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National and Regional Contests
Germany Contests
Bundeswettbewerb Mathematik
1973 Bundeswettbewerb Mathematik
1973 Bundeswettbewerb Mathematik
Part of
Bundeswettbewerb Mathematik
Subcontests
(4)
4
2
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|Same gender on the right| = |Different gender on the right|
n
n
n
persons sit around a round table. The number of persons having the same gender than the person at the right of them is the same as the number of those it isn't true for. Show that
4
∣
n
4|n
4∣
n
.
Multiples of 2^n with n digits, all of them being 1 or 2
Prove: for every positive integer there exists a positive integer having n digits, all of them being
1
1
1
's and
2
2
2
's only, such that this number is divisible by
2
n
2^{n}
2
n
. Is this still true in base
4
4
4
or
6
6
6
¿
3
2
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Fractional parts of square roots can be anything
Given
n
n
n
digits
a
1
,
a
2
,
.
.
.
,
a
n
a_{1}, a_{2},..., a_{n}
a
1
,
a
2
,
...
,
a
n
in that order. Does there exist a positive integer such that the first
n
n
n
decimal digits after the dot of that number's square root are exactly those given digits¿ Give reason for your answer.
Tiling rectangular rooms with 2×2 and 4×1.
For covering the floor of a rectangular room rectangular tiles of sizes
2
×
2
2 \times 2
2
×
2
and
4
×
1
4 \times 1
4
×
1
were used. Show that it's not possible to cover the floor if there is one plate less of one type and one more of the other type.
2
2
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Star shape region with centers A, B => all of [AB] centers
In a planar lake, every point can be reached by a straight line from the point
A
A
A
. The same holds for the point
B
B
B
. Show that this holds for every point on the segment
[
A
B
]
[AB]
[
A
B
]
, too.
Divide and append, all problem's solution
We work in the decimal system and the following operations are allowed to be done with a positive integer: a) append
4
4
4
at the end of the number. b) append
0
0
0
at the end of the number. c) divide the number by
2
2
2
if it's even. Show that starting with
4
4
4
, we can reach every positive integer by a finite number of these operations
1
2
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All but one digit is 5 => no perfect square
A positive integer has 1000 digits (decimal system), all but at most one of them being the digit
5
5
5
. Show that this number isn't a perfect square.
Square with 51 points => 3 covered by disk
In a square of sidelength
7
7
7
,
51
51
51
points are given. Show that there's a disk of radius
1
1
1
covering at least
3
3
3
of these points.