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Problems
Contests
National and Regional Contests
Austria Contests
Austrian MO Regional Competition
2018 Regional Competition For Advanced Students
2018 Regional Competition For Advanced Students
Part of
Austrian MO Regional Competition
Subcontests
(4)
4
1
Hide problems
d(n -1) + d(n) + d(n +1) <= 8 , d(n) is no. of all positive divisors of n>=2
Let
d
(
n
)
d(n)
d
(
n
)
be the number of all positive divisors of a natural number
n
≥
2
n \ge 2
n
≥
2
. Determine all natural numbers
n
≥
3
n \ge 3
n
≥
3
such that
d
(
n
−
1
)
+
d
(
n
)
+
d
(
n
+
1
)
≤
8
d(n -1) + d(n) + d(n + 1) \le 8
d
(
n
−
1
)
+
d
(
n
)
+
d
(
n
+
1
)
≤
8
.Proposed by Richard Henner
3
1
Hide problems
no of 3-elements subsets of {1,2,...,n} such one is arithmetic mean of other 2
Let
n
≥
3
n \ge 3
n
≥
3
be a natural number. Determine the number
a
n
a_n
a
n
of all subsets of
{
1
,
2
,
.
.
.
,
n
}
\{1, 2,...,n\}
{
1
,
2
,
...
,
n
}
consisting of three elements such that one of them is the arithmetic mean of the other two.Proposed by Walther Janous
2
1
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triangle $CSE$ is equilateral [Austria Regional 2018, Problem 2]
Let
k
k
k
be a circle with radius
r
r
r
and
A
B
AB
A
B
a chord of
k
k
k
such that
A
B
>
r
AB > r
A
B
>
r
. Furthermore, let
S
S
S
be the point on the chord
A
B
AB
A
B
satisfying
A
S
=
r
AS = r
A
S
=
r
. The perpendicular bisector of
B
S
BS
BS
intersects
k
k
k
in the points
C
C
C
and
D
D
D
. The line through
D
D
D
and
S
S
S
intersects
k
k
k
for a second time in point
E
E
E
. Show that the triangle
C
S
E
CSE
CSE
is equilateral.Proposed by Stefan Leopoldseder
1
1
Hide problems
If a+b<2, prove that sum 1/(1+a^2) \leq 2/(1+ab)
If
a
,
b
a, b
a
,
b
are positive reals such that
a
+
b
<
2
a+b<2
a
+
b
<
2
. Prove that
1
1
+
a
2
+
1
1
+
b
2
≤
2
1
+
a
b
\frac{1}{1+a^2}+\frac{1}{1+b^2} \le \frac{2}{1+ab}
1
+
a
2
1
+
1
+
b
2
1
≤
1
+
ab
2
and determine all
a
,
b
a, b
a
,
b
yielding equality.Proposed by Gottfried Perz