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Problems
Contests
National and Regional Contests
Germany Contests
German National Olympiad
2001 German National Olympiad
2001 German National Olympiad
Part of
German National Olympiad
Subcontests
(7)
4
1
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In how many ways can the Nikolaus House be drawn
In how many ways can the ”Nikolaus’ House” (see the picture) be drawn? Edges may not be erased nor duplicated, and no additional edges may be drawn. https://cdn.artofproblemsolving.com/attachments/0/5/33795820e0335686b06255180af698e536a9be.png
5
1
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for which n can a Fibonacci number end in n 9-s in the decimal system.
The Fibonacci sequence is given by
x
1
=
x
2
=
1
x_1 = x_2 = 1
x
1
=
x
2
=
1
and
x
k
+
2
=
x
k
+
1
+
x
k
x_{k+2} = x_{k+1} + x_k
x
k
+
2
=
x
k
+
1
+
x
k
for each
k
∈
N
k \in N
k
∈
N
. (a) Prove that there are Fibonacci numbes that end in a
9
9
9
in the decimal system. (b) Determine for which
n
n
n
can a Fibonacci number end in
n
n
n
9
9
9
-s in the decimal system.
6 (12)
1
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line BD is tangent to the circumcircle of ADZ
Let
A
B
C
ABC
A
BC
be a triangle with
∠
A
=
9
0
o
\angle A = 90^o
∠
A
=
9
0
o
and
∠
B
<
∠
C
\angle B < \angle C
∠
B
<
∠
C
. The tangent at
A
A
A
to the circumcircle
k
k
k
of
△
A
B
C
\vartriangle ABC
△
A
BC
intersects line
B
C
BC
BC
at
D
D
D
. Let
E
E
E
be the reflection of
A
A
A
in
B
C
BC
BC
. Also, let
X
X
X
be the feet of the perpendicular from
A
A
A
to
B
E
BE
BE
and let
Y
Y
Y
be the midpoint of
A
X
AX
A
X
. Line
B
Y
BY
B
Y
meets
k
k
k
again at
Z
Z
Z
. Prove that line
B
D
BD
B
D
is tangent to the circumcircle of
△
A
D
Z
\vartriangle ADZ
△
A
D
Z
.
6 (11)
1
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ratio between the volumes of the pyramids
In a pyramid
S
A
B
C
D
SABCD
S
A
BC
D
with the base
A
B
C
D
ABCD
A
BC
D
the triangles
A
B
D
ABD
A
B
D
and
B
C
D
BCD
BC
D
have equal areas. Points
M
,
N
,
P
,
Q
M,N,P,Q
M
,
N
,
P
,
Q
are the midpoints of the edges
A
B
,
A
D
,
S
C
,
S
D
AB,AD,SC,SD
A
B
,
A
D
,
SC
,
S
D
respectively. Find the ratio between the volumes of the pyramids
S
A
B
C
D
SABCD
S
A
BC
D
and
M
N
P
Q
MNPQ
MNPQ
.
3
1
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game rectangle with 60 rows and 40 columns
Wiebke and Stefan play the following game on a rectangular sheet of paper. They start with a rectangle with
60
60
60
rows and
40
40
40
columns and cut it in turns into smaller rectangles. The cuttings must be made along the gridlines, and a player in turn may cut only one smaller rectangle. By that, Stefan makes only vertical cuts, while Wiebke makes only horizontal cuts. A player who cannot make a regular move loses the game. (a) Who has a winning strategy if Stefan makes the first move? (b) Who has a winning strategy if Wiebke makes the first move?
1
1
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x^4 -40x^2 +q = 0 has 4 roots in arithmetic progression
Determine all real numbers
q
q
q
for which the equation
x
4
−
40
x
2
+
q
=
0
x^4 -40x^2 +q = 0
x
4
−
40
x
2
+
q
=
0
has four real solutions which form an arithmetic progression
2
1
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Placing points in a rectangle
Determine the maximum possible number of points you can place in a rectangle with lengths
14
14
14
and
28
28
28
such that any two of those points are more than
10
10
10
apart from each other.