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Contests
National and Regional Contests
Germany Contests
Bundeswettbewerb Mathematik
1992 Bundeswettbewerb Mathematik
1992 Bundeswettbewerb Mathematik
Part of
Bundeswettbewerb Mathematik
Subcontests
(4)
2
2
Hide problems
n = b + c + d +... =b * c * d * ... , good positive integer
A positive integer
n
n
n
is called good if they sum up in one and only one way at least of two positive integers whose product also has the value
n
n
n
. Here representations that differ only in the order of the summands are considered the same viewed. Find all good positive integers.
algorithm to create a seuqence of 2^n words created by n gifits of {0,1}
All
n
n
n
-digit words from the alphabet
{
0
,
1
}
\{0, 1\}
{
0
,
1
}
considered. These
2
n
2^n
2
n
words should be in a sequence
w
0
,
w
1
,
w
2
,
.
.
.
,
w
2
−
1
w_0, w_1, w_2, ..., w_{2^-1}
w
0
,
w
1
,
w
2
,
...
,
w
2
−
1
be arranged that
w
m
w_m
w
m
from
w
m
−
1
w_{m-1}
w
m
−
1
by changing of a single ornament (
m
=
1
,
2
,
3
,
.
.
.
,
2
n
−
1
m = 1, 2, 3, ..., 2n-1
m
=
1
,
2
,
3
,
...
,
2
n
−
1
). Prove that the following algorithm achievesthis : a) Start with
w
0
=
000...00
w_0 = 000... 00
w
0
=
000...00
. b) Let
w
m
−
1
=
a
1
a
2
a
3
.
.
.
a
n
w_{m-1} = a_1a_2a_3 ... a_n
w
m
−
1
=
a
1
a
2
a
3
...
a
n
with
a
i
∈
{
0
;
1
}
a_i \in \{0; 1\}
a
i
∈
{
0
;
1
}
,
i
=
1
,
2
,
3
,
.
.
.
,
n
i = 1, 2, 3, ..., n
i
=
1
,
2
,
3
,
...
,
n
. Determine the exponent
e
(
m
)
e(m)
e
(
m
)
of the highest power of two dividing
m
m
m
and set
j
=
e
(
m
)
+
1
j = e(m)+1
j
=
e
(
m
)
+
1
. In
w
m
−
1
w_{m-1}
w
m
−
1
replace the ornament
a
j
a_j
a
j
with
1
−
a
j
1-aj
1
−
aj
. this is now
w
m
w_m
w
m
.
1
2
Hide problems
2 player game with stones, 1 bowl of p stones and one bowl of q stones
There are two bowls on the table, in one there are
p
p
p
, in the other
q
q
q
stones (
p
,
q
∈
N
∗
p, q \in N*
p
,
q
∈
N
∗
). Two players
A
A
A
and
B
B
B
take turns playing, starting with
A
A
A
. Who's turn:
∙
\bullet
∙
takes a stone from one of the bowls
∙
\bullet
∙
or removes one stone from each bowl
∙
\bullet
∙
or puts a stone from one of the bowls into the other. Whoever takes the last stone wins. Under what conditions can
A
A
A
and under what conditions can
B
B
B
force the win? The answer must be justified.
m in positive integer n such that f(n) = 2n, f(1992) = 2199, f(2000) = 200
Below the standard representation of a positive integer
n
n
n
is the representation understood by
n
n
n
in the decimal system, where the first digit is different from
0
0
0
. Everyone positive integer n is now assigned a number
f
(
n
)
f(n)
f
(
n
)
by using the standard representation of
n
n
n
last digit is placed before the first. Examples:
f
(
1992
)
=
2199
f(1992) = 2199
f
(
1992
)
=
2199
,
f
(
2000
)
=
200
f(2000) = 200
f
(
2000
)
=
200
. Determine the smallest positive integer
n
n
n
for which
f
(
n
)
=
2
n
f(n) = 2n
f
(
n
)
=
2
n
holds.
3
2
Hide problems
3 spheres touch plane of a triangle at it's vertices, tangent in pairs
Given is a triangle
A
B
C
ABC
A
BC
with side lengths
a
,
b
,
c
a, b,c
a
,
b
,
c
. Three spheres touch each other in pairs and also touch the plane of the triangle at points
A
,
B
A,B
A
,
B
and
C
C
C
, respectively. Determine the radii of these spheres.
convex equilateral pentagon
Provided a convex equilateral pentagon. On every side of the pentagon We construct equilateral triangles which run through the interior of the pentagon. Prove that at least one of the triangles does not protrude the pentagon's boundary.
4
2
Hide problems
i+a for i = 1, 2, 3, ..., k$ is even for set {a_i}
A finite set
{
a
1
,
a
2
,
.
.
.
a
k
}
\{a_1, a_2, ... a_k\}
{
a
1
,
a
2
,
...
a
k
}
of positive integers with
a
1
<
a
2
<
a
3
<
.
.
.
<
a
k
a_1 < a_2 < a_3 < ... < a_k
a
1
<
a
2
<
a
3
<
...
<
a
k
is named alternating if
i
+
a
i+a
i
+
a
for
i
=
1
,
2
,
3
,
.
.
.
,
k
i = 1, 2, 3, ..., k
i
=
1
,
2
,
3
,
...
,
k
is even. The empty set is also considered to be alternating. The number of alternating subsets of
{
1
,
2
,
3
,
.
.
.
,
n
}
\{1, 2, 3,..., n\}
{
1
,
2
,
3
,
...
,
n
}
is denoted by
A
(
n
)
A(n)
A
(
n
)
. Develop a method to determine
A
(
n
)
A(n)
A
(
n
)
for every
n
∈
N
n \in N
n
∈
N
and calculate hence
A
(
33
)
A(33)
A
(
33
)
.
connected sequences revisited
For three sequences
(
x
n
)
,
(
y
n
)
,
(
z
n
)
(x_n),(y_n),(z_n)
(
x
n
)
,
(
y
n
)
,
(
z
n
)
with positive starting elements
x
1
,
y
1
,
z
1
x_1,y_1,z_1
x
1
,
y
1
,
z
1
we have the following formulae: x_{n+1} = y_n + \frac{1}{z_n} y_{n+1} = z_n + \frac{1}{x_n} z_{n+1} = x_n + \frac{1}{y_n} (n = 1,2,3, \ldots) a.) Prove that none of the three sequences is bounded from above. b.) At least one of the numbers
x
200
,
y
200
,
z
200
x_{200},y_{200},z_{200}
x
200
,
y
200
,
z
200
is greater than 20.