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Bundeswettbewerb Mathematik
1974 Bundeswettbewerb Mathematik
1974 Bundeswettbewerb Mathematik
Part of
Bundeswettbewerb Mathematik
Subcontests
(4)
4
2
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Bundeswettbewerb Mathematik 1974 Problem 1.4
All diagonals of a convex polygon are drawn. Prove that its sides and diagonals can be assigned arrows in such a way that no round trip along sides and diagonals is possible.
Bundeswettbewerb Mathematik 1974 Problem 2.4
Peter and Paul gamble as follows. For each natural number, successively, they determine its largest odd divisor and compute its remainder when divided by
4
4
4
. If this remainder is
1
1
1
, then Peter gives Paul a coin; otherwise, Paul gives Peter a coin. After some time they stop playing and balance the accounts. Prove that Paul wins.
3
2
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Bundeswettbewerb Mathematik 1974 Problem 1.3
Let
M
M
M
be a set with
n
n
n
elements. How many pairs
(
A
,
B
)
(A, B)
(
A
,
B
)
of subsets of
M
M
M
are there such that
A
A
A
is a subset of
B
?
B?
B
?
Bundeswettbewerb Mathematik 1974 Problem 2.3
A circle
K
1
K_1
K
1
of radius r_1 = 1\slash 2 is inscribed in a semi-circle
H
H
H
with diameter
A
B
AB
A
B
and radius
1.
1.
1.
A sequence of different circles
K
2
,
K
3
,
…
K_2, K_3, \ldots
K
2
,
K
3
,
…
with radii
r
2
,
r
3
,
…
r_2, r_3, \ldots
r
2
,
r
3
,
…
respectively are drawn so that for each
n
≥
1
n\geq 1
n
≥
1
, the circle
K
n
+
1
K_{n+1}
K
n
+
1
is tangent to
H
H
H
,
K
n
K_n
K
n
and
A
B
.
AB.
A
B
.
Prove that a_n = 1\slash r_n is an integer for each
n
n
n
, and that it is a perfect square for
n
n
n
even and twice a perfect square for
n
n
n
odd.
2
2
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Bundeswettbewerb Mathematik 1974 Problem 1.2
Seven polygons of area
1
1
1
lie in the interior of a square with side length
2
2
2
. Show that there are two of these polygons whose intersection has an area of at least 1\slash 7.
Bunndeswettbewerb Mathematik 1974 Problem 2.2
There are
30
30
30
apparently equal balls,
15
15
15
of which have the weight
a
a
a
and the remaining
15
15
15
have the weight
b
b
b
with
a
≠
b
a \ne b
a
=
b
. The balls are to be partitioned into two groups of
15
15
15
, according to their weight. An assistant partitioned them into two groups, and we wish to check if this partition is correct. How can we check that with as few weighings as possible?
1
2
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Bundeswettbewerb Mathematik 1974 Problem 1.1
Find the necessary and sufficient condition that a trapezoid can be formed out of a given four-bar linkage.
25 Points are given on the plane
Twenty-five points are given on the plane. Among any three of them, one can choose two less than one inch apart. Prove that there are 13 points among them which lie in a circle of radius 1.