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National and Regional Contests
Germany Contests
Bundeswettbewerb Mathematik
2015 Bundeswettbewerb Mathematik Germany
2015 Bundeswettbewerb Mathematik Germany
Part of
Bundeswettbewerb Mathematik
Subcontests
(4)
4
2
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Many friends in a social network
Many people use the social network "BWM". It is known that: By choosing any four people of that network there always is one that is a friend of the other three. Is it then true that by choosing any four people there always is one that is a friend of everyone in "BWM"?Note: If member
A
A
A
is a friend of member
B
B
B
, then member
B
B
B
also is a friend of member
A
A
A
.
Sum of distances to sides => perpendicularity
Let
A
B
C
ABC
A
BC
be a triangle, such that its incenter
I
I
I
and circumcenter
U
U
U
are distinct. For all points
X
X
X
in the interior of the triangle let
d
(
X
)
d(X)
d
(
X
)
be the sum of distances from
X
X
X
to the three (possibly extended) sides of the triangle. Prove: If two distinct points
P
,
Q
P,Q
P
,
Q
in the interior of the triangle
A
B
C
ABC
A
BC
satisfy
d
(
P
)
=
d
(
Q
)
d(P)=d(Q)
d
(
P
)
=
d
(
Q
)
, then
P
Q
PQ
PQ
is perpendicular to
U
I
UI
U
I
.
3
2
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Weird angle condition results into concurrence
Let
M
M
M
be the midpoint of segment
[
A
B
]
[AB]
[
A
B
]
in triangle
△
A
B
C
\triangle ABC
△
A
BC
. Let
X
X
X
and
Y
Y
Y
be points such that
∠
B
A
X
=
∠
A
C
M
\angle{BAX}=\angle{ACM}
∠
B
A
X
=
∠
A
CM
and
∠
B
Y
A
=
∠
M
C
B
\angle{BYA}=\angle{MCB}
∠
B
Y
A
=
∠
MCB
. Both points,
X
X
X
and
Y
Y
Y
, are on the same side as
C
C
C
with respect to line
A
B
AB
A
B
. Show that the rays
[
A
X
[AX
[
A
X
and
[
B
Y
[BY
[
B
Y
intersect on line
C
M
CM
CM
.
Coloring 1,2,...,n with rules
Each of the positive integers
1
,
2
,
…
,
n
1,2,\dots,n
1
,
2
,
…
,
n
is colored in one of the colors red, blue or yellow regarding the following rules: (1) A Number
x
x
x
and the smallest number larger than
x
x
x
colored in the same color as
x
x
x
always have different parities. (2) If all colors are used in a coloring, then there is exactly one color, such that the smallest number in that color is even. Find the number of possible colorings.
2
2
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Integer sum equals 100000
A sum of
335
335
335
pairwise distinct positive integers equals
100000
100000
100000
. a) What is the least number of uneven integers in that sum? b) What is the greatest number of uneven integers in that sum?
7143 somewhere in decimal expansion => huge denominator
In the decimal expansion of a fraction
m
n
\frac{m}{n}
n
m
with positive integers
m
m
m
and
n
n
n
you can find a string of numbers
7143
7143
7143
after the comma. Show
n
>
1250
n>1250
n
>
1250
.Example: I mean something like
0.7143
0.7143
0.7143
.
1
2
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Coins on the table form regular polygons
Twelve 1-Euro-coins are laid flat on a table, such that their midpoints form a regular
12
12
12
-gon. Adjacent coins are tangent to each other. Prove that it is possible to put another seven such coins into the interior of the ring of the twelve coins.
Coloring squares in a rectangle: Who can win?
Let
a
,
b
a,b
a
,
b
be positive even integers. A rectangle with side lengths
a
a
a
and
b
b
b
is split into
a
⋅
b
a \cdot b
a
⋅
b
unit squares. Anja and Bernd take turns and in each turn they color a square that is made of those unit squares. The person that can't color anymore, loses. Anja starts. Find all pairs
(
a
,
b
)
(a,b)
(
a
,
b
)
, such that she can win for sure.Extension: Solve the problem for positive integers
a
,
b
a,b
a
,
b
that don't necessarily have to be even. Note: The extension actually was proposed at first. But since this is a homework competition that goes over three months and some cases were weird, the problem was changed to even integers.