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Problems
Contests
National and Regional Contests
Germany Contests
German National Olympiad
2013 German National Olympiad
2013 German National Olympiad
Part of
German National Olympiad
Subcontests
(6)
3
1
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Three circles with some intersections
Given two circles
k
1
k_1
k
1
and
k
2
k_2
k
2
which intersect at
Q
Q
Q
and
Q
′
.
Q'.
Q
′
.
Let
P
P
P
be a point on
k
2
k_2
k
2
and inside of
k
1
k_1
k
1
such that the line
P
Q
PQ
PQ
intersects
k
1
k_1
k
1
in a point
X
≠
Q
X\ne Q
X
=
Q
and such that the tangent to
k
1
k_1
k
1
at
X
X
X
intersects
k
2
k_2
k
2
in points
A
A
A
and
B
.
B.
B
.
Let
k
k
k
be the circle through
A
,
B
A,B
A
,
B
which is tangent to the line through
P
P
P
parallel to
A
B
.
AB.
A
B
.
Prove that the circles
k
1
k_1
k
1
and
k
k
k
are tangent.
4
1
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Length of paths on a cube.
Let
A
B
C
D
E
F
G
H
ABCDEFGH
A
BC
D
EFG
H
be a cube of sidelength
a
a
a
and such that
A
G
AG
A
G
is one of the space diagonals. Consider paths on the surface of this cube. Then determine the set of points
P
P
P
on the surface for which the shortest path from
P
P
P
to
A
A
A
and from
P
P
P
to
G
G
G
have the same length
l
.
l.
l
.
Also determine all possible values of
l
l
l
depending on
a
.
a.
a
.
5
1
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Forming a further commission is possible
Five people form several commissions to prepare a competition. Here any commission must be nonempty and any two commissions cannot contain the same members. Moreover, any two commissions have at least one common member. There are already
14
14
14
commissions. Prove that at least one additional commission can be formed.
6
1
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A sequence approximating \sqrt{2}
Define a sequence
(
a
n
)
(a_n)
(
a
n
)
by
a
1
=
1
,
a
2
=
2
,
a_1 =1, a_2 =2,
a
1
=
1
,
a
2
=
2
,
and
a
k
+
2
=
2
a
k
+
1
+
a
k
a_{k+2}=2a_{k+1}+a_k
a
k
+
2
=
2
a
k
+
1
+
a
k
for all positive integers
k
k
k
. Determine all real numbers
β
>
0
\beta >0
β
>
0
which satisfy the following conditions:(A) There are infinitely pairs of positive integers
(
p
,
q
)
(p,q)
(
p
,
q
)
such that
∣
p
q
−
2
∣
<
β
q
2
.
\left| \frac{p}{q}- \sqrt{2} \, \right| < \frac{\beta}{q^2 }.
q
p
−
2
<
q
2
β
.
(B) There are only finitely many pairs of positive integers
(
p
,
q
)
(p,q)
(
p
,
q
)
with
∣
p
q
−
2
∣
<
β
q
2
\left| \frac{p}{q}- \sqrt{2} \,\right| < \frac{\beta}{q^2 }
q
p
−
2
<
q
2
β
for which there is no index
k
k
k
with
q
=
a
k
.
q=a_k.
q
=
a
k
.
2
1
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Nested Radical Sum
Let
α
\alpha
α
be a real number with
α
>
1
\alpha>1
α
>
1
. Let the sequence
(
a
n
)
(a_n)
(
a
n
)
be defined as
a
n
=
1
+
2
+
3
+
…
+
n
+
n
+
1
α
α
α
α
a_n=1+\sqrt[\alpha]{2+\sqrt[\alpha]{3+\ldots+\sqrt[\alpha]{n+\sqrt[\alpha]{n+1}}}}
a
n
=
1
+
α
2
+
α
3
+
…
+
α
n
+
α
n
+
1
for all positive integers
n
n
n
. Show that there exists a positive real constant
C
C
C
such that
a
n
<
C
a_n<C
a
n
<
C
for all positive integers
n
n
n
.
1
1
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Diophantine equation - Complete squares
Find all positive integers
n
n
n
such that
n
2
+
2
n
n^{2}+2^{n}
n
2
+
2
n
is square of an integer.