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Bundeswettbewerb Mathematik
1979 Bundeswettbewerb Mathematik
1979 Bundeswettbewerb Mathematik
Part of
Bundeswettbewerb Mathematik
Subcontests
(4)
3
2
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Bundeswettbewerb Mathematik 1979 Problem 1.3
In base
10
10
10
there exist two-digit natural numbers that can be factorized into two natural factors such that the two digits and the two factors form a sequence of four consecutive integers (for example,
12
=
3
⋅
4
12 = 3 \cdot 4
12
=
3
⋅
4
). Determine all such numbers in all bases.
Bundeswettbwerb Mathematik 1979 Problem 2.3
The
n
n
n
participants of a tournament are numbered with
0
0
0
through
n
−
1
n - 1
n
−
1
. At the end of the tournament it turned out that for every team, numbered with
s
s
s
and having
t
t
t
points, there are exactly
t
t
t
teams having
s
s
s
points each. Determine all possibilities for the final score list.
2
2
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concurrent wanted, 2 squares related
The squares
O
A
B
C
OABC
O
A
BC
and
O
A
1
B
1
C
1
OA_1B_1C_1
O
A
1
B
1
C
1
are situated in the same plane and are directly oriented. Prove that the lines
A
A
1
AA_1
A
A
1
,
B
B
1
BB_1
B
B
1
, and
C
C
1
CC_1
C
C
1
are concurrent.
Bundeswettbewerb Mathematik 1979 Problem 2.2
A circle
k
k
k
with center
M
M
M
and radius
r
r
r
is given. Find the locus of the incenters of all obtuse-angled triangles inscribed in
k
k
k
.
1
2
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Bundeswettbewerb Mathematik 1979 Problem 1.1
There are
n
n
n
teams in a football league. During a championship, every two teams play exactly one match, but no team can play more than one match in a week. At least, how many weeks are necessary for the championship to be held? Give an schedule for such a championship.
Problem-3(last problem)
The plane is painted in red or blue color. Prove that you have a rectangle with the corners of the same color.
4
2
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Almost Divisible!
Prove that the polynomial
P
(
x
)
=
x
5
−
x
+
a
P(x) = x^5-x+a
P
(
x
)
=
x
5
−
x
+
a
is irreducible over
Z
\mathbb{Z}
Z
if
5
∤
a
5 \nmid a
5
∤
a
.
Bundeswettbewerb Mathematik 1979 Problem 2.4
An infinite sequence
p
1
,
p
2
,
p
3
,
…
p_1, p_2, p_3, \ldots
p
1
,
p
2
,
p
3
,
…
of natural numbers in the decimal system has the following property: For every
i
∈
N
i \in \mathbb{N}
i
∈
N
the last digit of
p
i
+
1
p_{i+1}
p
i
+
1
is different from
9
9
9
, and by omitting this digit one obtains number
p
i
p_i
p
i
. Prove that this sequence contains infinitely many composite numbers.