MathDB
Problems
Contests
National and Regional Contests
Germany Contests
German National Olympiad
1967 German National Olympiad
1967 German National Olympiad
Part of
German National Olympiad
Subcontests
(5)
2
1
Hide problems
F_n sequences
Let
n
≠
0
n \ne 0
n
=
0
be a natural number. A sequence of numbers is briefly called a sequence “
F
n
F_n
F
n
” if
n
n
n
different numbers
z
1
z_1
z
1
,
z
2
z_2
z
2
,
.
.
.
...
...
,
z
n
z_n
z
n
exist so that the following conditions are fulfilled: (1) Each term of the sequence is one of the numbers
z
1
z_1
z
1
,
z
2
z_2
z
2
,
.
.
.
...
...
,
z
n
z_n
z
n
. (2) Each of the numbers
z
1
z_1
z
1
,
z
2
z_2
z
2
,
.
.
.
...
...
,
z
n
z_n
z
n
occurs at least once in the sequence. (3) Any two immediately consecutive members of the sequence are different numbers. (4) No subsequence of the sequence has the form
{
a
,
b
,
a
,
b
}
\{a, b, a, b\}
{
a
,
b
,
a
,
b
}
with
a
≠
b
a \ne b
a
=
b
.Note: A subsequence of a given sequence
{
x
1
,
x
2
,
x
3
,
.
.
.
}
\{x_1, x_2, x_3, ...\}
{
x
1
,
x
2
,
x
3
,
...
}
or
{
x
1
,
x
2
,
x
3
,
.
.
.
,
x
s
}
\{x_1, x_2, x_3, ..., x_s\}
{
x
1
,
x
2
,
x
3
,
...
,
x
s
}
is called any sequence of the form
{
x
m
1
,
x
m
2
,
x
m
3
,
.
.
.
}
\{x_{m1}, x_{m2}, x_{m3}, ...\}
{
x
m
1
,
x
m
2
,
x
m
3
,
...
}
or
{
x
m
1
,
x
m
2
,
x
m
3
,
.
.
.
,
x
m
t
}
\{x_{m1}, x_{m2}, x_{m3}, ..., x_{mt}\}
{
x
m
1
,
x
m
2
,
x
m
3
,
...
,
x
m
t
}
with natural numbers
m
1
<
m
2
<
m
3
<
.
.
.
m_1 < m_2 < m_3 < ...
m
1
<
m
2
<
m
3
<
...
Answer the following questions: a) Given
n
n
n
, are there sequences
F
n
F_n
F
n
of arbitrarily long length? b) If question (a) is answered in the negative for an
n
n
n
: What is the largest possible number of terms that a sequence
F
n
F_n
F
n
can have (given
n
n
n
)?
3
1
Hide problems
sum a_i/ *s - a_i) >=n /(n-1)
Prove the following theorem: If
n
>
2
n > 2
n
>
2
is a natural number,
a
1
,
.
.
.
,
a
n
a_1, ..., a_n
a
1
,
...
,
a
n
are positive real numbers and becomes
∑
i
=
1
n
a
i
=
s
\sum_{i=1}^n a_i = s
∑
i
=
1
n
a
i
=
s
, then the following holds
∑
i
=
1
n
a
i
s
−
a
i
≥
n
n
−
1
\sum_{i=1}^n \frac{a_i}{s - a_i} \ge \frac{n}{n - 1}
i
=
1
∑
n
s
−
a
i
a
i
≥
n
−
1
n
5
1
Hide problems
5x + 2y + z = 10n
For each natural number
n
n
n
, determine the number
A
(
n
)
A(n)
A
(
n
)
of all integer nonnegative solutions the equation
5
x
+
2
y
+
z
=
10
n
.
5x + 2y + z = 10n.
5
x
+
2
y
+
z
=
10
n
.
6
1
Hide problems
PQ <PP_i => n<15 in 3D
Prove the following theorem: If there are
n
n
n
pairs of different points
P
i
P_i
P
i
,
i
=
1
,
2
,
.
.
.
,
n
i = 1, 2, ..., n
i
=
1
,
2
,
...
,
n
,
n
>
2
n > 2
n
>
2
in three dimensions space, such that each of them is at a smaller distance from one and the same point
Q
Q
Q
than any other
P
i
P_i
P
i
, then
n
<
15
n < 15
n
<
15
.
1
1
Hide problems
locus of R, PQR equilateral, P inside square ABCD, Q on sides
In a plane, a square
A
B
C
D
ABCD
A
BC
D
and a point
P
P
P
located inside it are given. Let a point
Q
Q
Q
pass through all sides of the square. Describe the set of all those points
R
R
R
in for which the triangle
P
Q
R
PQR
PQR
is equilateral.