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National and Regional Contests
Germany Contests
Bundeswettbewerb Mathematik
1972 Bundeswettbewerb Mathematik
1972 Bundeswettbewerb Mathematik
Part of
Bundeswettbewerb Mathematik
Subcontests
(4)
3
2
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2^(n-1) subsets, every three have a common element
2
n
−
1
2^{n-1}
2
n
−
1
subsets are choosen from a set with
n
n
n
elements, such that every three of these subsets have an element in common. Show that all subsets have an element in common.
Arithmetic mean + interchange digits = geometric mean
The arithmetic mean of two different positive integers
x
,
y
x,y
x
,
y
is a two digit integer. If one interchanges the digits, the geometric mean of these numbers is archieved. a) Find
x
,
y
x,y
x
,
y
. b) Show that a)'s solution is unique up to permutation if we work in base
g
=
10
g=10
g
=
10
, but that there is no solution in base
g
=
12
g=12
g
=
12
. c) Give more numbers
g
g
g
such that a) can be solved; give more of them such that a) can't be solved, too.
2
2
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Rolling beer mats: number of rotations
In a plane, there are
n
≥
3
n \geq 3
n
≥
3
circular beer mats
B
1
,
B
2
,
.
.
.
,
B
n
B_{1}, B_{2}, ..., B_{n}
B
1
,
B
2
,
...
,
B
n
of equal size.
B
k
B_{k}
B
k
touches
B
k
+
1
B_{k+1}
B
k
+
1
(
k
=
1
,
2
,
.
.
.
,
n
k=1,2,...,n
k
=
1
,
2
,
...
,
n
);
B
n
+
1
=
B
1
B_{n+1}=B_{1}
B
n
+
1
=
B
1
. The beer mats are placed such that another beer mat
B
B
B
of equal size touches all of them in the given order if rolling along the outside of the chain of beer mats. How many rotations
B
B
B
makes untill it returns to it's starting position¿
79 integers => 13 divides one of their digit's sum's
Prove: out of
79
79
79
consecutive positive integers, one can find at least one whose sum of digits is divisible by
13
13
13
. Show that this isn't true for
78
78
78
consecutive integers.
1
2
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Minimum sum if lower bound for crosses is given
There's a real number written on every field of a
n
×
n
n \times n
n
×
n
chess board. The sum of all numbers of a "cross" (union of a line and a column) is
≥
a
\geq a
≥
a
. What's the smallest possible sum of all numbers on the board¿
How many fields must a knight walk to...
Given an infinity chess board and a knight on it. On how many different fields the knight can be after
n
n
n
steps¿
4
2
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Greatest integer function
Which natural numbers cannot be presented in that way:
[
n
+
n
+
1
2
]
[n+\sqrt{n}+\frac{1}{2}]
[
n
+
n
+
2
1
]
,
n
∈
N
n\in\mathbb{N}
n
∈
N
[
y
]
[y]
[
y
]
is the greatest integer function.
p players, n games, out of 3, 2 haven't played => n<=p²/4
p
>
2
p>2
p
>
2
persons participate at a chess tournament, two players play at most one game against each other. After
n
n
n
games were made, no more game is running and in every subset of three players, we can find at least two that havem't played against each other. Show that
n
≤
p
2
4
n \leq \frac{p^{2}}4
n
≤
4
p
2
.