
a
+
b
4
a
. When does equality hold?(Walther Janous)
3
1
Hide problems
filling a 3xn table with numbers 1 to 3n, sum of rows and columns related
For a given integer
n
≥
4
n \ge 4
n
≥
4
we examine whether there exists a table with three rows and
n
n
n
columns which can be filled by the numbers
1
,
2
,
.
.
.
,
,
3
n
1, 2,...,, 3n
1
,
2
,
...
,,
3
n
such that
∙
\bullet
∙
each row totals to the same sum
z
z
z
and
∙
\bullet
∙
each column totals to the same sum
s
s
s
. Prove: (a) If
n
n
n
is even, such a table does not exist. (b) If
n
=
5
n = 5
n
=
5
, such a table does exist.(Gerhard J. Woeginger)
4
1
Hide problems
s(x) d(x) = 96, sum and no of positive divisors
For a positive integer
n
n
n
we denote by
d
(
n
)
d(n)
d
(
n
)
the number of positive divisors of
n
n
n
and by
s
(
n
)
s(n)
s
(
n
)
the sum of these divisors. For example,
d
(
2018
)
d(2018)
d
(
2018
)
is equal to
4
4
4
since
2018
2018
2018
has four divisors
(
1
,
2
,
1009
,
2018
)
(1, 2, 1009, 2018)
(
1
,
2
,
1009
,
2018
)
and
s
(
2018
)
=
1
+
2
+
1009
+
2018
=
3030
s(2018) = 1 + 2 + 1009 + 2018 = 3030
s
(
2018
)
=
1
+
2
+
1009
+
2018
=
3030
. Determine all positive integers
x
x
x
such that
s
(
x
)
⋅
d
(
x
)
=
96
s(x) \cdot d(x) = 96
s
(
x
)
⋅
d
(
x
)
=
96
.(Richard Henner)