MathDB
Problems
Contests
National and Regional Contests
Germany Contests
Bundeswettbewerb Mathematik
1994 Bundeswettbewerb Mathematik
1994 Bundeswettbewerb Mathematik
Part of
Bundeswettbewerb Mathematik
Subcontests
(4)
1
1
Hide problems
Bundeswettbewerb Mathematik 1994 Problem 1.1
Given eleven real numbers, prove that there exist two of them such that their decimal representations agree infinitely often.
3
2
Hide problems
Bundeswettbewerb Mathematik 1994 Problem 2.3
Let
A
A
A
and
B
B
B
be two spheres of different radii, both inscribed in a cone
K
K
K
. There are
m
m
m
other, congruent spheres arranged in a ring such that each of them touches
A
,
B
,
K
A, B, K
A
,
B
,
K
and two of the other spheres. Prove that this is possible for at most three values of
m
.
m.
m
.
Bundeswettbewerb Mathematik 1994 Problem 1.3
Given a triangle
A
1
A
2
A
3
A_1 A_2 A_3
A
1
A
2
A
3
and a point
P
P
P
inside. Let
B
i
B_i
B
i
be a point on the side opposite to
A
i
A_i
A
i
for
i
=
1
,
2
,
3
i=1,2,3
i
=
1
,
2
,
3
, and let
C
i
C_i
C
i
and
D
i
D_i
D
i
be the midpoints of
A
i
B
i
A_i B_i
A
i
B
i
and
P
B
i
P B_i
P
B
i
, respectively. Prove that the triangles
C
1
C
2
C
3
C_1 C_2 C_3
C
1
C
2
C
3
and
D
1
D
2
D
3
D_1 D_2 D_3
D
1
D
2
D
3
have equal area.
2
1
Hide problems
Bundeswettbewerb Mathematik 1994 Problem 2.2
Let
k
k
k
be an integer and define a sequence
a
0
,
a
1
,
a
2
,
…
a_0 , a_1 ,a_2 ,\ldots
a
0
,
a
1
,
a
2
,
…
by
a
0
=
0
,
a
1
=
k
and
a
n
+
2
=
k
2
a
n
+
1
−
a
n
for
n
≥
0.
a_0 =0 , \;\; a_1 =k \;\;\text{and} \;\; a_{n+2} =k^{2}a_{n+1}-a_n \; \text{for} \; n\geq 0.
a
0
=
0
,
a
1
=
k
and
a
n
+
2
=
k
2
a
n
+
1
−
a
n
for
n
≥
0.
Prove that
a
n
+
1
a
n
+
1
a_{n+1} a_n +1
a
n
+
1
a
n
+
1
divides
a
n
+
1
2
+
a
n
2
a_{n+1}^{2} +a_{n}^{2}
a
n
+
1
2
+
a
n
2
for all
n
n
n
.
4
2
Hide problems
Bundeswettbewerb Mathematik 1994 Problem 1.4
Let
a
,
b
a,b
a
,
b
be real numbers (
b
≠
0
b\ne 0
b
=
0
) and consider the infinite arithmetic sequence
a
,
a
+
b
,
a
+
2
b
,
…
.
a, a+b ,a +2b , \ldots.
a
,
a
+
b
,
a
+
2
b
,
…
.
Show that this sequence contains an infinite geometric subsequence if and only if
a
b
\frac{a}{b}
b
a
is rational.
Need help with Graph problem in space
Let S be a set of
n
≥
3
n\geq 3
n
≥
3
points in space. We color some line segments having two points in
S
S
S
as their endpoints red, let
r
r
r
be the number of edges colored red (color
r
r
r
edges red). We know no two colored segmentes have the same length. Proof there is a path of red edges increasing in size of length at least
⌈
2
r
n
⌉
\Bigg\lceil\frac{2r}{n}\Bigg\rceil
⌈
n
2
r
⌉
.