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Contests
National and Regional Contests
Austria Contests
Austrian MO Regional Competition
2008 Regional Competition For Advanced Students
2008 Regional Competition For Advanced Students
Part of
Austrian MO Regional Competition
Subcontests
(4)
4
1
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39th Austrian Mathematical Olympiad 2008
For every positive integer
n
n
n
let a_n\equal{}\sum_{k\equal{}n}^{2n}\frac{(2k\plus{}1)^n}{k} Show that there exists no
n
n
n
, for which
a
n
a_n
a
n
is a non-negative integer.
3
1
Hide problems
39th Austrian Mathematical Olympiad 2008
Given is an acute angled triangle
A
B
C
ABC
A
BC
. Determine all points
P
P
P
inside the triangle with
1
≤
∠
A
P
B
∠
A
C
B
,
∠
B
P
C
∠
B
A
C
,
∠
C
P
A
∠
C
B
A
≤
2
1\leq\frac{\angle APB}{\angle ACB},\frac{\angle BPC}{\angle BAC},\frac{\angle CPA}{\angle CBA}\leq2
1
≤
∠
A
CB
∠
A
PB
,
∠
B
A
C
∠
BPC
,
∠
CB
A
∠
CP
A
≤
2
2
1
Hide problems
39th Austrian Mathematical Olympiad
For a real number
x
x
x
is
[
x
]
[x]
[
x
]
the next smaller integer to
x
x
x
, that is the integer
g
g
g
with
g
≦
<
g
+
1
g\leqq<g+1
g
≦
<
g
+
1
, and
{
x
}
=
x
−
[
x
]
\{x\}=x-[x]
{
x
}
=
x
−
[
x
]
is the “decimal part” of
x
x
x
. Determine all triples
(
a
,
b
,
c
)
(a,b,c)
(
a
,
b
,
c
)
of real numbers, which fulfil the following system of equations:
{
a
}
+
+
{
c
}
=
2
,
9
\{a\}++\{c\}=2,9
{
a
}
+
+
{
c
}
=
2
,
9
{
b
}
+
[
c
]
+
{
a
}
=
5
,
3
\{b\}+[c]+\{a\}=5,3
{
b
}
+
[
c
]
+
{
a
}
=
5
,
3
{
c
}
+
[
a
]
+
{
b
}
=
4
,
0
\{c\}+[a]+\{b\}=4,0
{
c
}
+
[
a
]
+
{
b
}
=
4
,
0
1
1
Hide problems
39th Austrian Mathematical Olympiad
Show: For all real numbers
a
,
b
,
c
a,b,c
a
,
b
,
c
with
0
<
a
,
b
,
c
<
1
0<a,b,c<1
0
<
a
,
b
,
c
<
1
is: \sqrt{a^2bc\plus{}ab^2c\plus{}abc^2}\plus{}\sqrt{(1\minus{}a)^2(1\minus{}b)(1\minus{}c)\plus{}(1\minus{}a)(1\minus{}b)^2(1\minus{}c)\plus{}(1\minus{}a)(1\minus{}b)(1\minus{}c)^2}<\sqrt{3}.