MathDB
Problems
Contests
National and Regional Contests
Germany Contests
Bundeswettbewerb Mathematik
1985 Bundeswettbewerb Mathematik
1985 Bundeswettbewerb Mathematik
Part of
Bundeswettbewerb Mathematik
Subcontests
(4)
1
2
Hide problems
64 dice in 8x8 fomation, wanted is all face 1 on top
Sixty-four dice with the numbers ”one” to ”six” are placed on one table and formed into a square with eight horizontal and eight vertical rows of cubes pushed together. By rotating the dice, while maintaining their place, we want to finally have all sixty-four dice the "one" points upwards. Each dice however, may not be turned individually, but only every eight dice in a horizontal or vertical row together by
9
0
o
90^o
9
0
o
to the longitudinal axis of this row may turn. Prove that it is always possible to solve the dice by repeatedly applying the permitted type of rotation to the required end position.
no perfect power in sequence 11, 111, 1111, ...$
Prove that none of the numbers
11
,
111
,
1111
,
.
.
.
11, 111, 1111, ...
11
,
111
,
1111
,
...
is a square number, cube number or higher power of a natural number.
2
2
Hide problems
4 projections of foot of altitude are collinear
Prove that in every triangle for each of its altitudes: If you project the foof of one altitude on the other two altitudes and on the other two sides of the triangle, those four projections lie on the same line.
r1 + r2 + r3 + r4 = 2r for radii of inspheres of terrahedron
The insphere of any tetrahedron has radius
r
r
r
. The four tangential planes parallel to the side faces of the tetrahedron cut from the tetrahedron four smaller tetrahedrons whose in-sphere radii are
r
1
,
r
2
,
r
3
r_1, r_2, r_3
r
1
,
r
2
,
r
3
and
r
4
r_4
r
4
. Prove that
r
1
+
r
2
+
r
3
+
r
4
=
2
r
r_1 + r_2 + r_3 + r_4 = 2r
r
1
+
r
2
+
r
3
+
r
4
=
2
r
4
2
Hide problems
each point of the 3-dimensional space is coloured
Each point of the 3-dimensional space is coloured with exactly one of the colours red, green and blue. Let
R
R
R
,
G
G
G
and
B
B
B
, respectively, be the sets of the lengths of those segments in space whose both endpoints have the same colour (which means that both are red, both are green and both are blue, respectively). Prove that at least one of these three sets includes all non-negative reals.
512 persons meet at a meeting
512
512
512
persons meet at a meeting[ Under every six of these people there is always at least two who know each other. Prove that there must be six people at this gathering, all mutual know.
3
2
Hide problems
nice BWM sequence problem
Starting with the sequence
F
1
=
(
1
,
2
,
3
,
4
,
…
)
F_1 = (1,2,3,4, \ldots)
F
1
=
(
1
,
2
,
3
,
4
,
…
)
of the natural numbers further sequences are generated as follows:
F
n
+
1
F_{n+1}
F
n
+
1
is created from
F
n
F_n
F
n
by the following rule: the order of elements remains unchanged, the elements from
F
n
F_n
F
n
which are divisible by
n
n
n
are increased by 1 and the other elements from
F
n
F_n
F
n
remain unchanged. Example:
F
2
=
(
2
,
3
,
4
,
5
…
)
F_2 = (2,3,4,5 \ldots)
F
2
=
(
2
,
3
,
4
,
5
…
)
and
F
3
=
(
3
,
3
,
5
,
5
,
…
)
F_3 = (3,3,5,5, \ldots)
F
3
=
(
3
,
3
,
5
,
5
,
…
)
. Determine all natural numbers
n
n
n
such that exactly the first
n
−
1
n-1
n
−
1
elements of
F
n
F_n
F
n
take the value
n
.
n.
n
.
rays in space
From a point in space,
n
n
n
rays are issuing, whereas the angle among any two of these rays is at least
3
0
∘
30^{\circ}
3
0
∘
. Prove that
n
<
59
n < 59
n
<
59
.