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Contests
National and Regional Contests
Germany Contests
Bundeswettbewerb Mathematik
1986 Bundeswettbewerb Mathematik
1986 Bundeswettbewerb Mathematik
Part of
Bundeswettbewerb Mathematik
Subcontests
(4)
4
2
Hide problems
a_{n+1} = 1/16 (1 + 4a_n +\sqrt{1 + 24a_n})
The sequence
a
1
,
a
2
,
a
3
,
.
.
.
a_1, a_2, a_3,...
a
1
,
a
2
,
a
3
,
...
is defined by
a
1
=
1
,
a
n
+
1
=
1
16
(
1
+
4
a
n
+
1
+
24
a
n
)
(
n
∈
N
∗
)
.
a_1 = 1\,\,\,, \,\,\,a_{n+1} =\frac{1}{16}(1 + 4a_n +\sqrt{1 + 24a_n}) \,\,\,(n \in N^* ).
a
1
=
1
,
a
n
+
1
=
16
1
(
1
+
4
a
n
+
1
+
24
a
n
)
(
n
∈
N
∗
)
.
Determine and prove a formula with which for every natural number
n
n
n
the term
a
n
a_n
a
n
can be computed directly without having to determine preceding terms of the sequence.
subsets of a set with m elements, related
Given the finite set
M
M
M
with
m
m
m
elements and
1986
1986
1986
further sets
M
1
,
M
2
,
M
3
,
.
.
.
,
M
1986
M_1,M_2,M_3,...,M_{1986}
M
1
,
M
2
,
M
3
,
...
,
M
1986
, each of which contains more than
m
2
\frac{m}{2}
2
m
elements from
M
M
M
. Show that no more than ten elements need to be marked in order for any set
M
i
M_i
M
i
(
i
=
1
,
2
,
3
,
.
.
.
,
1986
i =1, 2, 3,..., 1986
i
=
1
,
2
,
3
,
...
,
1986
) contains at least one marked element.
2
2
Hide problems
x_n = n/(n+a)
Let
a
a
a
be a given natural number and
x
1
,
x
2
,
x
3
,
.
.
.
x_1, x_2, x_3, ...
x
1
,
x
2
,
x
3
,
...
the sequence with
x
n
=
n
n
+
a
x_n = \frac{n}{n+a}
x
n
=
n
+
a
n
(
n
∈
N
∗
n \in N^*
n
∈
N
∗
). Prove that for every
n
∈
N
∗
n \in N^*
n
∈
N
∗
, the term
x
n
x_n
x
n
can be represented as the product of two terms of this sequence , and determine the number of representations depending on
n
n
n
and
a
a
a
.
right triangle exradii criterions
A triangle has sides
a
,
b
,
c
a, b,c
a
,
b
,
c
, radius of the incircle
r
r
r
and radii of the excircles
r
a
,
r
b
,
r
c
r_a, r_b, r_c
r
a
,
r
b
,
r
c
: Prove that: a) The triangle is right-angled if and only if:
r
+
r
a
+
r
b
+
r
c
=
a
+
b
+
c
r + r_a + r_b + r_c = a + b + c
r
+
r
a
+
r
b
+
r
c
=
a
+
b
+
c
. b) The triangle is right-angled if and only if:
r
2
+
r
a
2
+
r
b
2
+
r
c
2
=
a
2
+
b
2
+
c
2
r^2 + r^2_a + r^2_b + r^2_c = a^2 + b^2 + c^2
r
2
+
r
a
2
+
r
b
2
+
r
c
2
=
a
2
+
b
2
+
c
2
.
1
2
Hide problems
ways that polyline be segments of n points on circle doesn't cut itself
There are
n
n
n
points on a circle (
n
>
1
n > 1
n
>
1
). Denote them with
P
1
,
P
2
,
P
3
,
.
.
.
,
P
n
P_1,P_2, P_3, ..., P_n
P
1
,
P
2
,
P
3
,
...
,
P
n
such that the polyline
P
1
P
2
P
3
.
.
.
P
n
P_1P_2P_3... P_n
P
1
P
2
P
3
...
P
n
does not intersect itself. In how many ways is this possible?
numbering edges of cube from 1 to 12
The edges of a cube are numbered from
1
1
1
to
12
12
12
, then is calculated for each vertex the sum of the numbers of the edges going out from it. a) Prove that these sums cannot all be the same. b) Can eight equal sums result after one of the edge numbers is replaced by the number
13
13
13
?
3
2
Hide problems
triangle construction given 3 points, one at each side
The points
S
S
S
lie on side
A
B
AB
A
B
,
T
T
T
on side
B
C
BC
BC
, and
U
U
U
on side
C
A
CA
C
A
of a triangle so that the following applies:
A
S
‾
:
S
B
‾
=
1
:
2
\overline{AS} : \overline{SB} = 1 : 2
A
S
:
SB
=
1
:
2
,
B
T
‾
:
T
C
‾
=
2
:
3
\overline{BT} : \overline{TC} = 2 : 3
BT
:
TC
=
2
:
3
and
C
U
‾
:
U
A
‾
=
3
:
1
\overline{CU} : \overline{UA} = 3 : 1
C
U
:
U
A
=
3
:
1
. Construct the triangle
A
B
C
ABC
A
BC
if only the points
S
,
T
S, T
S
,
T
and
U
U
U
are given.
non-periodic sequence
Let
d
n
d_n
d
n
be the last digit, distinct from 0, in the decimal expansion of
n
!
n!
n
!
. Prove that the sequence
d
1
,
d
2
,
d
3
,
…
d_1,d_2,d_3, \ldots
d
1
,
d
2
,
d
3
,
…
is not periodic.