MathDB
Problems
Contests
National and Regional Contests
Germany Contests
Bundeswettbewerb Mathematik
1984 Bundeswettbewerb Mathematik
1984 Bundeswettbewerb Mathematik
Part of
Bundeswettbewerb Mathematik
Subcontests
(4)
3
2
Hide problems
a^2 + b^2 + c^2 = d^2 if ab is even or odd
Let
a
a
a
and
b
b
b
be positive integers. Show that if
a
⋅
b
a \cdot b
a
⋅
b
is even, then there are positive integers
c
c
c
and
d
d
d
with
a
2
+
b
2
+
c
2
=
d
2
a^2 + b^2 + c^2 = d^2
a
2
+
b
2
+
c
2
=
d
2
; if, on the other hand,
a
⋅
b
a\cdot b
a
⋅
b
is odd, there are no such positive integers
c
c
c
and
d
d
d
.
a_{n+1} = a_n - b_n and b_{n+1} = 2b_n if a_n >=b_n, or else ...
The sequences
a
1
,
a
2
,
a
3
,
.
.
.
a_1, a_2, a_3,...
a
1
,
a
2
,
a
3
,
...
and
b
1
,
b
2
,
b
3
,
.
.
.
b_1, b_2, b_3,...
b
1
,
b
2
,
b
3
,
...
suffices for all positive integers
n
n
n
of the following recursion:
a
n
+
1
=
a
n
−
b
n
a_{n+1} = a_n - b_n
a
n
+
1
=
a
n
−
b
n
and
b
n
+
1
=
2
b
n
b_{n+1} = 2b_n
b
n
+
1
=
2
b
n
, if
a
n
≥
b
n
a_n \ge b_n
a
n
≥
b
n
,
a
n
+
1
=
2
a
n
a_{n+1} = 2a_n
a
n
+
1
=
2
a
n
and
b
n
+
1
=
b
n
−
a
n
b_{n+1} = b_n - a_n
b
n
+
1
=
b
n
−
a
n
, if
a
n
<
b
n
a_n < b_n
a
n
<
b
n
. For which pairs
(
a
1
,
b
1
)
(a_1, b_1)
(
a
1
,
b
1
)
of positive real initial terms is there an index
k
k
k
with
a
k
=
0
a_k = 0
a
k
=
0
?
2
2
Hide problems
sum of squares of lengths of different segments of regular n-gon
Given is a regular
n
n
n
-gon with circumradius
1
1
1
.
L
L
L
is the set of (different) lengths of all connecting segments of its endpoints. What is the sum of the squares of the elements of
L
L
L
?
subsets of plane wanted, at least 1 of 2 two semicircular arcs of AB
Determine all bounded closed subsets
F
F
F
of the plane with the following property:
F
F
F
consists of at least two points and always contains two points
A
A
A
and
B
B
B
as well as at least one of the two semicircular arcs over the segment
A
B
AB
A
B
. Definitions: A subset of the
F
F
F
of the plane is said to be closed if: For every point
P
P
P
of the plane that is not an element of
F
F
F
, there is a (non-degenerate) disc with center
P
P
P
that has no elements of
F
F
F
.
1
2
Hide problems
s(k) = 1 + z + z^2 + ...+ z^k divisible by n
The natural numbers
n
n
n
and
z
z
z
are relatively prime and greater than
1
1
1
. For
k
=
0
,
1
,
2
,
.
.
.
,
n
−
1
k = 0, 1, 2,..., n - 1
k
=
0
,
1
,
2
,
...
,
n
−
1
let
s
(
k
)
=
1
+
z
+
z
2
+
.
.
.
+
z
k
.
s(k) = 1 + z + z^2 + ...+ z^k.
s
(
k
)
=
1
+
z
+
z
2
+
...
+
z
k
.
Prove that: a) At least one of the numbers
s
(
k
)
s(k)
s
(
k
)
is divisible by
n
n
n
. b) If
n
n
n
and
z
−
1
z - 1
z
−
1
are also coprime, then already one of the numbers
s
(
k
)
s(k)
s
(
k
)
with
k
=
0
,
1
,
2
,
.
.
.
,
n
−
2
k = 0,1, 2,..., n- 2
k
=
0
,
1
,
2
,
...
,
n
−
2
is divisible by
n
n
n
.
2 player game with 2n-digit num divisible by 9
Let
n
n
n
be a positive integer and
M
=
{
1
,
2
,
3
,
4
,
5
,
6
}
M = \{1, 2, 3, 4, 5, 6\}
M
=
{
1
,
2
,
3
,
4
,
5
,
6
}
. Two persons
A
A
A
and
B
B
B
play in the following Way:
A
A
A
writes down a digit from
M
M
M
,
B
B
B
appends a digit from
M
M
M
, and so it becomes alternately one digit from
M
M
M
is appended until the
2
n
2n
2
n
-digit decimal representation of a number has been created. If this number is divisible by
9
9
9
,
B
B
B
wins, otherwise
A
A
A
wins. For which
n
n
n
can
A
A
A
and for which
n
n
n
can
B
B
B
force the win?
4
2
Hide problems
total length of the irrigation ditches > 70, in square 12x12
In a square field of side length
12
12
12
there is a source that contains a system of straight irrigation ditches. This is laid out in such a way that for every point of the field the distance to the next ditch is at most
1
1
1
. Here, the source is as a point and are the ditches to be regarded as stretches. It must be verified that the total length of the irrigation ditches is greater than
70
70
70
m. The sketch shows an example of a trench system of the type indicated. https://cdn.artofproblemsolving.com/attachments/6/5/5b51511da468cf14b5823c6acda3c4d2fe8280.png
quadrilateral is space
A sphere is touched by all the four sides of a (space) quadrilateral. Prove that all the four touching points are in the same plane.