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Contests
National and Regional Contests
Austria Contests
Austrian MO National Competition
2011 Federal Competition For Advanced Students, Part 1
2011 Federal Competition For Advanced Students, Part 1
Part of
Austrian MO National Competition
Subcontests
(4)
4
1
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Minimal sum of distances in a tetrahedron
Inside or on the faces of a tetrahedron with five edges of length
2
2
2
and one edge of lenght
1
1
1
, there is a point
P
P
P
having distances
a
,
b
,
c
,
d
a, b, c, d
a
,
b
,
c
,
d
to the four faces of the tetrahedron. Determine the locus of all points
P
P
P
such that
a
+
b
+
c
+
d
a+b+c+d
a
+
b
+
c
+
d
is minimal and the locus of all points
P
P
P
such that
a
+
b
+
c
+
d
a+b+c+d
a
+
b
+
c
+
d
is maximal.
3
1
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Arithmetic and harmonic three-element Sets
A set of three elements is called arithmetic if one of its elements is the arithmetic mean of the other two. Likewise, a set of three elements is called harmonic if one of its elements is the harmonic mean of the other two.How many three-element subsets of the set of integers
{
z
∈
Z
∣
−
2011
<
z
<
2011
}
\left\{z\in\mathbb{Z}\mid -2011<z<2011\right\}
{
z
∈
Z
∣
−
2011
<
z
<
2011
}
are arithmetic and harmonic?(Remark: The arithmetic mean
A
(
a
,
b
)
A(a,b)
A
(
a
,
b
)
and the harmonic mean
H
(
a
,
b
)
H(a,b)
H
(
a
,
b
)
are defined as A(a,b)=\frac{a+b}{2} \mbox{and} H(a,b)=\frac{2ab}{a+b}=\frac{2}{\frac{1}{a}+\frac{1}{b}}\mbox{,} respectively, where
H
(
a
,
b
)
H(a,b)
H
(
a
,
b
)
is not defined for some
a
a
a
,
b
b
b
.)
2
1
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Maximum of f_k(x,y) with x^2+y^2=1
For a positive integer
k
k
k
and real numbers
x
x
x
and
y
y
y
, let f_k(x,y)=(x+y)-\left(x^{2k+1}+y^{2k+1}\right)\mbox{.} If
x
2
+
y
2
=
1
x^2+y^2=1
x
2
+
y
2
=
1
, then determine the maximal possible value
c
k
c_k
c
k
of
f
k
(
x
,
y
)
f_k(x,y)
f
k
(
x
,
y
)
.
1
1
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Integer triplets such that x^4+x^2=7^zy^2
Determine all integer triplets
(
x
,
y
,
z
)
(x,y,z)
(
x
,
y
,
z
)
such that x^4+x^2=7^zy^2\mbox{.}